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so much more time but I'm gonna do I wanted to definitely do a follow-up on on the cue cat paper q cat paper that the pharmacist design thing so I basically if I pull out my notes from last time II understand Vicki so I like I I think I did on the video at the end sort of a bit of a the XX icing coupling gate and in the meantime some of the author's from off the paper actually reached out and said they are open for you know helping me answer questions and and stuff like that which I really appreciate I want to go through some of the stuff solo first because I think that's what helps me learn and I think what's gonna help people watching is well enjoy the experience more but I'll definitely I'll definitely get in touch with the the group I think that's I think there's a lot to do there I'm okay I want to start taking a sneak peek at vqd and what is these and how is it different from pqe so and probably there was a probably reference on the paper I quantum assisted design or something if I just search for that there's actually a quantum aided design that's the paper that's the paper so if I take a look at the cute I search for vqd it's nothing new here Oh in a second that's not a paper that's for the that's the one for the optics hardware i won the one for it's probably this one here you go so the original find deflation algorithm fifty-two of excited states let's take a look at this so is this gonna be any how findable in our key in archives excited states that seems to be the paper did I click on it I click on it someone this is week here we go nice make my face smaller let's just apply through the paper see if oh before that let me pause for a second because I want to just make sure I'm freaking out these days that my sound is on you know because it happened that I did a one-hour session without any sound and that cannot happen again so I'm kind of having a little widget app here that tells me what that my microphone is working I'm gonna pause this for a second you should well one note is don't go back in here that makes my face lighter as well it's like almost like having a an actual lamp in here so what is about the calculation of excited state energies of electronic structure Hamiltonians has many important applications and is the calculation of optical spectra and in reaction rates while low depth quantum algorithms such as the very you know quantum eigenvalues solver vqe have been used to determine ground state energies methods for calculating excited state currently involved the implementation of high depth controlled unit or ease or a large number of additional samples here we show ok so there's I I don't know your show nabob is la he will show how how overlap estimation can be used to deflate eigenstates once they are found enabling the calculation of excited state images and their degeneracies we propose inclusion that requires the same number of key pizzas vqe and at most twice the circuit depth our method is robust to control errors is companies compatible with our mitigation strategies that can be implemented on near-term quantum computers still use the deflate eigenstates okay so there's the the algorithm in here and then this some mathematical proof and detailing x' error accumulations discussion sampling const destructive destructive swap test and that's the that's the overlap test the harm are overlap test am i mk2 so there's a better way to do that than what I'm done what I'm using for the vql is thing it's a modular multiplication okay once for microsimulation I can show me the references I want to take a look at the algorithm itself try to understand the diagram first maybe it's not all the way up so what have we caught in here so our prepares these from the usual state zero expectation estimation of animal is p1 p2 and PN and overlap estimation then classical adder calculates the overlap of severe the calculating the overlap of something lambda 0 and lambda K lambda 1 lambda K okay so there is a lambda K that's prepared I know what those lambdas are and then classic a lot of calculates the actual expectation value of the Hamiltonian and then there's an optimizer that updates lambda K which I guess is a set of angles okay to minimize and there's a cost function that is the energy for the energy for these set of angles plus the overlap okay plus how much overlap so there seems to be two components to these let me see if I can get my I'll pause for a second click on here whenever I'm having some itches again with that with a pen but that's that's what it that's that's what it seems to do or so why I don't know probably have to read the text but it says it's some expectation estimation and in some overlap but this expectation is this going to be the classic this is this just what vicki with us or or what's the what's the essential part in here or what's the it would stick let's see i'm gonna skip danger for now maybe it's gonna then should help though but for the special master because there's some I can fella prawns are equally small size of Jeannie girl's PageRank algorithm alone has had a significant impact on modern society and its course solves an eigenvalue problem associated with the stochastic matrix describing the world wide web another important example is principal communal assist which is once per application in bioinformatics neuroscience image processing etc the time-independent schrodinger equation provides yet another example of fundamental eigenvalue problem its numerical solution enables properties of atoms molecules materials to be predicted with far-reaching applications in materials designed by Discovery Center a great transition of excited state energies of molecules is required to predict charge and energy transfer processes in photovoltaic materials or thundersense some chemical reactions suggest another however classical methods such as density functional board okay chronic computers have the potential to solve these these and other prongs significantly faster than a non-american interation of car migrants solver introduced in reference 30 is the first algorithm designed to find the lowest eigen value of a Hamiltonian on a near-term non fault or and chronic computer EQ is based on the variational principle and utilizes the fact that quantum computers can store quantum states using an extremely fewer resources than required classically vqe uses parameterize point circus to prepare trial wave functions and compute their energy in a classical computer to to find the parameters minimizing these energy the low circuit depth of the QE has like 200 K since it's obstruction modification has been suggested to enable Vikki to find excited state energies exam for example a folded spectrum method which requires finding the expectation of the square handle tone Ian with chronically more turns or symmetric symmetry based methods which are non isometric again oh I know about that such situations have been more recently superseded by two proposals a method that minimizes the phone annoy Minh entropy and the quantum subspace expansion method requires the large number of high depth control unit Ares and the quantum subspace expansion method currents all our traditional okay we can dive into those examples maybe later as the Curiosity as well our algorithm extends the qe2 sister that's what I thought it kind of extends the key to systematically find excited states that almost no extra cost we achieve this by adding overlap terms into the optimization function okay so it's as simple as adding overlap terms in networks in order to exploit the fact that the hermitian matrices it made a complete set of orthogonal eigenvectors exploding further the fact that Vicki retains the classical parameters of on that states that enable their their repression low depth quantum circuits can then be rarely used to calculate these overlap terms let's try understand that so in theory the real primer is lambda for the ends at state are classically optimized with respect to the expectation value of this Hamiltonian which is decomposed by my I mean I spent 20 years and so it's not a big deal okay we know this we know these are method extensity to calculate the case let's stay by instead optimizing the parameters for the answer state such that the cost function is relieved so that works just you just add the overlap but what's the overlap here what is this calculating you're minimizing the overlap so you want something orthogonal to what what is lambda I okay so the case excited state buying centers in the parameters for the states such as the cost function is minimized this can be seen as minimizing the energy subject to the constraint that that lambda K is orthogonal to the states this is orthogonal tool and what are these the all the other ones so lambda 0 would be and here we show how choosing sufficiently large means minimum of everything is going to be the energy of the case state provided enhances so that's this mean that you first find the the lowest energy and then you find an orthogonal one so you're you're kind of finding the next one and then you do this over and over again that would be cool that is a genius idea this is a well the first term can be used in key the second term is the sum of overlaps of the answered state with states 0 to K minus 1 and can be computed efficiently in front of you using one of the methods given in section 4 so computing that occurs knowledge of lambda 0 to lambda K minus 1 and so and interactive procedures required to calculate the K eigen value that's all I said yeah okay first first time this here is calculated ok we're done with Vicki T we're done it's nice I like it that's them smart come on it's said of something kind of fencing it's like just this it's read more okay so you calculate the lowest energy and then you see QE by minimizing then lambda 1 is calculus jeans then has it's so simple the concept I mean I guess it's it's probably tricky but I mean how to come up with that answer minimizing because the idea is that because they are the eigenvalues they're all all orthogonal to each other right for K 1 then k 2 and how do you find so how do you consider the overlap how do you do that so how do you consider how do you accumulate how do you make sure that it's orthogonal to all of those because the first iteration seems easy to me but how do you do that for for example because if you wanna you wanna find for K 3 or K 2 you want something orthogonal to both k0 and k1 right schematic for rational expression is in figure 1 showcased a progression circuit some period and use the results it's gonna be ok so we know this almost figure why subscribe I'm sorry that's the console optimize updates you can minimize how do you calculate these how do you make sure that when K is bigger than 1 you're the overlap now that I know the overlap test makes makes so this destructive test makes it easier to understand the whole concept here so but where how do I make sure that it's orthogonal to all of the previous ones that's that's the point here the circus is reportedly to compute each of the expectation values and overlap terms for I lower than K the overlap terms activities in circumcision for let's go to section for a low death method formation I mean I know over I know is that trial stay Patricia soon for the universe of the preparation circuit for the eyes proves the computed state your obvious there's a low depth method for our overlap estimation proposed in reference in the Russians twelve can be seen by writing the overlap as [Music] so you're using this wait a second so you're using the you're just making sure that it's orthogonal to the previous one you found how this is going to you that you can find another one because you're always minimizing or want because R so R is the circuit that is the various your enzymes right all of the signs to me with precision by the fraction of all zero bit strings when measuring this state the computational basis listen these are true eigenstates was possibly non distinct energies Smith requires knowing the inverse of the preparation circuit for each previously computed stayed well these inverses of known in theory by inverting gate in the given position of the original version so the device errors may mean that the information is inaccurate in practice it will be fine to the optimum parameters originally found to prepare the e the eyes state using Big D then it's inverse can be found by fixing these and varying the trial state parameters so that the overlap is maximized system you can able to retain the robustness to control error Sanskrit rest okay this I don't get much why do you need the inverse as you're not minimizing for humanizes the expectation value of the of the overlap are in the universe that the Europe doesn't require that you okay well it does if you it does if you don't want to have double the amount of qubits right because you would literally have to have double amount of qubits to do one to do that destructive side because the same number of qubits in Iran I think you and around twice a circuit in the fenders really describe an alternative method again that seems good to have Vicki but twice an hour of qubits because you're doing the destructive swab test okay okay so this is another so what was described in here as I just keep all this stuff okay so what's described in here not in here this is another method okay that the overlap equals these and so this is what's in reference 12 in reference laceration reference 12 it's a nature okay what not gonna by isn't gonna pay for this [Music] well so but I'm happy with the altar of the alternate method method I just wanna understand intuitively what's what's what's going on but I think I think I've got it already so vqd requires it's okay so the surface is mm on a qubits if a larger gate that is available then yeah okay then you can actually do phase estimation can be used to reduce the total runtime of overlap estimation so you have different techniques in here to calculate these so can never create alpha cheaply what is alpha Q P is this like a what is alpha q is they say this is a thing actually what a reference for D accelerated rational quantum eigen solver April 2019 so it's really recent that's so cool oh I had it open already it's alpha v QE I don't mean alpha v QE can evolve I even keep in alpha keeping me okay these I probably might want to take a look at that corner phase estimation on steroids of simulation we similarly Vicki Dion that's hydrogen are one and the same bases for a range of inter nuclear separations and compared to exact diagonalization astronomic to using fry so this is the example they run an example the discussion on air accumulation in general we cannot assume perfect state preparation suppose okay but I think I got the basic idea I will deep dive into this problem later on I just want to make sure that I kind of go back to the other paper to the original one about these simulation and that stuff okay but I get that I get the gist of it so it's the idea is you use vqe to compute the first the lowest eigenstate then you kind of add a change your cost function to consider kind of minimizing the overlap between that state and instead of the next step right I give a white and and the next so you're the the next eigen state you're looking for yeah because that means that it's if you're minimizing your overlap it means that it's orthogonal and you do that iterative lien so you're getting all the you can get up to you know kind of basically get all the different eigen sites yeah I get it I think it's a I'll dive into these probably later on but it's choice of effective Hamiltonian because there's a lot of new ones I guess to the Hamilton you choose and or how do you do all the star sampling cost and then the destructive swap test and these we know about these but I also want to dive into all these part because maybe I could use that actually for for my other project what is these they it's the bitwise product I guess that's what it means is it because I'm you should use maybe that's how do i this is the be twice product headline message for mental simulation bonds her air accumulation our mediation signature constrains okay we'll dive into these into the error correction partly that air mitigation part lender I just wanted to know where is these post-processing explained probably somewhere here Seeger for the softest neighbors overlap of two states to determine precision pretty measures means after applying a circuit upon register in the state Wyler's also tests acting on 2n cubed states required and and cylinder control so upstate leading to ins notion that the same outcome description can be obtained more efficiently without and ancillary using parallel they bail basis measurements so that actually has a name it's a bail basis measurement oh yeah it makes sense because it's a bail based measurement so it's telling you whether they're both zeros or not actually that makes sense the bail basis measurement interesting so in classical logic the so called SRK subtests turns your foreo cards just to two n qubits and that's okay but it they don't explain it that's the post-processing is just that and then one and so if this is one it's not - right so you're really only gonna get I don't know well maybe there's another way to find that what what is the post processing in here but I'll deal with this in the other video anyway I get the gist I think the idea is I think I know enough to just go back to to the original paper here the actual simulation stuff so I can take that off the least so far so that's done what else is done this is done and this is for later and so I think I think there's no advantage into using an msk dress it controls it gate but I won't understand people get more why's that enough those are different types of I mean the MS gate is native to anion computer that's also wanna in the mail that I got from the authors they mentioned that and kind of makes sense I but I don't know if I'm happy with these explanation so but but I think my next would be to understand the design of the answers is it be better yeah good it was better than I expected to be honest I thought it we have I would have a harder time could be some happy I hope you enjoyed the video |
not not into about a round so um i'm still struggling to do the video thing because i think it would be great but the problem is i've read that because i'm using a headset a bluetooth headset and i think the obs when i'm using the microphone here which is what i care the most so you can you guys can actually hear um hear proper voice um then i think the whole headset is set as a headphone or something like that and then the audio from the desktop does not come through um but maybe there's a way that i can tweak the the actual desktop settings so that the audio goes through the speakers and not through my headset you know what i mean so that the windows coupling or pairing with headset is just microphone and then obs will just do symbols yeah i have to still play with this a little bit so i might do some test streams um in the coming days but for now i'm still a bit confused because i i'm so yeah maybe maybe this is stupid peachy to draw right but actually i thought there was uh i discovered this like a uh paint i think there was like a no no was that this no was there it was a um online paint or something like that i don't remember just paint was it that james paint yeah that's that's the thing so i don't even have to open the um so does this make sense at all anyhow is it easy to work with whatever actually probably doesn't make much sense but anyway um so we we've been doing lagrangian mechanics um let's try to paint a picture so we started with quantum mechanics and then kind of we realized that okay so this i realized that there's something worth taking a look at with the path integral stuff it's awful actually path integral stuff and then there's also the schrodinger stuff stuff that kind of led me to the lagrange mechanics i suspect this leads me to hamiltonian mechanics in terms of the concept because i think this and some other stuff that i've been reading this week that these is the stronger equation is nothing else than like an equivalent of uh you know potential energy plus kinetic energy or i don't know what the the usual terms for these are but which is kind of what they're trying to get which is quite kind of what hamiltonian sense right um but like what is so what is outside of that are these the two only things that they need to dive in and then it's just about like grinding through um the schroinger equation derivations and maybe seeing how kind of like these two things interconnect like i i'm a bit lost in terms of mapping like what else is uh what else is there uh that i what else is there that i could or that i should be aware of because then this whole thing with quantum field theory that maybe i should take a look at some quantum field theory and it's like where does this feat is this another approach to quantum mechanics um and then obviously we have the whole quantum computing staff which is just a specific usage of quantum mechanics um so this is more just a use case or like specifics of the use case and but but there's still a lot of things that i haven't like take a look at the quantum so maybe maybe what if i so quantum mechanics introduction uh what does wikipedia tell me in terms of so fundamentals effects experiments formulations equations interpretations advanced topics oh actually so that that is that is a pretty good um guide through okay so background so there's a background causing all crown theory bracketation hamilton interference so i think this background is something that i've more or less covered up uh than this fundamentals bone rule wavelength coherency coherence complementarity um energy level entanglement hamiltonian the problem i have with this is there's a lot of different isolated topics that i'm not so um they feel so scattered that i don't know they just feel too scattered um cubits pin super positions and just fundamentals then effects siemens effects stark effect boom effect landau quantization experiments i mean going through experiments there will be this is probably a great thing to to do and maybe i should go through these next kind of like going through all the experiments maybe one procession or you know kind of probably multiple sessions per experiment and then kind of try to go um go up that uh you know kind of see where things end up connecting the bell test stuff uh and so this is more the experimental thing okay and then we've got the what is this for example on the davidson electron scattered by the surface of a crystal of nickel metal this displayed a diffraction pattern this confirmed the hypothesis advanced by the broccoli of wave particle duality and was an experimental milestone in the creation of quantum mechanics so maybe maybe the wave particle duality would be an interesting um thing to take a look at but i don't see anything of things like elementary particles and stuff like these like oh look at this so elementary particles because i'm missing a ma like i'm literally missing a map of of what to explore right i'm feeling a bit lost so there's this whole thing with quarks and leptons and what not to the elementary particles that could be an interesting thing to take a look at so we've got element elementary particles and we've got um the experiments um okay but then you've got experiments we've got formulations so this dynamic is schrodinger face space or okay so this is two so this is formulation so what i've been playing with currently is two formulations which i i really understand them at a high level like i don't of course know the details i i by any chance be able to make any derivation of that um and then there's this matrix things the matrix mechanic so that's missing okay so we could let's do that today matrix mechanics see how how different that is because this is listed as a different formulation okay but i like that actually i that's a good map the quantum mechanics series in the wikipedia as a sort of a skeleton um because there's dynamical pictures so this is the seismic interaction and shorting it so these are okay actually different pitches and then there's phase space which is sum over history which is path integral oh no that's another one face space ah okay cool yeah let's okay so let's let's make let's let's do this so i'll focus then i'll focus on the next the next um i'll focus the next streams and finishing up the formulations part of things and we won't be getting into the details of each form of the formulations just kind of fly over the formulations um and then uh and then we'll go and do experiments and maybe from the experiments we'll take a look at the elementary particles but those are different formulations cool and then we've got equations so i guess those things will pop up as part of the formulations and these interpretations and advanced topics so there's advanced topics quantum computing quantum chaos density matrix one field theory okay that is listed as a comp as an advanced type of quantum gravity um a bunch of things and then scientists okay categories quantum mechanics okay cool but that's uh i like that so i'm this type of person that likes to have a bird's a bird's eye view on a topic um while working through these but okay so let's take a look at let's take a let's go back to formulations then um and i should actually just bookmark these i think because this takes me to the bookmarks that i've put in here at the end of the day they're just part of the same thing right so um let's just let me just you know what i will basically get rid of these this get rid of these what is this oh that is the actual um the actual category um cool i mean i might keep that but then i will just keep these uh yeah so this is uh i will call that basically table of contents or something cool um okay let's go through so we know as i know roughly the hamiltonian stuff corresponds to um sorry the the uh schrodinger uh formulation is basically a um it's basically a hamiltonian kind of concept where you know hamiltonian mechanics i guess the idea of this just to make sure is that so it's still about action i guess momenta and simple interpretation of other mechanics comes from this application to one dimensional system so the value of of the hamiltonian and uh given the position momentum i guess this is of the hamiltonian is the total energy of the system the sum of kinetic and potential energy space chronic then what it says so this is the hamiltonian and then function of p alone well his function of q alone so in this example the time derivative of the momentum equals to the newtonian force so the first hamilton equation means that the force equals the negative gradient of potential energy the time derivative of q is a velocity and so the second hamiltonian equation means that the particle's velocity equals to the derivative of its kinetic energy ah okay so i guess that's the gist of it saying that if you've got the energies you can find the velocities and you can find the um the forces which then can yeah because it's basically the hamilton mechanics is telling you that um that the potential energy is related to the force and that the kinetic energy is related to the velocity which kind of the the later the latter makes sense the potential energy being related to the forces i guess it's the force yeah well it is right because so if you think about the potential energy as an example coming from the from gravity yeah well that tells you what the force is right so if i am um if i'm really like if i'm up a hill or a building like yeah the the sort of the floor the the how's that call was it like the normal force i think it was called so the normal force is actually what's keeping me on the phone on the ceiling of the building but then the gravitational force is what wants to pull me down and and that directly relates to potential energy right because the potential energy tells me that at that point if i'm just static standing at the building i have a certain potential energy just because of the gravity so um that seems to check or adding intuitive level right i don't know what other ways to define potential energy to be honest like what are all the ways to define potential energy because it's positioned relative to other objects stresses within itself its electric charge or other factors okay so that may be a bit more complicated but i guess the stressing part it is easy to relate intuitively to forces correlation center mass elastic potential energy okay for the spring case yeah so if you're contracting in the spring then you can think of the forces that this that the object is experiencing because of that position relative to kind of i guess relative to the relaxing position uh of that spring with the electric potential energy of an electric charge in electric field uh there's extra potential energy so that results from conservative coulomb forces and is associated with the configuration of a particular set of point charges with definite system an object may have an electric potential energy by virtue of two key elements its own electric charge and its relative position to another electrically charged object okay cool the whole electric stuff is something that i want to dive in at some point as well because i i have not touched that like ever electromagnetism in general just as a node to myself okay but that's that those are potential um these are potential energies so it's always it's funny that you can just reduce it to position right it's it feels awkward like it feels awkward you can reduce this to position but yeah i guess it's positioned relative to other objects that's the point right and that's kind of that is the interaction that that generates the force so but okay so basically what what the hamilton mechanics are telling us is you if you know the energies of a system you can know what the velocities are and you can know what the um uh what the positions are right because if you know the potential energy you can know the position and if you know the um the relative position and if you know the kinetic energy you can know the velocity um sorry the potential you know the forces and so if you know the forces you know the mass you can do the acceleration and all this kind of stuff um [Music] i think that checks enough for me right now so so this is the schrodinger stuff what is about this heisman attraction thing so okay but then we have the sum of histories concept which is basically the path integral formulation which is here is a bit of a different picture here we we've done here we keep track of the action so this means not the total energy but like how has the system changed energy wise and and that's why you have that difference where it's the potential minus the kinetic minus the potential because that's a specific definition that allows us to define this concept of um stationary action i think was it called right so that that this is minimize this isn't it's a minimal function when velocity is zero and everything is you know on potential energy um yeah exactly and and with these you can know then the stuff but again we're we're still and and this is so this is then again i haven't this is all checking stuff from the classical mechanic stuff still right so i i haven't it's not that i have checked in the schroedinger equation nor the path integral for quantum mechanics in depth i guess the idea of the path integral here is that that is just the sum over all possible paths but i don't know the details there yet of the formalism so in classical kind of mechanic to just have one path or it will just give you one path the minimal change minimal action path and and in quantum mechanics it will give you probably a set of paths that will represent that minimal concept as well that's the guess that's the guess so in quantum mechanics dynamical pictures are the multiple equivalent ways to mathematically formulate the dynamics of a quantum system so we've got heisenberg interaction and schrodinger so let's take a look at them quickly heisenberg interaction schrodinger trying to go with now right but heisenberg pitcher as the formulation largely due to vienna heisenberg of quantum mechanics which the operators incorporate the dependency on time but the state vectors are time independent an arbitrary fixed basis rigidly underlying the theory and its tension contraception in a picture which the operators are constant instead and the states evolve in time ah interesting i never thought about this this way so in the heisenberg picture the states are time independent and the operators evolve so this means that um yeah that in a given time your whatever whatever operator position operator momentum operator would be different and so it would affect the same state differently while what heisman is doing is like no no the state is the states evolve through time okay um weird i guess it's a matter of taste the two pictures only differ by basis change with respect to time dependency which corresponds to the difference between active and passive transformations the hyzerbaby's formulation of ma the heisman pitch is the formulation of matrix mechanics in an arbitrary basis in which a hamiltonian is not necessarily diagonal whatever that means in it further search to define a third hybrid picture the interaction picture okay but that's actually here the interaction picture of a direct picture is an intermediate representation between schrodinger and heisenberg whereas in the other two pictures either the state vector operators carry time dependence in the interaction picture both carry part of the time dependence of observables the interaction pitch is useful in dealing with changes to the wave functions and observables due to interactions most field theoretical calculations use the interaction representation because they construct the solution to the many body showing equation as a solution to the free particle problem plus some unknown interaction parts so equations that include operators acting at different times which hold in the interaction picture don't necessarily halt in the schrodinger or the heisman features because time-dependent unitary transformations relate operators in one picture to the to the analogous operators in the others these just which applied hypotonian and state vectors operators and state vectors in the interaction pitch are related by change of bases to those same operators and state vectors in the schrodinger picture to switch to the interaction picture we divide the schrodinger pitcher into parts h0s plus h1s any possible choice of parts will yield a valid interaction picture but in order for the contribution to be useful and simplifying the analysis of problem the parts will typically be chosen so that 0s is well understood and exactly solvable while one has contained some harder to analyze perturbation to the system if the handling has explicit time dependence for example if the quantum system interacts with an applied external electric field that there is in time it usually be advantages to include explicitly time-dependent terms with the with h1s leaving h0 as time independent okay so time-dependent state vectors the the way so the way most of the quantum computing stuff is being done out there right now then it's the it's the that's the um yeah it's the highest back pitcher right because everywhere and all this quantum computing stuff that's out there like you always assume that the states are time independent so this is a bit of a hybrid okay like we're treating we're getting a hamiltonian and we're like treating part of it as in the time independent part of it as time depend that's pretty crazy state vector is p time dependent state vector is showing the picture a state vector in the interaction pitch is defined as an initial initial time pending unit charge information the ears [Music] does not really changes in the state of systems energy in the system by relating changes in the state of a system to the energy in the system given by no product called hamiltonian therefore at one time is known the time dynamics are in principle now and all that remains is to plug in the hamilton into the shrug equation and involve the system state as a function of time even with the computer such technique is to apply a neutral transformation to hamiltonian doing so can result in a simplified version of the stronger equation which is the same a unitary transformation of frame change can be expressed in terms of time of a time dependent hamiltonian and a unitary operator and this change the hamiltonian transforms as okay applies solutions untransformed transform equations are also related by you especially i guess with these i guess i uh exactly i guess the point of this is you transform it into an easier one and then you undo the transformation so that's why that's what that's what these means right yeah exactly because the unitary matrix if multiplied by itself is the identity or it's one so so then you can recover the original stuff by just multiplying it by the just undoing the change okay so you change the frame to the stuff and then change it again so you apply the unitary to simplify the hamiltonian then calculate what i'm going to calculate and then revert and change back the bases that's just the basis change idea that's what unitary transformation means um operators have okay so that's the same thing here is it time evolution boom the term interaction representation was invented by schwinger and this new mixed representation the state vector is no longer constant in general but it is constant if there are no coupling between fields the change of representation leads directly to the tomo nagashvinga equation where the hamiltonian is in the case is is the qed interaction hamiltonian but it can also be a generic interaction and uh passing through the point x the rate of formula represents a variation of the phase the purpose of the interaction pitch is to shunt all the time dependence due to h zero onto the operators this is allowing them to evolve freely and leaving only h1 to control the time evolution of the state vector the interaction picture is convenient when considering the effect of a small interaction term being added to the hamiltonian of a solved system by utilizing the interaction between one can use time-dependent perturbation theory to find the effect of okay so these are all just different ways that different tools that will help you solve the same problem but it's the same as a dynamical picture so it's like you have operators right that act on states and transform these things and so if you have the operators then you can you know you you know and if these operators are time dependent and you know how they'll change with time and if you're going to calculate how the set is going to look like in you know t plus 5 from now then you just you just plug in the stuff and you get it well the path integral is like you get a so the path integral you're just having i mean what is the difference right some of the histories is that you don't have the operators it's a it's a it's a for it's a formalism it's a formulation that doesn't have the operator concept does it it just has probabilities of evolution i think that's the idea just has probabilities of evolving into uh you know in another state and i think these are the transition amplitudes okay but then what is and then and then so what is matrix matrix mechanics it's a formulation from mechanical spread by heisenberg uh okay but that's the same okay so that's the heisenberg picture isn't it so it interprets the physical properties as mattresses that evolve in time it is equivalent to the schrodinger wave formulation of quantum mechanics is measured by annotation in some countries the way from relation produces spectra okay that is titus matrix mechanics that's just this the three fundamental papers okay that's the so when when it was introduced by verna heisenberg max paul it was not immediately accepted it was a source of controversy at first later introduction of wave mechanics was greatly favored it was not mathematical language by um this great energy stays uh the br the heart is discriminates matrix mechanics on other hand became from the boar school which was concerned with the screen energy states and quantum jumps boys followers did not appreciate physical models that pictured electrons as waves or as anything at all they preferred to focus on the quantities that were directly connected to experiments in atomic physics spectroscopy gave observational data on atomic transitions arising from the direction okay yeah but uh again i think that's the same thing and then phase space but isn't done then the path integral phase i think phase space so places the position and momentum variables on equal footing in phase space in contrast to showing a picture uses the position or momentum representations the two key features of the phase space formulation are that the quantum state is described by a quasi probability distribution instead of a wave function okay that's different than state vector or density matrix quasi probability distribution and operator multiplication is replaced by a star product the theory was fully developed by um hilbran grono walt and definitely by joe moyer oh yeah that's where the weakness equation thing comes in right that's what's i think then use for the cv qc the chief advantage of the phase space formation is that it makes quantum mechanics appear as similar to hamiltonian mechanics as possible by avoiding the operator formalism thereby freeing the quantization of the burden of the hilbert space this formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistics statistical mechanics enabling natural comparison between the two yeah yeah yeah in face space is often favored in certain quantum optics applications or in the study of the coherence and a range of specialized technical problems so that's why it's in the xanadu model right it's less employed in practical solutions in phase pay the phase-based distribution of a quantum state is a quasi-probability distribution with a quasi-probability distribution why isn't that why isn't that link linked the yield expectation values respect is certainly all while at the additivity axiom because regions integrated under them to not represent probabilities of mutual exclusive states to compensate some quasi-probability distributions also counterintuitively have regions of negative probability ah okay that's what it means it means that they also have negative probabilities and stuff like this mm-hmm uh the phase-based distribution may be treated as the fundamental primitive description of the quantum system without any reference to wave functions or density matrices there are several different ways to represent the distribution all interrelated the most noteworthy is within a representation yeah um okay now i can okay now i'm connecting some of the dots with what the xanadu people are doing so the state here is represented by the by the the the wigner vikna zavigna vikna wigner um the phase space distribution possesses properties akin to the probability density in a two n-dimensional phase space for example it is a real value and like the general complex value of the wave function we can understand the probability of lying within a position interval for example by integrating the weakness function overall momentum and overall position interval okay yeah so you integrate a specific region and you get a probability but it's more yeah it's it's all really a continuous feature right it's being drawn in here actually in all the in the path integral as well um so if axp is an operator presenting an observable it may be mapped to phase space through the wigner transform conversely its operator may be recovered by the veil transform so there is a way to constrain from an operator to something that can act on the weakness stuff it could take negative values even for pure states with a unique exception of optimally squeezed coherent states it's not allowed precisely within face space region is more than the planck's constant i've keep coming back to what i mean i never remember what the planck's constant really is or means cannot be a retro function or as it could be localized in terms of a small region of faith space okay there's a lot of detail in here but a star priority non-community binary operator in the face space formation that replaces standard operator multiplication is star product represented by the symbol star each representation of the phase space distribution has different characteristic star product what is a star product though left and right derivatives okay that's its own thing jesus okay it's all impressive defense convolution the energy i can say versions are known as star target states [Laughter] favorite word of the day uh come on don't be so slow i just want to tweet awesome um i love how some of these names just pop up like that they look so uh where was i there tarjan functions so gen functions okay yeah look there's so much stuff to take a look at that it's just those are different formulations each of these things is like it's its own it's its own big big big big big rabbit hole okay so the face-based stuff is a continuous quantum mechanics thing [Music] um of competing stuff okay cool and this is everything right in terms of formalisms and like okay so here's a nice summary new kind of theoretical developments mathematical structure also here the postulates probably i should do the postulates pictures of dynamics yeah probably i should do that as well maybe to kind of close up these and then and then and then i think that we can just go then i'd like to go through some of the fundamental particles and then i'd like to go through experiment stuff and and and i guess at some point we'll just pick a formalism and deep dive into this uh deep dive into it um i just what is planck's constant that's an energy thing right it's a photon is equal to its frequency multiplied by the planck's constant okay so it's a it's a c it's a constant that relates the energy with the frequency and also the mass to frequency um okay and what is the reduced planck's constant this has to still sink in a little bit bar h bar it's just a quantum of angular momentum okay so this is related okay i can i see that frequency by frequency being like denoting sort of the region yeah yeah so it's kind of like so so okay so so this is something that relates it's constant that relates then the different energy jumps that you can do and that's why it's like there's nothing like we don't know what happens in between i think that's kind of what you come up with a lot of the uh what time is it how much time we have okay we've got to be more time um so where are we here the the formulations of quantum mechanics so the postulates i wanted to the postulates right um but here's like okay so the old quantum theory a bit of history i guess uh derived black body spectrum which was later used to avoid the classical ultra catastrophe by making the americas assumption that in the interaction could you change this crit units called quantum planck's postulate the direct proportionality between the frequency of radiation and the quantum energy and that's the planck's constant i had it right there um quantum were actual particles which were laid adopted photons uh the new quantum theory so eisenberg's matrix mechanics is the first successful attempt of replicating the observed quantization of atomic spectra later okay later developments pattern creator phase-based formulation quantum field theory jesus christ c star algebra so there's even another formalism okay but yeah let's take a look at the postulates the following summary the mathematical framework of quantum mechanics can be partly traced back to dirac uh neumann axioms postulates are canonically presented in six statements though there are many important points to each um yeah but that those are postulates of then of of the heisenbach version or the schrodinger version i mean whatever the dynamical version right or what a physical system is generally described by three basic ingredient states observables and dynamics a group of physical symmetries classical description can be given a fairly dark way by phase-based model sympathetic phase space observables are real valid functions on it time evolution is given by a one parameter group of similar okay this is just to abstract a quantum description number consists of a hill with space states yeah okay but if we talk about hilbert space we're talking about some concrete formulation here right okay but you can master so this is postulates description of a state of a system postulate one the state of an isolated physical system is represented at a fixed time d by a state vector belonging to a hilbert space called the state space yeah but that's that is so again that is yeah that is for the dynamical picture and the path integral but that's not for phase space right but you can map it okay um state vector postulate to every measurable physical quantity a is described by a hermitian operator a acting on the state space this operator is an observable meaning that its eigenvectors form a basis for h for the helibert space it's an observable that can be measured okay position momentum so but these things are defined as physical quantities but again that's also the dynamic feature right because in the path indicator you don't have operators and i kind of forgot why the hermitian operator was the hermitian part was important partial through the result of measuring the physical quantity must be one of the eigenvalues of the corresponding observable a portion of four when the physical quantity a is measured on a system in a normalized state the probability of attaining an eigenvalue is given by amplitude squared of the product by function so that's okay that's the the bond rule isn't it partial at five if the measurement equals on the system is maybe it gives the result a and then the status immediately after the measurement is the normalized projection of this onto the eigen subspaces okay so that's like the measurement problem right postulate 6 and the time evolution of the state vector is governed by the schrodinger equation where is an observable associated with the total energy of the system called the hamiltonian yeah but that's again that's all the implications of the postulate so these are all the postulants pitches of dynamics as we know heisenberg dirac picture and then we have time as an operator spin police principles representations the problem of measurement that is the problem a list of mathematical tools okay but that doesn't do a good job to the path interval stuff right oh related elements okay to electromagnetism resulted in quantum field theory which was developed starting in 1930s chemical theory has driven the development of more surgical from electrical mechanics which uh okay so and quantum theory to all electromagnetism so electromagnetism is something that i also would like to check in general um but i'm getting uh i'm getting a good sense of a map here maybe maybe we could do a bit of the exercise to actually have a bit of a kind of map that we build somehow um but yeah okay cool i think it's good enough for today's session i got a bit of an overview on all the formulations already so i think we can start next with i'll probably let's do the next one let's do the fundamental particles and maybe the wave particle duality thing that's a bit confusing as well and then we can go through experiments and i think with experiments we can take a more of a deep down so we can just really go and actually take a look at say the you know bell's inequality for example and try to look at it from different formulas perspective as well and see what we learn there yeah i think that's one of those so the next one is going to be maybe a bit of a or i don't know maybe i'll do the maybe i'll do that offline i'll see the the elementary elementary particles um and and then we'll just jump to the experiments but that's it formulations is nothing else than these so you've got all these formulations and yeah of course each of them is a its own rabbit hole and i mean i'm not even talking about advanced topics like the quantum field theory stuff and quantum gravity which is that's what's i think that's kind of the frontier of trying to connect with the quantum staff and the classical stuff right awesome that was a good session i like kind of the overview stuff have a good day |
well this was supposed to be part 2 from the Yellow Submarine project review but Miho reached out to me saying man you're crazy etc us in like there's a bunch of time it's really complicated is gonna take a lot of hours here we go I mean really care I I think I think it's worth spending the hours but I'm happy that I just picked that problem totally in totally that the project early randomly and then it turned out this is like certify you open the box to a completely different type of chronic computing which is what sana do I hope I'm pronouncing it correctly it's doing apparently so okay gotta figure it not so what I'm gonna try to do is I'm gonna try to browse a little bit here gonna spend probably 10-15 minutes taking a look at what that paradigm is and then how this house is different from from regular quantum computing just really try to get to the basics and then see if we can just surface around the project maybe not do a super deep dive into the project but at least understand how is that different from regular chronic computing and by regular I mean what we would you see most of the videos in my channel right so the gate model that is permanently implemented by the IBM Q experience guys and so well let's just take a look see whether that's really so what is this it's literally the first time I take a look at this stuff so that's what I do in all my videos I don't really check anything a priori so blah blah the future of integrated photonics quantum photonic processors will solve today Stathis business problems locations my machine learning good are there any learning materials software and Elaine strawberry fields interactive so me how mentioned I should get familiar with strawberry fields if I want to really understand the project well [Music] so the it's listing the first dedicated machine learning platform for quantum computers are so that's an actual machine learning platform okay so this has to be really specific because those seem like gates as well and it's like a circuit thing but there's definitely some weird things like FK VR which ooh white paper Jones joined slack field from machine learning sorry feels interactive documentation let's take a look at the documentation [Music] mr. Barry fields oh wait a second actually I did check one of these pages it was what I was doing I was doing something and I was checking something when during one of my videos from ado but I think I was not aware that this is something totally different so features strawberry fields getting started as usual continuous okay so that's CV quantum computing that's what I didn't I don't know if he mentioned he mentioned it so me I mention didn't one of his comments or messages he sent me but so this is basically continuous variable quantum computing okay introduction that sounds like the good place to start right and the physical systems are interesting to continuous with a light-being example such systems reside in infinite dimensional hilbert space mr. Hilbert is everywhere offering a paradigm for quantum computation which is distinct from the qubit from the cubic model okay interesting the continuous variable model takes its name from the fact that the quantum quantum operators underlying the model have continuous spectra the CV model is a natural fee for simulating personick systems electromagnetic fields harmonic oscillations phonons haven't understood indeed understand any of those words but and for setting where continuous corner but I guess I guess what this is trying to say is that this is sort of truly in nature ten years in terms of the way you manage the information but I mean really is it really different than the cubed model because in the cubic model you can also sort of operate in a continuous sort of I mean probabilities that so the amplitude in the face is not you have zero and we assure you sure most of the stuff you use are certain certain states like zero to one the plus the minus etc but that's probably okay so there's a table here high level comparison so you've got Q modes instead of qubits okay and an information in it is one beaten here's relevant operators quadrature operators more operators have poly operators the x y&z okay common States common States coherent state squeezed States okay so now um so now I know where the squeak with this squeezed things come because I was one of the things that I say here in the and the code right squeeze squeeze guy was like what the hell is that okay number States so you've got different types of states okay come on states and then poly eigenstates and this is where we know 0 1 plus minus and then the complex face thing common gates rotation displacement squeezing beamsplitter cubic phase okay phase shift Haram artsy not tea gay yeah so those are we're familiar with this is the first time I see this exquisite in displacement rotation common measurements hormone time heterodyne foreign counting poly bases measurements yeah so you can measure on the Z base on the X base on the Y base I think that's what it means cubed based competitions can be embedded into the CV picture the most elementary CV system is the persona criminal oscillator it defined with the canonical mode operators this and this dissatisfied the well-known commutation bull okay it is also common to work with the quadrature operators what the self eternal producer is this way to abstract at the moment we can picture a fixed harmonic oscillator mode say with an optical fiber as a single wire in a quantum circuit these cue modes are the fundamental information carrying units a civvy quantum computers by combining multiple cue modes each with corresponding operators and interacting them with a few sequences of suitable quantum gates we can implement a general CV quantum computation so the D comment Academy between cubed and CV systems is perhaps most evident in the basis expansion of quantum states so Q is this and okay so it's sort of a linear combination of those two things while a cue mode is this stuff what I need to grow off whatever for Cuba's we use the discrete set of coefficients for civil systems we can have a Kentucky's with a discrete set of coefficients for civil systems we can have a continuum the states the states X are the eigenstates of the X quadrature of X being a real number these quadrature States are special States for more general family of civvy states the Gaussian States which we now introduce so our starting point is the vacuum state all the states can be that seems like this the same thing with classical a classical ok with regular chronic computing other states I mean regular it's not regular it's just the gate model or the qubit model let's call it all the states can be created by evolving the vacuum state according to this where H is a personick hamiltonian so here we are again with the Hamiltonians and t is the evolution time i mean at the end of the day it's not that it's a different thing essentially it's just it's a different sort of framework right I mean it's pretty cool to see that it feels like that's really the language level at the same time right so it's like when you've got like Python in classic Olimpia Python Java Script and all this kind of stuff and then each of them is a bit of in a different nature I mean at the end of a what they do is they manipulate the same love the same staff down under the hood and then this is just sort of like a different mathematical framework for for expressing the same was like you're messing you're messing with stuff at the quantum level that's what everything everyone's doing but then the d-wave people are doing something different the Sunnah do people are doing something different and then IBM is doing something different small squad right akin to operators for a single cue mode cause Gaussian states are parameterize by two continuous complex variables a displacement parameter and a squeezing parameter okay let's as always that's the theme of the channel is intuition right so I'm not I don't want on this than that I'm not going to understand all that I just want to gain intuition about that and I think that shouldn't be complicated I mean okay so you've got these things and then you've got like a displacement parameters quizzing parameter often expressed as these Gaussian stairs are so named because we can identify each Gaussian state through its displacement and squeezing parameters with the corresponding Gaussian distribution the names displacement is squeezing maybe come from the fact that that's what you're doing to the underlying whatever Thorin's the displacement gives the center of the Gaussian while the squeezing that reminds the variance and rotation of the distribution so many important pure states in the CV model our special States coherent state displacement and squeezing zero squeezed States ok displacement is zero but this but there's some squeezing so this is so this looks like those are ok the sir dimensions and I mean these are the parameters right so you can displace it exquisite and ok displaced squeezed States it's like eigenstates I can stay vacuum stayed number States okay so that this number States Gaussian States we talk to a Gaussian States ok number States complimentary to the continuous Gaussian States are the discrete number states or States these are eigenstates of the number of the number States are discrete countable basis people I'm really not understanding most of the stuff here this each of the Gaussian States considered in the previous section can be expanded in the number of say for example coherent States mixed mixed states mixed states mixed Gaussian states are also important in the CV picture for instance a thermal state this which is parameterize to the mean photon number this and Alex is pure states by applying quadratic order Hamiltonians of thermal States cv Gades unitary operations can always be associated with generating hamiltonians via the recipe this but this is pretty similar I mean really I might be totally wrong but this is pretty similar to the way that you can kind of Express and build any kind of gate in the in the qubit model right at least there was something like that in a paper it's like this like yeah it's it's e to the something for convenience you can classify unit or unit R is in the degree of they're generating Hamiltonians Gaussian gates one mode and two mode gates which are a so this is one qubit and two qubits I guess are quadratic in the motor paraders displacement rotation squeezing and beam splitter gates these are equivalent I find it funny how they keep going back they keep making an analogy with with a cubic model so you kind of understand clifford group of gates from the qubit model always done cliff for gates which ones were these ones which one's worth this ones image will tell us non cliff ranae its cliff cliff or gates XY okay you see with image with image search you will always get what you want XYZ the harm art SS okay it's all you see you see e to the minus I PI whatever um so those are non-gaussian gates are Gaussian gates and non Gaussian gates the Gaussian gates are sort of the equivalent to the Clifford gates non-gaussian gates are single-mode gates which is degree 3 or higher cubic face gates are : to the monthly for gates in the cubic model what a non non Clifford gates tell me tell me what are the non Clifford gates maybe I was too optimistic nan Clifford gates okay I guess it's whatever it's not a plea for polygroup this we know the Clifford group okay any gate from the forum okay it's not a cliff oh okay yeah good so now then you've got different gates displacement rotation squeezing beam splitter cubic phase and this is what they do so what what I assume in here is it seems like this is pretty similar so you basically if God but this is constant states you've got displacement and squeezing as parameters number States I don't know mixed age I don't know okay because basically okay big face seems the those kids just play with those parameters right so displacement probably I'm guessing the spring displacement displacement probably adds displacement it's squeezing at squeezing whatever those effects are right but then okay and then you've got measurements [Music] okay so the gas in longest measurement citizen class consists of two continuous types on line third line measurements while keen on Gaussian measurements is photon counting how modern measurements and measurements foreign counting so this is basically this is basically what you okay so the measurements you can do all but essentially essentially I might be totally wrong but intuitively that seems like nothing not so much different than the cubed model but it's like a slightly different so you've got to be the different different types of states different things you can play with and probably that's because you're playing with photons I I'm not a hard process sure but I think that's the idea behind this right that you're playing with photons and that makes it different okay conventions and formulas boom the nice thing about standards is that you have so many to choose from an intersection we provide definitions of various corporations used by strawberry fields measurements gates okay so here you can actually deep dive into those things States for bases vacuum state coherent state squeezed state thermal state cod state displacement squeezing so squeezing I don't know squeezing I just like the word what my god I have indeed opened up a box that I don't know if I want to open the squeeze gate affects the position and momentum operators the bases the composition of displacement in squeezing operators was our eyes Mike Crowell and the following quantity was calculated the important special cases of the last formula are obtained one this and this on the other hand okay Chinese artists even to deduce that your blackness what-what-what is squeezing strength s Kaede the position and momentum operators what else we've got displacement we obtained the position and momentum operators position and momentum the matrix elements of this place interpreted in the basis someone needs to work on these people so that this is more understandable rotation we write the phase space rotation operator us it rotates the position the momentum quadratic face it cheers the phase space preserving position beam splitter they will soon the operators according to this against a 50-50 beam splitter two modes quizzing it can be the component to opposite local squeezers sandwiched between 50% beam splitters so control to XK the control test gate also known as the addition gain or the sound gate is a control displacement in position so it seems controlled faces resolution the face or addition of the second mode in the position basis so this it seems it's basically touching the momentum and the position it seems like those operators are defined define like that okay but maybe maybe what I should do next is I should take a look at some of the quantum algorithms so maybe maybe maybe what is in order maybe that helps me understand that lost channels thermal lost channels so that's probably the same thing like do coherence and stuff like that the compositions Wilson the composition whatever regular the composition so far I don't care about this but it seems like it's a little bit more complicated in the sense they've got different types of states so you've got different types of states and then you've got different gates up take a look at what I'll do is I'll take a look at some quantum algorithms in my next video okay and maybe that's gonna help me understand a bit more the basics of this because it can't be that complicated really probably if you want to know the details it just can't be that complicated and that different from from what the other one stuff let's see and then we'll go back to the algorithm to the project actually I really want to go back to the project don't want this to be sort of a tangent I I think it's worse definitely exploring the the whole concept behind wanna do and behind this the photonic so that the CV quantum computing my intuition tells me it's just a slightly different model cool Stadium from more I didn't expect that to end up this way |
this is what are we doing today today is a bit of a different topic than my typical um live research today i'm gonna be taking a look at an idea that i've been um that have been sort of like entertaining for a while which is like i've been i've been i've been doing all this kind of you know playing with the idea of like raw videos and role you know kind of like role research like how kind of how do you you know um capture that like honest and oh guys give me a second someone's at the door yes now cool so uh oh sorry i gotta switch the scene back there you go cool so um what i was gonna say i was gonna say that um yeah i've been playing with with this idea of doing all this kind of like raw video and stuff like that right i think i mean ultimately why is this useful um it's a good exercise i think it's a good thing it's it's an interesting experience for myself as well but then i also kind of realized that you know it might be useful something else it might be useful um to actually kind of train an ai in the future like kind of use that as training data right as in like raw data of like some you know kind of showing machine um what learning and so the research truly uh looks like i mean i don't know it's actually it's it's interesting right because one one could think about like what is it that makes something um conscious smart human whatever it is right but like how do we learn this stuff i mean and if you take a look at things like this gpd3 story and all this kind of stuff you you do end up thinking like you know maybe all that all that really is there is like it's a language model right in a way like it's it's a sort of a symbolic way of understanding the world i mean that's what we do even when we think right it's just in our head to steal language but it's in our heads and um you know watching keats grow is and explore the world is a it's a super interesting experience to kind of that actually kind of in in a way reinforces that that vo on on consciousness um just seeing how like a child just actually tries to understand things but just kind of like imitating uh you know what they what what they what they perceive what they kind of see right it's like it's it's really interesting i mean there's of course everything that goes into like you know learning how to build our how to kind of like operate our body but then sort of the the interaction with the world is like the language and how do you think and stuff like that it's just super interesting but anyway it's it's one of the things where it's like well you know how do we how do we learn how do we learn to to learn right like we we learn this by um essentially seeing others do that essentially kind of uh searching for four things and and when we think about okay how do we how do i learn about a complex um how do i learn about a complex problem how do i learn about something well you look for someone to teach it to you you look you're i mean that's kind of one way to do things right so you kind of um you're looking to be trained in a way right so you're you're looking for some material that you can just go through that you can process whether it's text video whatever it is so that you so that you can basically adapt your model right and so this is like th this is this is this is what learning is in a way but then you kind of sometimes sometimes think about like cool so what what you know what is then self learning when you know you're not really i mean essentially there's nothing that's not really like self learning a lot self learning this is interesting right how how would you how do you come up with your own conclusions or how do you kind of do research is interesting because in a way it's like you're not really it's like unsupervised learning in a way right so um or the classical traditional learning is like um you're basically you're basically doing in a supervised learning in a way right so i mean let me just am i am i saying i'm not saying or not um let me just uh can i just start any cognitive one it's just like supervised learn i think this is what when you're providing data with this to a system and you're actually telling no no ah and you're actually telling the system it also known as supervised machine learning is the subcategory it is defined by its use of label data set so you're actually basically telling the system what is the correct answer just many many many times and that is just that is how a lot of the you know human teaching works as well right you just assume what the teacher is telling you or what the material that you're looking at is telling you you're just assuming that assume that the labels are correct and so you're you just you know you're just gonna go through these but then unsupervised learning is when you have that um you you have that kind of you know just raw data there's like no you know it there's no guidance in terms of whether something is correct or or or incorrect you just kind of uh so the hope is that through mimicry which is an important mode of learning people the machine is forced to build compact internal representations of its world and then generate the imaginative content from it inc okay within a stacked exhibit self-organization that captures patterns as probability densities or combinations of now feature preferences the levels of supervision spectrum so it it this is essentially i guess something that you could use from a like a as a dichotomy between sort of research and and actually learning research is when you're kind of you know you've got some basics some base knowledge but then you're trying to learn something new on your or discover something new on your own you're not just like you don't you you you know you just don't stumble upon these on the world you just you try to understand make sense of make sense of data do experiments do you know whatever research stuff search for things right and so in a way um you know but even even if you're relying on you know existing things like let's say i want to learn quantum mechanics but i want to learn about some so i want to learn about quantum mechanics it's kind of like you have you have knowledge there that you can use for super for sort of learning stuff in a supervised way right um but still like you i guess one way of doing that is a bit of that kind of mixture of you're getting things and then you're trying to discover things you're um your yourself in an unsupervised way i don't know how to explain that i have the feeling that there's this kind of you know dual mode of of of learning you know in in a way you kind of have you know you have the staff that's the body of knowledge uh that exists out there and wikipedia is a great example but then you kind of have that like okay so let's try to discover or let's try to answer this question without actually finding you know well it's actually understanding it right like how do you you know build build the understanding right like you know give an example it's like how do how do i learn uh what a fourier transform is well i can go and look you know look at the answer but it's when it comes to this kind of topics the answer is there's no just unique answer like one liner or something right it's just there's there's a bunch of things around a concept that you kind of have to discover if you want to understand the first principles of it and that you can then apply later on so it's actually you know this is what i'm exploring with the whole uncertain systems right so how do how do we learn this kind of stuff how do like how does it look like this process kind of exploring this a little bit and then you know with a very specific instantiation of the problem which is quantum mechanics um but in general i guess that applies to everything so one of the things that i um that i've been entertaining in my hand in the past for the past you know months is why you know if if we just kind of build a huge body of this kind of data will be you know you build a great data set to kind of teach a machine how to do that how to actually um how to actually how to actually teach itself in a sort of in this kind of mixed mode like supervised unsupervised way right and and and so you know i thought why not like encourage other people to do that and that would be a great idea sort of a bit of a little little web 3d project where you kind of have a contract a sort of smart contract or something and you would upload say a video or a document or a tweet whatever it is like whatever it is that you've used to capture that raw process uh around a specific and it doesn't have to be self-contained like it just it's gonna show you exploring stuff right or it's gonna show how do you explore you know um whatever it is that you're exploring and capture that and then basically you know give to the contract and then the contract will kind of give you um you know sort of coins in exchange or something like that like whatever it is like a made-up you know knowledge currency or something like that and uh you know of course that you know the the the idea is to use the this kind of like uh blockchain web three economical model where you know these coins would allow uh you know what kind of build up intrinsic value in a way uh would allow you to do other stuff i don't know and i just kind of wanted to play with these especially because i've recently discovered replete as i was building as i was building or playing with you know some of the um senpai stuff and i kind of uh came across these uh the other day where basically apparently there's a apparently there's a really easy way to build smart contracts and test them and play with these within replay so i think it would be great because i'm a lazy person and that is basically um you know a lot of the stuff that is difficult about web3 is how to get all that infrastructure ready in terms of like how do you build a contract how to deploy stuff and i'm talking about some playing with some very sort of the prototype would be very simple i i don't even care about like upload it's more like i i'd assume you know you have you know you kind of connect your wallet and then you basically give the thing a um uh an ip fs link an interplanetary file system link uh and um so you know i kind of assume you've uploaded whatever it is already in the ipfs uh peer peer peer-to-peer network and you just give it to the contract and then the contract gives you some kind of currency in exchange of course the question is uh what how do you how do you make sure so that's just a folder what is these and and then you know of course i think these are just examples i i hadn't seen that before can i just delete these folders oh actually um i i did try something a while ago but i i forgot that this is it was in here what is that is that an example and solidity but yeah so basically i mean there's still the problem of like how would you validate that i'm not just uploading like a cat video or something so you'd still need a human you'd still need a human to do that in a way so you depend on someone else you know we need to incentivize someone else to actually maybe validate your yeah that seems to be examples token simple token wallet simple wallet nft so there's some examples um i'll give you a second i'll be back in a second sorry i had to run to the door again um oh god i should exercise a bit more anyway uh essentially that's kind of what i um what i had in mind so basically build that contract and uh yeah kind of encourage people to just you know um upload the video so upload the content upload the stuff get the coins uh as a reward and uh you know by sort of hoping that by doing these we encourage more people especially professional researchers for example to you know kind of upload their own you know their own ways of doing things they're their own ways of of doing research yeah oh i'm gonna go for some water and then we'll get started with the coating i mean we'll see my goal for these is try to spend the next 20 minutes half an hour or so and trying to kind of understand how could i how can i do that in replit should i just go and create something so literally starter and i'll just call it um probably call it ancies do you call us like aunties or something and this coin or um whatever i don't know uh [Music] as always naming is the the most complicated since when is ansys columnar i'll just go answer this coin and then see what happens can i get something to drink and be back in a second cool so what do we have here um yeah it feels i think that's exactly the same i had in the uh in the other folder did i delete that uh i just want to make sure i have stuff clean so um i have these folders here you can just delete these and then delete this yeah and can i just delete folder stops existing okay ans is going cool um we have here tools so welcome to the world theorem or three i'm sorry here's better easier to what read that whatever um so press the run button uh you should only need to do this once and might take like 15 seconds to install okay so this will install the installs everything replica is really cool but i have to still get to you get used to like how how does this actually work you hit the the run button and it actually what is it that it's that it's running ready contract simple storage files those are all the dependencies interesting whoa whoa it actually opens up some kind of ui that's cool reloading page oh ah she can connect a wallet deployed contracts interesting okay um yeah but i want to see i want to kind of open the this thing here how can i how can i okay actually okay it just runs it like that no this is the room i one okay so um you should only need to do this once and might install the run tools and uh start up the contract develop the the contract deployment ui and compile your contract soul file okay so that uh-huh well so this will automatically compile whenever you edit it and all your contracts inside of this file will be available to deploy from the ui pressing control s will reload the ui we have pre-installed packages from open zeppelin contracts to install other solidity packages that are distributed in npm make sure you install them using the package installer in the sidebar i think that's here cool examples we include a few example uh contracts in the examples folder this will not be automatically deployed or accessible in the ui but you can copy paste them into your main contracts all um import them they're there for you to reference feature work okay cool lsp support i don't know what lsp is is it linting integration with um hard hat for local in rebel testing um an actual solidity route to quick prototyping uh testing functions are lines feedback okay cool seems uh yeah but how do you how do you deploy contracts do i actually have to connect my wallet um like how do i test without like how can i test in a sort of a dummy network or something like that because that would be definitely the the best right i don't want to start deploying things on ethereum and pay all the money for nothing i'll get one youth for testing so there's actual there's some actual testing stuff that you can do ethereum ethereal mainnet ah let's see what happens so what if i connect the wallet sorry guys uh why it's in german i have uh i don't want to what i don't want to do is uh oh because i'm in no i'm not an incredible let me just move that away from the screen here see if i can just uh reconnect because i um i already have a wallet but [Music] so give me a second okay because i have to figure this out um you know just in case i mean i i in this wallet i just have probably like 0.018 or something so not a big deal but maybe i'll just i'll just put the beer back screen just in case and uh because i i when i installed that i did not connect my wallet again so i'll do that in a second i'm still here hopefully you can hear me but what i need to do is i need to actually find the fine you know i have stored my keyphrase in a super ultra secret way in place [Music] it's a secret i can't remember where here maybe come on or ah or wade maybe maybe they are here so let's see if that if i imported the right uh have i imported the right it looks like yeah okay cool so now so i have 0.02 eth in this wallet seventy dollars currently that is and and i have another wallet but i don't know what's in there anyway back in here so what if i connect the wallet now what will happen will it okay so that's the account right and i want to connect it with this account connect it okay cool so 0.02 e to have here that's my address okay so i can switch to a test network and i can use this one this is pretty cool replicas net so replica gives me a test net like this is nice so i can get a need for testing okay so with these i can actually deploy contracts that's cool okay um i now now that i have the wallet connected i can choose the contract and actually deploy it so what do we have here so we have contract simple storage and then we have a math test um and we have examples what have i done here examples nft token i think it's token is what i want right because that i want basically i want to kind of have uh so this okay so um so that's an it's an ethereal ethereum example um and so basically i hope you guys can read that because i don't want to zoom in too much uh because then my stuff gets messed up in here but maybe i'll just i'll just i don't know i can't can't i hide these no i can't um yeah maybe i'll actually exactly i'll do that and close it here so i i have these oh and i have to connect the wallet again let me connect that's not coinbase i know i'm convinced i want to connect that one and i want to switch that switch to test network i'm okay i'm not in main net right because i don't have a need let me just refresh that okay whatever i have an ease so i'm i'm because i don't have an eth in real life so i assume that i'm on the test that's that's wrong i'm assuming it's probably a bug but anyway um cool so this way i can just kind of you know get rid of these for a second and focus on that so this is okay so an address is comparable to an email address okay so blah blah blah so i have so there's a token supply i mean again this token wouldn't really have a supply right because in a way it's it's just you want to um reward people for uploading the stuff but i mean it feels it feels somehow uncomfortable like if all it if all you need is if all you need is um a video to get that like a coin without supply doesn't look like it doesn't sound like a great idea but let's ignore that for a second so here it's basically you map the balances um okay so you get the addresses and you see how many uh how many how many yeah basically and then the constructor okay so the deploying the owner is is the person who deploys the contract and then it say set the token balance of the owner to the total token supply so the owner has all the supply and then sends the amount of tokens from any caller to any address so basically this method um basically uh okay so requires blah blah and then what it does is it takes the amount from here and adds it here um so what this does is i guess it's just a contract where i can't i can send or anyone actually anyone can send any money to anyone right there's no i could of course require that the sender is the owner so only only the owner can actually send the staff um but that's not uh i guess that's not the point but that's a good basics a good basis and then wallet what is the what is the wallet doing so you have an owner lock deposit log withdrawal hmm withdrawal okay so only the owner of this wallet can withdraw so the the sender must equal the owner uh and then the balance must be there recipient transfer what that is what is what is these doing uh there are events events allow for logging of activity on the blockchain software applications can listen for events in order to react to contract state changes ah interesting what is address payable recipient boo there's a lot of stuff that i didn't know about that payable solidity this modifier allows the function to receive ether okay there's definitely quite some stuff that i need to learn um yeah but that will be and an nft simple nft uh whoa whoa whoa what is that ownable token counter okay so whatever disease i don't know what is that is it some sort of kind of standard or something it's the non-financial token standard okay so it reduces the standard for nft in other words it's the type of talk this type of token is is unique and can have different value than another token from the same smart contract maybe due to its age rarity or even something else like it's visual but visual yes i actually have a variable called token id so for any contract the pair contract address must be globally unique that said a dab can have a converter that uses the token id as input and outputs an image of something cool like zombies weapon skills or amazing kitties okay yeah okay so that's just an nft standard the nft standard and uh mintik is essentially creating an nft so who opens it what is the tokens uri which is a link to an image hosted on ipfs and then um okay so that is already cool so that's already all built into the standards so there's sort of a mean function that puts it into the blockchain you said that's pretty easy okay so there's a lot of stuff that's been done already cool but what do we need i mean let's think about these what do we because i'm like so i am uh torn between two ideas right so one is that experience that you're uploading you know you turn it into an nft because essentially that that's what it that's what is it it's a um it's sort of a unique experience right so that's uh that's what it actually is but then but then what's the value right so you you you mean that and then why would someone buy this well someone will buy that as in you know you you could buy nfts just because but yeah the thing is you know it would make sense that you kind of like turn special moments in nfts like for example let's imagine that you know a researcher kind of um goes and like you know disco sort of creates a new theory of everything that it turns out to be you know that that one right so the actual thing um and then captures data into a video and then you know then if these the video but then for this we have already nfts right so why would you um you know what would you encoura you want to encourage people to kind of upload all the raw data not just like oh that is kind of what led to that moment right that kind of aha moment so to say so you want to encourage people to just well um do that so in a way you want a token right you want to give people i i really think you want a token because you want to um you want to basically give give people something in exchange for it for for the video right um so i would i i would start with these simple storage store data okay so actually the contract has memory you know you just can have memory and they can store things and blah blah right but let's just ignore these let's just have let's just use that contract as a basis and say um well it's the ansys token and so yeah well it is the owner uh you know yeah you have you have the owner uh the owner it's gonna be me because i'm gonna deploy the contract so um i don't know about talking supply what i do want to have is a mapping between so it's the balances and then i also want to keep a i basically also wanna store do i wanna do i wanna keep ownership in terms of like track of ownership of like who uploaded certain things probably not like nobody cares um so in a way you want to have i probably would have just like removed right store data ah that's okay so i can declare whatever i want can i declare okay so can i just say um well i wanna keep track of the balances do i want to keep yes right because essentially that's that's the coin otherwise how how does this work though really like you did the smart contract needs to keep track on on who's how much of a token someone has then how the wallets actually work like do wallets have wallets have the log deposit lock withdrawal how do you get like how do you get a wallet to how to get a how do you get a token to be recognized um how how does dokin get recognized by um how does a token because i think the wallets really just do these i think i think that's the fun thing is like the wallets actually just lock deposits and withdrawals send ether from the function color to the simple wallet contract your own token um i guess there's a bunch of services that allow you to do things like that make my own okay so awesome okay um open metamask and click on the add token button select the custom token option and paste the contracts address in the first field okay so basically we'll face the token symbol and decimals automatically click on next and your token will be added to the wallet it will be available under the asset section in metamask okay so first of all but this is okay so there's a standard for these i guess what this means is it ensures there are certain functions that i need to implement there we go uh using contracts you can easily create our own contract which will be used to track gold an internal currency in a hypothetical game so uh okay so you you import stuff like that okay but i guess if i want to import things do i have to this is importing stuff like that and do i have to packages do i have to actually install it like that doesn't find it or what shouldn't find it automatically no okay um i often use the inheritance and we're using for both basic standard condition and the name symbol and decimals okay so this basically ensures that it has we'll we'll actually go with these so we will we will just go with these actually it's going to be ansys token oh actually i i do i do i have to um does this token need this um so these are the functions is the uh this thing must be like three letters i guess so symbol or can it be whatever i don't know why people make so much of a fuss out of these this is just very basic programming to be honest there's nothing really special about these other than like the mental model i guess simple is there any restrictions to the symbol um i'll call this the ansys token and and basically the name is anses and then this is going to be like what i mean isn't it is there any length symbol okay so there's different no there's no restriction it seems like because to be honest i'll just call it anses right um i mean initial supply so i have to give it an initial supply okay but i guess the supply can increase right it would be good to know how that actually works to be honest um but okay what is it what is it doing so you deploy the player's balance ah so there are different supply mechanisms okay let's do that uh it's funny because um what is that what is open zeppelin what is opensampling i guess you can just create okay it's it looks like a place where you can just create your contracts and stuff like that complete security products to build manage and inspect all the software long versions theorem projects okay yeah that might be an interesting thing to consider at some point but where was i so in this guide you'll learn how to create an erc20 token with a custom supply mechanism we'll showcase two automatic ways to use open separate contracts um okay but i don't wanna specifically use open zeppelin staff in here ah cool yeah sure of course those will be the miners right so the people actually providing the videos because it's like you're you're you're kind of you know working yeah that's actually cool okay modular is the mechanism um but i'm afraid this is open zeppelin specific so this is from ethereum.org so maybe that's better as the technical standard is a smart contraction blockchain for fungible token implementations defines a common list of rules that all fungible ethereum tokens should adhere to con consequently these tokens time empowers developers to full-time security predict how new tokens will function within the large ethereum system this simplifies these developers tasks because they can proceed with their work knowing that each and every new project will need to be redone every time a new token is released cool so um okay so these are the functions so total supply balance off okay so yeah that's and that's what wallets are using probably right so that's what wallets are using to kind of query the balance of of someone allowance transfer approve transfer from transfer approval total supply returns the amount of tokens in existence this function is a getter and does not modify the state of the contract keep in mind that there are no floats in solidity therefore most tokens adopt 18 decimals and will return the total supply um and other results as followed for one token okay not every token has 18 decimals and this is something you really need to watch for when dealing with tokens returns the amount of tokens owned by an address this getter returns a remaining number of tokens the spender will be allowed to spend on behalf of the owner this function is a getter and does not modify functions moves the amount from one address to another um events this event is emitted when the amount of tokens is sent from the front office to address so what's about code to base your from show ah okay so that is yeah okay ah so open zeppelin has a good implementation as well to supply okay so actually this has a total supply decimals 10 ether why is an ether it's just an example i guess and opens up link has another symbol total supply balance off okay so everything is well i mean i can just copy paste that and to be honest you just need to add probably uh okay so supply mechanism has to be added um in a drive to contract in a drive contract using mint um for generate mechanism c okay so that will probably just take me where i came from but i like that so i can implement a supply mechanism using mint and i think i think i'll just use that i think i'll just use that copy so i think i'll just use that basically and say cool so 0.2 but whatever it is so um string private symbol okay it's like a constructor the default value of uh 18 to select different value all are immutable they can only be set once during construction okay and how do i call the constructor what i call the constructor when i deployed or returns the name of the content of the symbol decimals total supply total supply uh where is mint in here though function mint so it creates amount tokens and assigns them to an account increasing the total okay so that increases the total supply that's okay that's interesting so it means the transfer with from set to the zero address okay nobody is your address ah so that's okay the zero account the zero address is my address i guess um and so it increases total supply okay ah okay so this means that i am then you know i can then just add more supply um okay but it actually does have these so what is it that it's uh missing in here ah what am i doing so there is a am i stupid that's the link probably how the mechanisms okay [Music] okay i probably want to read about these calmly and and then do that uh yeah but i think i'll probably go with something like that so it seems like i thought i thought these was already included in here i mean you have the mean function that adds total supply burn basically burn supply approve so what is this doing this is uh okay the thing with the spending whatever i have to go through this okay but that's a good probably good basis um what is this is this just that um connect the wallet okay so there's parsley resources not found oh so i actually have to import that probably i just have to import yeah i have to import this i guess okay so i'll get that sorted out next time but yeah let's do that essentially and i guess i guess all i'm missing here is to um to basically uh do i need to do any yeah i need to keep track right i want to keep track of uh sort of the uploaded videos right i need to keep track of that um and then um because essentially that's basically the yeah that's basically the you know i'm gonna i'm gonna be giving i'm gonna be giving yeah to be honest i don't know do i need to keep track of it in this contract because maybe that's just the meeting it's it's it's basically the supply it's it's basically the the the supply mechanism do i want do i um do i want to kind of have a supply that i just kind of give out or do i just want to say look every time someone basically uploads a video i'm gonna mint so i'm gonna uh i mean and i can keep it to videos i guess but i i because i don't know is there a way to define the value uh of of a specific thing it can be a long it can be short it can be like you know the amount of i guess it's just raw data right it doesn't matter what the size is because it does not determine the value of it necessarily um so maybe there's just a standard way to do that and then um you know i would just kind of do the upload but i should probably keep the list somewhere um because i then essentially i won i want another set of users to then basically say cool you know go in and check is that really uh is that really is that really or isn't it like is that really a valid video it's just like a cat video right um and uh how to make sure though that that's not you know that's that the approval mechanism it's going to be it's going to be tricky to keep it decentralized um but yeah if that's the case then then you get the coins and then uh yeah you go ahead what can you do these coins i don't know i'll have to think about these but i think that's an interesting idea basically yeah i don't know let's see and then of course more people to do that right um i kind of collect that dot um the database of uh of of raw data that uh could can potentially be used to train some sort of artificial intelligence in the future i hope so i guess cool then see you next time |
today or we'll do the quantum measurement part but um i'm aware that i'm flying over these just like at the speed of line pun intended no but uh like just flying flying over the staff without really you know sticking to a specific point in digging deep but that's like on purpose right that's kind of at least from my perspective the way that i um that i kind of like to to approach topics like these let me just because i can talk and do things at the same time i just wanted to tweet out that people can join if they want so twitch dot tv slash uncertainties there you go hopefully that's the right link yeah that's like so link perfect so yeah basically basically that's what i will do so uh let me open the chat awesome um so last time with the um and you know and i've and i've i haven't taken a look at all this stuff in here right so i will come back to these it's just the fourth session and and i'm just really you know the goal is to learn the quantum staff and and kind of learn quantum mechanics right and and just to kind of re refresh the goal of this whole project right is not to understand the project in its entirety it's it's you know my primary goal is to kind of learn more about quantum mechanics and i thought a great way to do this is by by kind of exploring this project because i know that their stuff they're doing is quite cutting edge and actually it it's what i like about this is that it's an exercise of um you know a big group of scientists are saying we've got quantum mechanics and we've got this model and kind of try to map it right so this mapping exercise is a great exercise to kind of i believe it's a great exercise for me it was a great chance for me to try to understand sort of what are the parts of quantum mechanics that are relevant and kind of like you know get a bit of a non-traditional view on on all this kind of stuff so and then you know just build up my knowledge base from there and i yeah i i thought i'm not taking notes or anything i'm just really this is just really the way that i that i do stuff so i know that it doesn't work for many of the most of the people but at least i record the sessions and i have the videos so if you ever want to come back to one of these learning sessions and then they are there in the youtube channel as well um uncertain systems and i'll organize that a little bit better because there's a lot of mixed content from from from the past as well but anyway today i wanted to spend some time on the quantum measurement stuff so we did the basics and we did quantum formalism which here in quantum formalism my last time i already learned a lot of stuff that i want to come back to which is basically um and then afterwards kind of after thinking about this be more depth i did realize that there is more to these that i have to definitely dig into right because i come from just pure strictly speaking quantum computing side of things where you have a state and then you have a quantum circuit which basically dictates how stuff evolves and i think the way that i was reading through some of these sections here would be misleading because in in essence you have this this is just a toy example right of a system that evolves and then each node is a is is a configuration of this is basically your state your system state and so this is the multi-way graph this means what this graph is telling you is that you know from this node this one rule you can apply which is the rule that turns it turns this state into the a b state from this point in time there's two rules you can apply so you apply one rule you'll turn the b into an a if you apply another rule you'll turn the a into an a b and then so you kind of have two possible alternatives and so i was having a hard time going through this because i think i was too much hardwired thinking about the actual quantums not like a quantum circuit which is not later on i kind of realized that's not what that's not the point the point here is if you think about this is a quantum mechanical system there's the concept of you know the the the the evolution the time evolution of the system and so that is described described by the hamiltonian and i think that what this is trying to say is that you know and also there's a there's a point in time a point here somewhere where they talk about like um the evolution amplitude or some some sort of like the probabilities of evolving from one state to another and i think that's what the hamiltonian is really encoding in here right it's instead of just being a circuit is it's just telling you how the system could evolve and so that's what the multi-way graph here is telling you so that's why there's a lot of stuff that i wanted to you know um that i definitely have to come back to and and deep dive into afterwards um there's also the whole thing with lagrangian stuff um and that i definitely want to take a look at so we'll we'll do again we'll we'll basically do a that that's the way that i do this right so just go end-to-end and then we'll just kind of go back and try to put pieces together and then deep dive into some of these aspects um but so what i took away from this section last time was basically this duality of you know you've got states and then you've got like the the way they map the formalism into the multi-weight graph and and we haven't talked about all the other hypergraph stuff because that is just everything that comes before these like and i think that it's more related to uh to you know general relativity and whatnot and and the quantum stuff is really happening at the multi-way graph level i guess where you can take a look at the possible evolution paths of your system and um and then there's another so that was the first half in the second half is like exactly talking about like how this evolution how can you reason about this evolution somehow and how how does this then somewhat mapped maps to to um to the standard quantum formalism but i literally did not take out anything else from that whereas like oh yeah that makes sense like it was totally you know at least i grasped i think i grasped the overall structure of the chapter and that's what matters um so let's go through quantum mechanical let's see how they map that and what's because that's that's that's where things get interesting as well right because it's um it's also where you have um some of the biggest questions around quantum like or open challenges around quantum mechanics in general right like the whole the whole measurement problem so let's get to it above we gave a brief summary of how quantum measurement can work in the context of our models uh did they here we give some more detail okay i i don't know i haven't i don't think i have read anything before about quantum measurement but could be in the dance in a sense the key to quantum measurement is reconciling our notion that the definite things happen in the universe with the formalism of quantum mechanics or the branching structure of a multivariate system but if definite things are going to happen what might they be here will be again considered the example of a string supposition system through the core of what we say also applies to the full hypergraph case consider the rule a aab ba we could imagine a simple classical procedure for evolving according to this rule in which we just do all updates we can say based on a left-to-right scan at each step yeah that was that was what was marked in the previous chapters just like the red notes right um the gender was it like something like the generational evolution or something like that um so you apply all the possible rules you can from right or left from left to right and so you get this path but if how we know that there are many other possibilities that can be represented by the multi-way system right so that that tells you that's just a path it's not even a path but that's just a subset of the nodes they don't even have to be consecutive or like happen at the same like consecutive levels um most of the states that appear in the multi-way system are however unfinished in the sense that there are additional that there are additional independent updates that can consistently be done on them for example with the rule a aba there are four separate updates that can be applied to aaa right but none of this depends on depending on the others so they can in effect all be done together giving the result put together would put another way all of these updates involve space like separated parts space like separated parts of this string so that they are all casually independent and cannot consistently be carried out at the same time discussed in 5.1 doing all this across the state together can be thought of as evolving in the generational steps that's what they said as one so you have generational states in in multi-way cases there might be a single sequence in some multi-way cases there may be a single con sequence of generational states so these are the ones marked in red and in other cases there can be several branches of generational states okay really why because if we consider these two rules a a b and b a so a maybe that's just another example but like i'd say yeah okay you can apply all the rules but there are different ways in which you can apply all the rules at the same time the presence of multiple branches is a consequence of having a mixture of space like in branch like separate events that can be applied to single state for example with the rule a b and a a to a b and a to b b um the first and second updates here are space like separated but the first and third are branch like separated okay let's try understand that the first and the second updates here are space like separated but the first and the third are branch-like separated i don't know what they mean by this so a b a b so these are oh the first these are the four possible updates you can have so uh yeah okay i get it so this means these two these two updates here the first two can be applied simultaneously but then these other two cannot a view of because they basically they they affect the same element in the string right um the view of quantum measurement is that that it is an attempt to describe multi-way systems in generational states sometimes there may be a new classical path sometimes there will be several outcomes for measurements of states okay so so the measurement has got to do with the generational states that is not what i expected maybe because i still have an incomplete picture of of the whole evolution concept in here i'm just probably too i'm just probably overthinking it too much but now let's let us consider the actual process of doing an experiment on a multi-way system okay but now let us consider a process right now or a quantum system our basic goal is as much as possible to describe the motorway system in terms of a limited number of generational states without having to track all different branches in the motorway system at some point in the evolution of a string substitution system we might see a large number of different strings but we can view them all as part of a single generational state if they in effect yield only space like separate events in other words this string should be assembled without branch like ambiguity they can be thought of as forming because this is international stages in a little traditional state right that's what they're saying here i guess it's like at any point in time you can see the state aa or abb but it doesn't matter because they all converge to this one if we think about this generational concept right um in the timeframes and quantum mechanics we can think of the state in the multi-wave system as being quantum states that's what they said in the previous chapter the construct we formed by assembling these states can be thought of as a superposition of states okay so [Music] casual invariants causal invariants like such a message basic word causal no causal invariance that implies that through the evolution of the multi-wave system any such superposition will then actually become a single quantum state the construct will form by assembling these states can be thought of as a superposition of the states there's something a bit off with these like it's still maybe i kind of coming back to the whole circuit thing and it's just i think that might be i might be just fooling myself but because when i'm building a quantum circuit i i am in a way designing the way the evolution is going to look like right so if i say apply a hard mark gate like i i know that i want to be getting into a superposition of a zero and a one it's not that i there's nothing missing in the air i don't understand i i can't i i just don't find a way to map the generational evolution concept with the quantum circuit example for example just take a simple one qubit circuit right like you have one cubit and then you apply a harmonic gate and so that takes you into a zero plus one state but i'm not 100 sure if that means because you know because i could think of you know i could think of like like the the hallmark gate as a rule would be put the zero into the zero plus one state put the one into the one into the zero minus one state um but there's just one update it can do so so that is already a generational state so but it's a it's a state that it's super position so that's why i'm not i'm i'm maybe i'm just making the wrong analogy that's or i'm just looking at the wrong example with a quantum circuit but that's still a bit hard so the constructive form of assembling the states can be thought of as a proposition of the state causal invariance that implies that through the evolution of the multi-wave system any superposition will then actually become a single quantum state in some sense the observer did nothing they just notionally identified the collection of states it was the actual evolution of the system that produced a specific combined state in describing a quantum system or a multi-wave system one must in effect define coordinates and in particular one must specify what foliation one is going to use to represent the progress of time and this freedom of freedom to pick a quantum observation frame is critical in being able to maintain a view in which one imagines defining things to happen in the system with a foliation like the following at any given time there is a mixture of different states okay so now these are mixtures are they like is the word mixture here used as in like like an actual um mixture of states are not like a superposition right so with the foliation like the following at any given time there is a mixture of different states and no attempt has been made to find a way to summarize what the system is doing because there's a proposition and a mixture are essentially different right consider i have a foliation like the following so each in this case each and this picture generally generational states have been highlighted and affiliation has been selected that essentially freezes time around a particular generational state in effect the observer is choosing a quantum observation frame in which there is a definite classical outcome for the behavior of the system freezing time around a particular state is something an observer can choose to do in their description of the system but the crucial point is that the actual dynamics of the evolution of the multivoid system cause the choice to have implications in particular in the case shown the original multiple system in which time is frozen progressively expands the choice the observer has made to freeze a particular state is causing more and more states to have to be considered as similarly frozen in the physics of quantum measurement one is just the idea of that for quantum measurement to be considered to have a definite result it must involve more and more quantum degrees of freedom this i didn't know quantum measurement to be considered to have a definite result it must involve one more quantum degrees of freedom what we see here is effectively manifestation it's found at this i i don't know what this means in facing time and sunlight inflation the picture of all we are effectively doing is creating a coordinate singularity and and defining our quantum resolution frame and there is an analogy to this journal to to do a freeze time of my friend once again first time in a relatively sticky reference frame for example as an object approaches the event horizon of a black hole its time is described by a typical coordinate system set up by an observer far from the black hole will become frozen and just like in our quantum case we'll consider this database stay fixed whatever i don't know but there's there is a complicated issue here to what extent is the singularity and the freezing of time a feature of our description and dual extent is something that really happens this depends in a sense in the relationship one has to to the system in traditional thinking about quantum measurement one is most interested in the impressions of observers who are in effect embedded in the system and first and for them the coordinate system they chosen in effect defines a reality but one can also imagine being somehow outside the system for example one might try to set up a quantum experiment or a quantum computer in which the construction of the system somehow makes it natural to maintain a frozen time foliation [Music] the picture below shows a toy example in which the motorway system by its very construction has a terminal state for which time does not advance but now the question arises of what can be achieved in the multi-wave system corresponding to the actual physical universe and where can we expect that and here we can expect that one will not be able to set up truly isolated states and that instead there will be continual inevitable entanglement one might have imagined could be maintained as a separate state will always become entangled with other states the picture below shows a slightly more realistic motorway system with an attempt to construct a foliation that freezes time god i need to understand that those foliation stuff would be better and we see there is in a sense the structured multiple graph limits the extent to which we can freeze time in effect multi-way system forces the coherence or entanglement just by its very structure we should note that it's not the case that there is just a single possible sequence directional stage because point this is a possible classical path here an example where there are four generational states that occur at a particular generational stance and and and that might be then the superposition really and now we can for example construct affiliation that at least for a while for this time for all of these generational states it is worth pointing out that if we try to freeze time for something that is not a proper generational state there will be an immediate issue a proper generational state contains a result of all space like separate events at a particular point in the evolution of a system i feel i feel i have to maybe step a little bit back and and and and really try to map like really try to understand that concept to be better of the generational states with like their like a proper example um if we try to freeze time for a state that did not include all space like separate events there would quickly be a mismatch with the progress of time for the excluded events or in effect the singularity of coin observation frame would spill over into singularity in the casual and the causal graph leading to a singularity in space-time in other words the fact that the states that appear in quantum measurement are generational states is not just a convenience but a necessity or put another way in doing quantum measurement we are effectively setting up a singularity in parental space and only if the states we measure are in effect complete in space time will the singularity be kept only in branches space otherwise it will also become a singularity in physical space-time or big words this is too much in general we'll talk about coin measurement we're talking about how an observer manages to construct a description of a system that in effect allows the observer to make a conclusion conclusion about what has happened in the system and what we have seen is the appropriate time freezing foliations allow us to do this and while there may be some restrictions uh principle possible to construct socializing motor system but in practice as the pictures above begin to suggest after a while the foliations have to we have to construct and get increasingly complicated effect what we're having to do in constrain in constructing deflation is to reverse engineer the actual evolution of the motorway system so that our elaborate description we're still managing to maintain time as frozen for a particular state to out-compute the system itself and so we will be asking the observer to do a more elaborate competition to maintain the description they are using and as soon as the computation required exceeds the capability of the observer the observer will no longer be able to maintain description it is worthwhile to compare the situation with what happens in in thermodynamic processes and in particular with the parent entropy increase in a reversible system it is always in principle possible to recognize say that the initial conditions for the system were simple and low entropy but it practiced the actual configurations of the system usually become complicated enough that this is increasingly difficult to do in traditional statistical mechanics one talks of coarse grain measurements as a way to characterize what an observer can actually analyze about the system in computational terms we talk about the computational capabilities of the observer and how computational irreducibility in the evolution of the system will eventually overwhelm the computational capabilities of the observer okay i i'm this is getting heavier so what do we what do we see here so what what is let's try to let's try to unwrap some of these it's again the generational states i'm missing a good example of these with a quantum circuit kind of like i have maybe no i was gonna say maybe it's got to do with noise and in general you know the mixtures that you know the evolution of the system is somewhat dictated by like a noisy thing but it's also not true because these nodes are basic states not just states they are basic states so so what this is trying to say is that oh god um what is it even trying to say that that the measurement in a measurement you can only see generational states i think that is at least something that i can say out of these and then and then making a measurement is picking a specific way of foliating the multi weight graph that's just your reference frame right like so maybe because i'm i'm inclined i'm a bit inclined to take a look at just the operators part because i because maybe that is going to help me understand or build an analogy with the circuit stuff so there are states and operators now models updating events are yeah okay updating events are corresponds to operators in the standard evolution of a multi-wave system all applicable operators are in effect automatically applied to every state oh okay so that is actually i think that is gonna oh that's short enough i think that's gonna help me build that bridge because that's that's what i was looking for right now so so there are states and they're operators right let's see what is the reference here oh come on um in our models the dating events are will correspond to operators so the operators are the rules right and and that's kind of yeah that's that's what i was saying right you can you can ride you can write rules that like specify the harmonar gate saying zero to the plus state one to the minus state in the standard evolution of the multi-way system all applicable operators are in effect automatically applied to every state to generate the actual evolution of the system that's what i don't understand but to understand the correspondence with standard quantum formalism we can imagine just applying particular operators by doing only particular updating events consider the string substitution system a b a b a so a b to a b a and b a to b a b and the system we are affected to operators so one and o two correspond to these two possible updating rules we can think about building up an operator algebra by considering the relations between different sequences of applications of these operators in particular we can study the commutator in terms of the underlying rules it's going to response to i don't know what a commutator is um uh the first header is also playing golden initial state are different we can then say that it stays related from the branch pair but then at the second step the branch panel resolves and the branches merge the same state and in fact we can represent this by saying that i wanted to commute um okay so that that whether they compute or not um that all branches can result in a single classical state just like in standard quantum formalism the computing operator is associated with seemingly classical behavior but there's a key point here even if even if causal invariance applies to branch pairs and will eventually resolve they might take time to do so and this is and this is delayed resolution that is the core of what leads to what we normally think of as quantum effects um once a branch pair is resolved there are no longer multiple branches and a single state has merged but before the branch pair is resolved there are multiple states and therefore what one might think of as quantum in determinacy in this case where branch pair is not a resolve the corresponding commutator will be non-zero in a sense the value of the commutator measures the branch like distance between the states so branch pair is not a resolve the corresponding commutator will be non-zero i don't think that's going to help anything you know so this is a brush like separations and there are other pictures in tongue one two stays on that if they are part of the same unresolved branch pair and does have a common ancestor the multi-way graph gives the full map of all entanglements but in any particular time corresponding to particular slice of or of foliation defined by a quantum observation frame and the branchial graph gives a snapshot that captures distant instantaneous comparison of the negative ones okay um it doesn't help me but i mean at least it confirms that the updating rules are the operators right or you can think of these as operators um but then this would mean that a cert that uh that a circuit is is picking a specific updating order you know so see what i don't understand is what what will be the analogy of that multi-way system like what is the multi-way system trying to because if i don't understand that then i i also don't know how to make sense of this whole generational stuff so for this i think i have to go back to the quantum formulas and stuff um because for a circuit right i have not only a specific set of rules right like the harmon the harmonic operator the control knob operator but i have a specific i have a specific order or i have a specific path it really is a specific path like a quantum circuit would be of a specific path in the multi-wave system because i know at every point in time what is the specific set of operators that i'm picking and where are they applied within the state right like you know in this case you have or somewhere below here the other stay you had a system with an example with like rules that both apply to so that'll be that that will be where is that there was an example here with the word exactly so you have these these two right so you could think of i could think of these as like let's say the harmon gate you actually have two rules no i mean it doesn't matter like whatever right like actually each operator is really each operator is really a set of rules right or would be i mean i would build it like that right so i'd say and and they're based on your computational basis although that feels somewhat wrong because i i i have the feeling that the computational basis got to do with the at the end of the defoliation ah that's maybe not true either i think the computational basis is at the end of the day that choice is the choice of language you're picking the choice of of notation you're picking if you're to pick zeros and ones then then so be it right um and so the rules are going to be defined like turn a zero into um the plus state or turn a one into the minus state but then there can be another say the x game right it's also going to have it's also going to share things like these so you're going to have a bunch of all different possible yeah evolutions but then what's the point so i think just a quantum circuit so i don't i don't really get maybe and as i said maybe it's because i'm missing that dynamic component of the quantum mechanics right let's say how is it that we describe a system evolving because for me i just you know coming from the circuit world just have that in my head i have a circuit which this model is just a specific path in here because i know exactly what rules i want to apply and where i want to apply them you know even if i could apply a rule in multiple spots in here rerun rule i might i'll choose to apply this one right um but i feel i think i really i think before before i go on i need to deep dive into this a bit more yeah before i even go to wave particle duality and stuff like that what is that even i mean that's also fairly short and quantum mechanics it's funny because that that is already kind of i don't know what when i see such short chapters i'm scared because it's probably just event horizons and singularities and space time and quantum mechanics cosmology expansion and singularities reversibility reversibility motion and special activity space and time structure space so i think i have to come back to this speed and the first part i kind of understand so we that that i'm i'm like i keep getting lost in the same space so each i was wrong so when i said that it's not a specific path but it's a specific sub graph because i might choose to apply a hard mark gate which might split into two is to me to two notes because i have a zero right let's try to do that again with the paint if i can manage to open the application without breaking the computer and the streaming that'll be awesome so so i start off let's say i start off with the state 0 0 right and and my set of operations are the hana mart and the x gate right but i have a specific circuit in mind which is you know apply a harmonic gate to this qubit and then apply an x gate to this qubit like if i would consider all the possible ways this can be done right like which is what these graphs seem to be doing it's like well okay one possibility is apply the hadamard rule to the first qubit but that actually leads into you know into these because this is these are really two rules but it could also just be no way yeah so but these two would merge right but i could also just apply the x-gate which in this case i have two possibilities no i also have the possibility of applying it everywhere and i also have the possibility of applying the hadamard gate to both oh god and the automar gate to so that would just grow right so i mean some of these merch and would not but then if i have a specific circuit in mind then i know i'm picking a specific subgraph in terms of the evolution i'm saying i want to go down this path and then next gate here so kind of like these are the two these are the two should change color but these are the the the two notes that i want to have everything else i don't care i've picked that specific i've picked that specific path that has led me to these having two separate notes i think this is i think i think this is probably what what's happening here probably but then why build that feature right like why why do you need to build a multi-way system for these um that's that's what i'm [Music] and that's kind of maybe the step that i'm missing towards like the quantum mechanical part of it right like to to kind of like um um let's go back to these so let's let's i think we have to unpack some of these things so here we talk about the states being quantum base the nodes being quantum basis states that makes currently sense based on what i on this feature i don't get so much the entanglement stuff because i wouldn't say these things are entangled even though they probably share some ancestry i mean they are correlated but it's not an entanglement correlation so they are like the thing is they are coral like the two things are correlated in a way right so each pair of states generated by a branching and its graph are considered to be entangled i don't care about the geometry geometry right now here we have the count of the paths let's say that we want to track what happens to some part of this branch like hypersurface each state undergoes updating events that are represented by edges in the multi-way craft and in general the path followed in the multi-weight graph can be thought of as geodesics in multi-way space so here they get into better worker down respond to energy the space time caught causal graph however it's just a projection of the full multiplied causal graph now note that every node in the multi-wave causal graph represents some event in the multiple graph but events are well produced branching and turns let's just zoom out turning power parallel energy in energy is identified with a hamiltonian h so what this says is that in our models we can expect transition amplitudes to have the basic form in agreement with the result quantum mechanics okay so let's start with transition amplitudes let's just dig into this so so so is your quantum mechanics transition amplitudes what are transition amplitudes path integrals let's do this extended sassy quantum mechanics transition amplitudes okay so this seems the right no uh so in quantum mechanics the physical system corresponds to a hilbert space states correspond to not in a one-to-one way to points in here with space and the physical postulate is that the transition amplitude a complex number from a state corresponding to v into state corresponding to u is given by where the physical meaning of the transition amplitude is that if you take the squared absolute value of its complex number you get the actual probability of the system going from the state corresponding to v okay so there's these dynamics are defined but i don't understand them so it's it's a postulate right but this is a probability amplitude that's that's that's just the measurement stuff so in quantum mechanics the probability amplitude is a complex number used in describing the behavior of systems the modulus squared of this is a probability density but probability amplitudes provide a relationship between the wave function of a system and the results of observations of the system but i don't want to that that's that's the that's what i don't want to have i don't want to have the the observation stuff i want to have the evolution stuff transition amplitude it keeps it keeps keeps taking me to the path integrals which i don't know what is it the traditional amplitude in the most commonly used operator approach the transition amplitude is expressed as the vacuum expectation value of the product of particle creation and annihilation operators these operators obey certain commutation relations so what is this oh quantum field theory so this is just a portion of these i'm not gonna no how can i even so what is what is the path integrals quantum mechanics what is the path integral the python formulation is a description of quantum mechanics that generalizes the action principle of classical mechanics it replaces the classical notion of a single unique classical trajectory for a system with a sum or functional integral over an infinity of quantum mechanically possible trajectories to compute a quantum amplitude this formulation has proven crucial to the subsequent development of theoretical physics because manifest lawrence covariance is easy to achieve than in operative formalism okay so this is a separate formalism than the operator formalism and i guess the operator formalism is just the staff with operators should probably dive into one of these god i think i've opened uh an unconventional text from our mechanics kind of feel theory starting mechanics i mean just symmetry principle without reference to classical mechanics and mathematical foundation the bridge conservation paper so i i i kind of get it that it's like so i guess what this might be trying to say is that in in in in the formalism itself there's just an obstruction that you to what you usually think classically which is like you have an object and it just has a trajectory and in this case have a quantum system which doesn't have a trajectory but it just has a collection of all the possible trajectories they can go through um and what is this but is this what the hamiltonian is defining or not that's that's my that's the point hamiltonian so what about connecting hamiltonian hamiltonian hamiltonian pathway does it disappear as a suggestion oh that's just a graph okay now hamiltonian hemapungen mechanics no hamilton mechanics hamiltonian mechanics there is a video and there's some lectures i was thinking about introducing or watching some videos as well uh also the iphone again with microsoft information of classical mechanics historically contributed to the formulation of statistical mechanics and quantum mechanics and melatonin mechanics was first formulated by william rowe and hamilton in 1833 starting from lagrangian mechanics the previous revolution of classical mechanics introduced by joseph luis de grange like lagrangian mechanics hamiltonian mechanics is equivalent to newton's laws of motion in the framework of classical mechanics so in hamiltonian mechanics the classical physical system is described by a set of canonical coordinates where each component of the coordinate is indexed to the frame of reference of the system the qi are called generalized coordinates and are chosen so as to eliminate the constraints or to take advantage of the symmetries of the problem and the time evolution of the system is uniquely defined by hamilton's equations where this is a hamiltonian which often corresponds to total energy of the system for a closed system is the sum of the kinetic and potential energy of the system in the training mechanics the time evolution is obtained by computing the total force being exerted on each particle of the system and from newton's second law the time evolution of both position and velocity are computed in contrast hamiltonian mechanics time evolution is obtained by computing the hamiltonian of the system into generalized coordinates and inserting it into hamiltonian equations this approach is equivalent to the one used in lagrangian mechanics the hamiltonian is the legendary transform of the lagrangian when holding q and t seeks blah blah blah no idea what i'm talking about the more degrees of freedom the system has the more complicated its time evolution is and in most cases it becomes chaotic cockle it in the hamilton from a lagrangian okay so this does actually take me down the classical path a little bit and then there's the crunch of mechanics stationary action principle that might be introduction it's funny because that's just from newtonian telegrams and mechanics that is super dense so what about solar cryo mechanics is there some sort of tutorial oh god lesson one basically branching mechanics uh that is quite wikipedia as well maybe i should just do this off of the wikipedia stuff which is pretty hardcore though but okay so so looking at the hamiltonian mechanics and then then there's this um but then isn't isn't it then schrodinger's equation right what defines the equation isn't isn't it what is this what actually defines the movement this is the hamiltonian operator the wave function is a linear partial differential equation that governs the wave function of a quantum mechanical system the series equation gives the evolution over time of a wave function the quantum mechanical characterization of an isolated physical system so that is that is what um preliminaries okay particle in a box example so here there's some examples with the harmonic oscillator quantum harmonic oscillator oscillator okay so here there's some examples of so this is this is maybe what then leads to these to the transition amplitudes right or like what if i google these two things like do they anyhow schrodinger equation [Music] transition amplitudes oh is the transition after this the path integral formulation that's then a different formulation so this relation between schrodinger's equation and the path integral formulation of quantum mechanics um this article relates to shrinkage equation with the pythagorean symbol non-releasing one-dimensional single particle hamiltonian composed of kinetic and potential energy so background schrodinger's equation the bracket notation is like these camber and stat that's the hamiltonian operator whatever i have no idea what i'm reading um the path integral formulation the bottom information states that the transition amplitude is simply the integral of the quantity over all possible paths from the initial state to the final state where s is the cl is the classical action so the reformulation of this transition amplitude originally due to dirac and conceptualized by phaman forms the basis of the path into the formulation part of the product formula says that for non-self-regeneration we have the transition amplitude can then be written as these although the kinetic energy and potential energy will produce do not commute the charter product formula cited both states it's a classical lagrangian god so they are related and then the the transition amplitudes is this is really it's got to do with that and so this is something that you can calculate as well from schroedinger's equation based on what i currently seem to have understand so these would be yeah so these would basically be a system that just evolves kind of i think i have to re-read these and then i'll probably have to do the lagrangian and the hamiltonian mechanics stuff i i feel like this this is going to be inevitable to actually understand why i mean we could we can do this just not starting from the lagrangian mechanic stuff it's just i don't want to dive too deep into these but that's quite complicated my grandchild mechanics i don't want to watch any of the quick videos here like the lagrangian method brilliant.org yeah that's also not intent mechanics doesn't it doesn't look better either anyway i think we'll just go from so an introduction to location mechanics maybe something like that it could be worth exploring oh that's a boat man i'm gonna buy a book yeah i think we'll start from lagrangian mechanics i want to try to get a sense of how to get how do you go from lagrange mechanics to hamiltonian mechanics to to then the schrodinger's equation sort of and then then this whole thing with a path integral formalism um and um but again it's so it's it's in in a way the hamiltonian tells you sort of how how the system is evolving and the system can evolve in multiple different ways right so you've got it can evolve you know in these and that right and that's kind of your yeah well in essence that's probably what the the wave function is telling you right it's like i'm either in this state or this like no not either in this state or in this state i'm saying i'm in a linear combination of states um are those equivalent then to these paths i i don't know too many open questions too much open questions that's complicated i i have to sort of tunnel a bit more because otherwise i'm going to get lost but basically i think at least i understood a little bit better the the function of these and then how the mapping between these and the quantum system and then what the what is then the analogy with a quantum circuit in there which is just pick a type graph of these that you specifically pick because you're telling the system how to evolve essentially you're programming the system you're telling the system what actions to to take what to do right how to evolve the system essentially but i'm not it's not entirely clear how the evolution maps to the rules right i think that's what they're trying to explain here is how these rules are mapping so probably i should read that again because they talk about the angles of this turnings and i think that is basically the rule applications right it's the action and oh look at this so this is the path in the performation of quantum mechanics i think i have to re-read that second part a little bit more next time and then maybe i should maybe i should maybe i should just to consider a path in the multi-way system going through some multi-way space to know how much turning to expect in the path we need an effect to integrate the lagrangian density along the path this will give us some form of but it's exactly what the started patting the formulation of quantum mechanics maybe i should just maybe i should just just go there just path integral formulation and see how that relates to schrodinger's equation yeah yeah yeah yeah probably also there's some some oh so there's some nice link that i can just follow here with references and stuff so we can start here awesome oh god that is actually that is actually a big kind of worms already for the next couple of years so far here this is heavy this is heavy okay but i better start to get a sense of these so there's just a formalism right in terms of how to study the the this evolution and i guess there are multiple formalisms the same that you've got multiform multiple formalisms in classical mechanics anyway have a good night |
If you can catch up to where the Wolfram Physics Project is right now who knows... many discovers await us all )))) |
Thanks for show. I watched some YouTube videos by Stephen Wolfram and it helped me with your video today. There was a foot note in this video you may want to study, [110] called Cellular Automata which he explains. Just a beginner but learning. Thanks. |
so this is just the I just have no words I mean I've been so I've been playing with IBM's circa composer until now for everything that I've done it's been crazy helpful to some extent it seems like it's it honestly to make an analogy it's like that's the class the IBM Q's like the IBM accuser composes like your classic calculator and that's like the scientific calculation and this is that's how I feel like when I see all the stuff in here no but I mean jokes aside it feels like the IBM Q experience just to give you just just to give you an sort of a comparison right let me just go go in there because it's a good tool it's a really great tool and it's really helped me to get where I am right now but this is just something from another world it feels like um I've basically gone through the video already um I haven't checked the how to use I guess it's an extensive guide Wow okay that's cool but we'll definitely go through that that's just amazing I'm just saying that because look at the insides so one of the things that I always kind of complained internally is sort of to myself with the IBM experience composer is that it's pretty cool because it has really it's a really intuitive interface what I really like about this is the fact that you can see as you build you can see different visualizations and what's happening right so you're at a Harmar here your Hardware here and kind of you see your states and everything and then you can also see the density matrix which I already explained in some videos that it is really useful because you can with the density matrix you can kind of see the state of your subsystems but it's held cumbersome to use here because really you can't do anything aside from we can see the values actually if you go here in each of those squares but you can't really calculate easily it's it's the density matrix of your whole system you can define registers you can define like systems and and and so it kind of it's not useful a hundred percent and then you can you know another useful thing here is you can ask you as you build you can definitely you can touch the console code here and then you can quickly modify your circuit which I'll at least it's a way that I used to work work around some limitations of this editor when I was like I don't know I'm just putting random stuff in here and it's like oh I what is the okay so what is the current if I now would like to know because now I know here in the visualization what's my current state vector and by the way I cannot fully I see I see what the I see I cannot you know you see the color coding which is of the face but you don't really see the value right and and you don't see in in a in proportion to like a 360 degrees rotation you didn't see how much this is always you know you kind of know that that's zero degrees you know that that's like a hundred eighty degrees or like that's a half away so to say right but you don't really and then if you say now I want to know what's my state here I first certainly thought you can put a brewery year and that helps you do that but I don't think so I don't think so so you kind of got to do this and then the way that I used to work around that is because you can quickly do that so you can just you know quickly delete that and then take a look at this and then go back and then copy paste again but basically it's kind of cumbersome to analyze a big circuit because you you you cannot see the steps in between while but it's kind of intuitive an author for learning really add the size if you don't go up more than like two or three cubits and then you're kind of learning the basics even like it helped me understand the quantum Fourier transform it helped me understand a lot of the stuff but here with these stuff so with queer the thing is and I've been playing a little bit here already but it's like I think the biggest advantage and I probably do a full review or full comparison between these two land experience and there's another tool that I've been playing around which is QPS quantum programming studio which is also a good one because it has a it also has a cool community section where you or discover you can actually see projects of other people so you can see people that like I've known to beat quantum other rather than you click here and actually you can see the circuit it can call you can fork it you can do a bunch of stuff you can also I if I run well exactly you can also just it gets automatically translated into like all those different languages which is honestly pretty cool so for example you can go to Kiske and then yeah so you get a kiss kid code for that it's pretty cool so and I'll probably do a full comparison you know sort of I'll try to do and then try to see the pros and cons of each of the tools but this feels like once you once you get um sort of you know the basic intuition level that is definitely the tool to go for at least that's what I feel I'll definitely explore that because and there's just a lot of stuff in here what I like a lot is this yellow things in here for example that's pretty cool because it's a sort of an animated it's a dynamic gate I don't know if those are really something you can implement in a computer but it's maybe just for you to understand the evolution of the circuit it's kind of cool because for example it's like a like a spinning so you let's say let's put an X guide here for example and let's maybe I like to get rid of those kind of tutorial things every time that I start but maybe there's a way to that and now I do these and and you kind of you see it's animated so you kind of see first of all you see block spheres breach of the cubed that's really cool you see sort of the amplitudes and the faces with a circle notation which is the same notation that is used by the O'Reilly O'Reilly book which is really interesting but you see everything look you see the value you see the complex part you see the face the amount of detail is awesome like it allows you to do a lot of debugging also I guess because if I would if I would just kind of load a load a complex circuit I'm just gonna like randomly choose I'm just gonna randomly choose something like that I don't know or maybe the quanta Fourier transform let's go for this right so you can see this is animated but I guess I can read I can get rid of these for now and and then kind of you know you can make up whatever input in here so that was just a way to kind of see all the inputs but we can do is you can sample at any point or that's what's more you can get a density matrix that is partial that is just awesome so you you kind of can resize it and say look I just want to take a look at the density matrix from that subsystem which is something you cannot do the IBM Q experience because you just get the encounter that sorta matrix of everything it feels kind of it's beautiful it's very nice but it's that that's kind of way more detailed and and so that's you know it's just and then you can take a look at the amplitudes as well from your just a subsystem I I got a really I got a really work through that because I don't know how that works at all like what the fixed means here this is just there's so much in here to explore so one thing that I'll definitely do with these tools I'll go through each of the example circuits definitely I mean we've done some of those in the channel but I like CRO research for example pretty interesting I don't know I think I think there's just a lot to play with in here and well yeah you can just share it look at this is you can just share it I mean you cannot see that because I cut it in the Edit of the video but you it's everything encoded in the URL so you can share the URL and then it just directly renders the circuit for you you can also export it there's an offline copy that's cool okay what does this do if they open it nice and it still works that's just amazing gotta check on that works yes so what I mean is I think I think this is awesome like definitely will work we'll go through the examples and it will see see what can we do because there's a lot of stuff in the tool boxes I don't even know where to start I feel like I'm in a candy shop really like a turns the one thing that I see kind of that I've seen cuz I've been just taking a roughly look at it is there is a difference here I'm kind of I don't I don't know how to do parametrized rotations I know there's this perma trance rotations here but I think they are implemented in a way that they're just really specific to a particular use case because in IBM you can do like literally said rotations and then you can just specify the angle right Y whereas here it seems like the permit transportations our permit rised based on an input a which is then it's a divided by two to the power of n with M being the number of qubits in input a so basically what this means is that if you have if you have say for example nothing gate no what is this say that you say that this is input a right so it's zero words like one zero one and now you add a rotation so what this rotation is Squa so this input is then here C 2 ^ 3 because you have three cubits basically so what this is doing is saying it's got your permit rising it in relation to the maximum number you can represent with your cubits in this case it's an 8 so if I would put an 8 so this means I will just put xxx face is 0 well that's seven sorry if I would get rid of this I don't know how it works but I have the impression this is what it's this is what it's doing it's kind of rotating the face you see the faces is different so if I put here a block amplitudes not sample but just here oh sorry has to be disarmed are exactly what am i doing I don't know I don't know what I do it but anyways it seems like you gotta encode these within the circuit to parameterize those gates it's a bit more complicated but I think there is a reason for that which is kind of yeah that's definitely crazy the one disadvantage this is maybe it's going to be difficult to then kind of get that implemented into into circ or into this kid or something else so you can actually run on a quantum computer but in terms of designing an algorithm I think that tool is just awesome it's way more powerful way way more powerful than everything else that I've seen so far but again it's just a tool right but it's it's it's definitely something that I wanna I wanna I want to get I want to get my hands dirty on this because I think that's definitely gonna help me in my journey farther to kind of investigate more and kind of you know it's like when you're going to build something or trying to understand a circuit you can basically debug step-by-step kinda because it can add the displace in the middle so say if we're just now doing all the stuff like that that's really cool you can always like tensity we know Jen's that's just nice so tells you oh I don't know what the log is here but it's just and um a shout out to Craig for for having build on so this tool has been built by Craig kidney I hope I'm pressing your name correctly this is the this is the block it's just um this just this is I'm gonna spend a lot of time in here definitely well yeah you can make gates it's pretty cool because you can actually directly put a matrix which is awesome because in some papers what I was doing is like it was yeah you know we have like a gate and they have deployment to Hamiltonian whatever ends like yeah how am I gonna do that with the IBM cue experience I mean you can do that because into the IBM cue experience you can implement some general gate with his u3 basically by giving it certain parameters but it's a bit more of a BA black bugs and here's more he's easy he is just like just giving the matrix but you can also have a circuit and then and then this just stays here look it's just here it's nice and it's also you know you can just share that then yeah it's pretty nice that's definitely that's definitely and if I'm going to spend some time in here in the next in the next weeks and I'll probably do a as I kind of learn the tool which I'm going to record because I definitely want to go through the different different example circuits in here then it will probably learn to use the tool a lot but I see here quarter turns eight turn six interns those are like standard standard gate model stuff pretty cool nice |
Such a rare video explainer. This is such a "baby" field , you hardly get public media explainer. Thank you ! |
You create an Rz with a specified angle by using the make gate menu. The parameterized gates are for when the angle has to depend on other qubits (measured or not). For example, this is useful for directly encoding phase estimation with a semi quantum Fourier transform, where you introduce output qubits one at a time and rotate by an amount that depends on previous output qubit measurements. It also happens to be useful when testing constructions that are supposed to work for all angles, making it easy to vary the angle in a controlled way., This all makes total sense! The very best tool I've seen so far! Amazing work! |
it's loading I guess in theory world let me see yeah I'll put it on your phone just exactly okay cool cool I just wanna because they change the they change them dammit my computer is lagging so much they changed the interface here and now I'm trying to see whether I can how can I get the there there used to be a video where I could see sort of the quality of the stream like you know if there's something going the stream health and exactly but now they've died and I just see I just kind of see the chat I see the preview which I don't want to see because it just bothers me maybe I have to switch to this thing yeah because it's really come on oh I definitely have to improve my setup I mean come on I can't even get like the yeah I guess it's fine so just two minutes two minutes in two minutes in I mean I'll just wanted to share the link as well yeah yeah that that's that's for sure like but I don't know yeah I mean I don't know so far people haven't really complained a lot and I always complain that on my end just feel me because my browser yeah for some reason yeah yeah no no no I mean either I mean like I'm just just waiting a little bit just if some zones as well and people would say as well so it's fine I mean I think the top is some of the the busiest lives I've had like a I don't know it's like about 15 people at once or something so it's still but so are you ready or what like how are you feeling I'll just wait a little yeah I mean like we yeah I will try to I'll try to you know I can give you hints or you know kind of comment on the way that you're approaching it but I'd like to really you know I'd like for you to kind of go through the exercise as well it's gonna be a bit of a different puzzle how familiar are you with the QFT okay that's okay that's great cuz it's gonna be uh oh sorry sorry I I accidentally I accidentally opened the stream and and I got some weird sound in there okay cool so I think we yeah I think yeah I think I think you can I think you can open it now just let's go ahead I think you can open it now I'll be observing the stream quality yeah I can see your screen I can see your screen I can see you fine now okay so let me explain you before you dive into it don't touch stuff let me explain you okay so so this is so this is basically it's basically an implementation of the QFT that that it's that it's buggy right so um in the goal for the puzzle is to fix that so you I'm gonna give you a couple of clues though because it's not I don't want to have this too open-ended but the idea would be you see you have four black boxes right like the first one is called the swap black box then you have three rotation layers right the rotation layer 123 and the idea is that you have to find which one of those are not working well and then and then basically replace them for the right one so it's not I'm not expecting that you fix the fix the algorithm by by adding or removing stuff like it's also fine if you say hey I think the rotation layer 3 is broke I'm gonna replace it with the right you know with the right type of of rotation now of course you can just go ahead and replace all of them right but that's not the point I want you to to guess which ones are broken that's kind of the goal is guess which ones are broken you have all the tools you can use all the tools I just removed some of the displays and I remove the custom gates but so you can use all the tools so you can go ahead oh wait a second I think they're saying it's like I'm getting in the chant that they are not hearing you that's sax I thought that crap I thought that that's AI but III thought that the sound was coming through so I mean that sucks let me see if I can fix that okay give me you hmm it's somehow not not really working or I done because worst case what we can do is you know I could just comment on what you're trying to do but I would like to really for the audience to really see I'm trying I'm trying I'm trying but my but exploit is just frozen that I hate this program I really hate this program man yeah I just froze literally I hate these people how can I sorry few people for the for those who are watching just hold on like a couple of minutes trying to get this fixed but well it doesn't seem like yeah we should have done a test yeah exactly exactly oh wait a second so now it opened a good audio default speakers all the preview systems sound so if I select the headphones maybe that will work as well I hate that the fact that this is not working I'll officially quit ex pleat after this [Music] it also just opens really bad and now and now I can't get rid of the of this damn window come on I'm such a noob man with this is the whole thing froze and I can't for some reason it doesn't even allow me it just it just vamping all the time I think I'm gonna I think I'm gonna just give us a thing I'm gonna just kill though the people if there's anyone watching just hold on five minutes I'll restart the broadcasting I'm just gonna I'm just gonna close the app and just gonna close the thing see if not at least I can maybe yeah exactly it's just the whole thing like so much that like I I have even trouble closing the yeah high CPU usage detect that you may experience performance issues its crew you'll explain it's telling me this now when I'm trying to close it and I'll just not have to war okay this is officially not reacting ow mm let me just so I'll just stop the actual stream see if that yeah I might have to reset the whole thing let me know so just give me a second I gotta kill this this but I don't want to finish the stream I just wanted them just close this crap piece of software which is not reacting to anything and you're ready you're ready go ahead and solving the puzzle right ah yeah yeah yeah make sense go ahead go ahead man go ahead like that's gonna be just [Music] yeah yeah don't worry don't worry yeah don't worry I'm just I'm just losing my I'm losing my faith in and I just keeps I keep hearing the densa unlike like some things like something's not working and I can get this thing I can't even get to open the task manager from Windows that's what I'm trying to do I'm trying to open the trying to open the okay finally the task manager is opening up why why is this using so much CPU that's just like this I want I won't ever again hmm the task manager is not even responding oh now kill this guy so now now we're back and nervous okay yeah hold on a second oh okay so let me let me see if I can open this thing again or a second do I have I have oh I have OBS I have never okay let's see let's try with OBS maybe don't kind of set it up so but okay it at least is catching at least is catching the monitor already yeah yeah I never know exactly study more start streaming so in theory display captured by the display is definitely too big let me just make it small because I otherwise not the full screen is visible and then where's my face VJ v yeah yeah no no I I'm trying to I'm trying to figure capture and that's the web camera it should be video capture but it's not somehow whatever screw screw that I'm not gonna put my thing in if I click on start streaming some video encoders that might be might be missing so I think I will require that I I probably can do just right away yeah so we'll try to well I'll I'll have to try with xsplit again let me see so let's open this thing up no I don't want to have to because it is I'm using the free version but that's enough like I don't know I should definitely I'll definitely truck yeah no but I mean it's uh I think the only thing that you don't get is you can get some of the extra staff or like the basic streaming functionality it should just be there but I guess what you mean that they might just open the seven your phone and check whether you're here at all at least although this seems to still be it seems to still be loading I don't know why yeah yeah yeah but I think that the thing is I'm not so sure I have I think I have enough almost time but it's not going to be I'm thinking how else can we how else could we stream yeah okay I'll call him and then see if we can figure things out on a separate one okay yeah yeah I mean cool yeah yeah |
It has my face , so i am chill having this on channel.., I look good :) |
one let's see is there a way for me to just quickly share the streaming will be cool to automatically share it in twitter but i don't think i can do that from twitch studio um although i can easily just go to tweet tweet dot com uncertainties i think this is right is this the correct thing no systems i don't even know my url there you go so tweeter.com let's go let's go so four transforms and senpai so um what are we doing so this is basically the stuff that is required for this exercise here and again i'm just tired of fighting with this one so i'm gonna be uh it's just because it does not some something does not work as i expected to work and i think i've done smaller examples and it seems to work well uh so if i just say python x2 and score one and let it run i forgot which is the one my guess is my guess this is somewhere to do with the sort of the the the the range of the integral you know um because these it's like i don't even know where the time that where they take that from where is pi popping up here um like even if you would say in you do infinite if you do infinite uh oh well great that doesn't work but if you just let it indefinite which i don't think is what we have to do because again let's remember the shape this has right it's like a decay thing like this it's even like the thing is that you know even r is not something you can get rid of like how how do i get rid of r how is that only dependent on alpha there's gonna be some step that i'm missing with some kind of constraint that i'm missing because again the actual integral is being done by senpai right but all i'm doing is i know this is real i know that r is real and positive we know n can be complex so we don't really know much anything about n and then i think that is the correct way of expressing these is n times e to the power of minus alpha times r we conjugate that um and so we we print we print these here and then basically we do the integral with respect to r [Music] maybe that maybe that's kind of the thing maybe there's something else in here but it goes with respect to r and we integrate that and that always gives us something that still depends on r because the exponential is its own integral at the end of the day right so this all makes sense and then and then i do i solve these for n right so again remember that's what we want we want to make sure that the uh integral is the area and the integral uh sorry whatever i'm saying the area under the probability density function so the integral uh that we just calculated um equals to 1 and we solve for n and i get that thing here for a equal not equal to zero this is square of two so i mean you know there's potential here maybe this is actually not that different from that right let's this is simplified let me see if i can get sir edge high sir edge um i'm basically following these stuff in here from this document this is all right away if you go to uncertain dot system slash roadmap i i linked the actual file in here and it's basically this pdf and so what i'm doing is i'm doing problem 2.1 but i'm trying to basically do it in a way that um so i try to actually learn simpai at the same time and try to solve this with simpai instead of just kind of doing all the stuff manually uh which i know it's a bit lazy but at the end of the day it's kind of i think it's it's sort of uh it's a more convenient tool and and i'm learning a lot of other stuff at the same time so it's good uh and then um you know it is important one of the things that i want to do with these is i truly understand you know why certain mathematical concepts are being used right so it's not like that's a solution that's what the solution should look like and let's go for it but it's more like you know i i actually spent some time a month ago because i i did a bit of a took a bit of a break for for another project but you know looking at things like the uh the momentum conversion from uh from space so from position space um and i'm trying to understand why a fourier transform is what it's used right so kind of really trying to go down sort of the first principles of why why this why these specific mathematical constructs are used for certain things but then not really diving too much into the highlights to the this integral by hand so uh using modern tools for this uh but again i'm i'm stuck since like five streams or something on just the first part here which is compute normalization factor of n which should be fairly straightforward because it's uh um you calculate the probability density and that's what normalization means right so you want wave functions that give you a normalized probability density function so this means that the these probabilities are kind of adding up to one in a way and so you do that by integrating the uh product of these and it's no and it's a conjugate and yeah and then kind of solving for for uh e for n when the whole expression equals one and the problem i'm having is this gives me this thing here which is not exactly let's see if i can uh open paint and try to maybe do something by hand a little bit because i don't know how to do that this in how can i manipulate expressions easily in simpli like this maybe i should learn that but so i mean square root of two is nothing else then two it's a power of like a half and then times and then we have minus a to the power of a half and then we have e to the power of a times r a is alpha nobody has nothing to do with these because here we have alpha so first of all alpha minus alpha to the yeah it just makes no sense if i do smaller if i do kind of that is the way to solve these kind of questions but for some reason that's my guess is i'm i'm doing the i i instead of doing an indefinite integral i could probably do a a finite integral because i know that for example for r for values below zero make no sense anyway um but then i don't really know where would be the other limit right even if i just put something like random that still does not give me any any good answer i think so n should be independent of r right since you should integrate over r from zero to infinite do i have to integrate from zero to infinite probably but i do zero the infinite because that's what makes sense because it kind of has like a decay shape right i mean it seems to me the thing is this breaks i don't know why it breaks maybe because maybe because i have sort of a kind of partial i don't know why these breaks well invalid and comparison i'll have to look into these i guess that this is the range that makes the most sense and i think i mean it's going to be definitely independent from r because because it's the co it's sort of it's it's it's the the normalization coefficient right so it's in a way it's just a weight um so i don't know i'll i'll i just to be honest i i i just kind of have today i wanted to kind of do i'm you know i'm doing short sessions so uh i wanted to spend some more time on uh digging into the how to do the transformation to the momentum representation in simpai at least try to start attacking that because i have no idea how to do that in simpai and so so whatever this sempai um and then so position position to momentum conversion python uh [Music] let's just leave that beat out i know it's an integral what i have to do but i don't even know where to start so suppose we have a three-dimensional wave function in position space that's what we have we can sorry we can write these functions as a weighted sum of orthogonal basis functions or in the continuous case as an integral okay space i'm not sure i understand this notation though it is clear that if we specify the set of functions psi k side sub k say as a set of eigen functions of the momentum operator the function holds all the information necessary to reconstruct psi r and is therefore an alternative description of the state psi so is it all what you're doing because i want to in quantum mechanics the momentum operator is given by by these so i'll have i definitely want to understand that either functions position operator so it's an integral but that's that's too abstract to digest for me at the moment let's try to understand that so there's a momentum operator i don't know what this case space is in here so transform from position to momentum space mathematically the duality position momentum is an example of pontriagine duality i think i read that so the fourier transform obtains the function in momentum space so i just have to do the fourier transform which is just the stuff okay so so this is position this momentum we can force this one from one to another we have the symmetric fourier transform okay so this is probably just the textbook hey that's how you do it right so you have your wave function uh what is p x i p x ah it's an integral okay so this is what you have to do these hormones are worth a little study if we define it we say with different momentum okay this is i'm definitely worth worth the read but it seems to be it seems to be what has to be done so this is a k2p so you're multiplying the function by these it's an integral so you're multiplying i forgot why this multiplication made sense it's because each of these components i'm trying to think about the not the not the real case but sort of the the sum case instead of an integral right each of these components is like you're trying to approximate this is like four years serious thing um right in a way oh yeah yeah i remember this is because so this is just notation right because what's because this translates to like a sine sine like a sine plus cosine like one of those components is complex and so it's just a way to say you like how kind of much weight like a pure sinus wave has into this and so kind of using these to to build up then the momentum representation uh yeah yeah and i did that a month ago and i forgot already for a year exactly because this is what you what you want to do you want to kind of have those cosine things and so yeah that's why having it with the exponential notation makes sense because then i think the complex parts cancel out yeah something like this uh but again um what we're trying to do what i'm trying to do is uh write these thing here and my and and sort of in the momentum version because what i want to do is that i want to show that the probability density of finding the particle with momentum h bar k is isotropic aka does not depend on the position but just the radius then maybe and i want to see i'm gonna sneak peek into at least the first step of the solution because i want to see how how what are they doing here which is funny because i already see that i i don't see what i would have expected okay so i do see this kind of stuff here which is i guess the same notation that you have here just k is replaced by p in here i don't see that part which bores me a bit and then this is and i also don't see oh yeah no it's correct it's cool because e ta e e to power of minus so this is the actual wave function here um but then it's times the same thing so it's just added oh i just see the two pi in here y and uh why why is that and so then it's just like they're they're going it's like oh that's proportional okay and so the rest i don't want to look like but let's yeah i should just try to do the the the b and then and then just do it but again i just talked too much today and two last doing i just don't have uh all the time i wanted to have so i'll just probably have to do this tomorrow but essentially duplicate and that's going to be b right so this is a and so with b well we have this expression right and so what we want to do is we wanna um just i'll i'll try to do something in like two or three minutes we wanna have we wanna follow these and again i'll i'll i'll really step back and try to understand why this all makes sense right oh hbar awesome so basically the the momentum wave function would be one divided square root of two you know what i should actually have these at least like that so i can switch tabs easily to pi is this a thing it's like h bar uh a thing in simpli constants h bar h bar just without the other score times and then is the integrate and essentially we're integrating from minus infinity to plus infinite and we are integrating these times these exponential i p x divided by i h bar um i p well x that will be probably r right so whatever disease that we're doing we're doing these and then let me see pre-print that does this compute at least or no it's not defined y is not defined simply there you go ah come on i don't know i'm copying as well i thought that would import it though why is i not there senpai complex oh it's just i okay it's just these p is not defined of course p is not defined what the f is p is b just i think p is just a symbol but again i don't know we can assume anything about this let me just pee and i'll leave it here i think uh just because i have to go i want to see if that prints well nice god okay um otherwise oh oh oh some things are going on here i just done is there a nicer way zero are these uh i hate these piecewise things this is just nothing this is impossible to read i have to work on that but okay i think that is the way you would write that and so um but what do you do with these this is then you turn it in uh cool uh yeah man it's cool i mean uh i'm archiving all these videos in a youtube channel called uncertain systems so just yeah basically youtube there you go so and i'm archiving all the sessions in here right away um so you can so you can you can kind of look at it like they have tons of videos um but basically yeah uh and you know that that's kind of again here it's kind of the roadmap for this year i'm really i'm really trying to kind of close that series of quantum mechanics homescratch basically learning the whole thing just by doing exercises and kind of exploring it live without really following any official material just kind of using the internet and what's there twitter and whatnot um so yeah man um happy happy you like it and um feel free to follow let's see let's see where that goes and kind of you know i think that's always useful to learn all these skills yeah i'll see you next time i'll be doing um i try to stream two through two to three times a week but then it always depends on the amount of time i have as this is really purely a hobby and it depends yeah see you next time |
is twitch studio better yeah i didn't know that something like that existed and i'm gonna be gonna be changing some of these kind of colors and whatnot and there's even a like a right away set up a beer back screen which is pretty cool okay but i see the mic the microphone is still but i can view it myself and then there's like a chatting one where i'm a bit bigger in there that's pretty cool i mean you can do that with obs as well obviously um but this is all nicely integrated with uh with twitch that seems to work seems like at least the twitch app is telling me that i'm live which is good and yeah what are we gonna do this week's schedule is gonna be messed up just because we have some some i have some kind of like personal stuff needs to be taken care of tomorrow today as well so it's gonna be a bit weird i think i'm not gonna do uh a full hour we'll see but i want to kind of keep keep banking my header against this thing here and i realized i said something really stupid last time which is that i was complaining about these and i was like well that's the point right that's not the point the point is like if you um if you have something like this because what i was doing i was doing this easier one yeah so so i was doing all these and then i said okay so how do so so how would this work right so you you can you can what about the cpu here it looks like a bit slow as well on my end 27 19 is performing quite well cool so i can see here chat that's cool okay so and i'll say okay so i can i can i can i can do i can just kind of like abstract this zero away or like factor the zero out and then what do i do with the rest and it's like well it's not you don't do anything with the rest you just um you you just kind of have like uh so what you will and what you would end up with is you would have one zero node that is connected to all the other hyper hyper edges and so you're gonna have it's gonna be part of four hyperedges and uh [Music] and then you know kind of the next step is like you'd say okay so for this note do i have because i think i think what i was doing last time was wrong i maybe i might have been just too quick to say that the just doing sort of just sort of doing the factoring is is wrong it's something that i don't want to do or something maybe that's just the way it's just i was doing this wrong before because before what i was doing is like saying okay so the algorithm was was was native because it was implemented with with uh just two notes in mind hypernodes and i was like okay so for three you just it's the same it's just the same algorithm just killed with three so the problem with that was that yeah then you kind of have the um the situation where you can end up if you do it like this there you go if you if you do this like by pairs right and say okay so this is this zero is equal here as well so we'll just gonna move it away from here you know you might as well just peek like this one wrongly which kind of then if you're ignoring these and then you take the other one you know that would lead you with say a zero and then a you know you can have a zero one and then you have a zero zero so that'll be one branch and then you might pick up so i pick up these and these and then i pick up this one and this one so you end up with like a zero and then a zero and a one and then a one and a one and okay but you can still merge these two right because now they are the same no that's not true because they are because you can't really merge them exactly remember that so i was just taking them i think that was just wrong i was taking the things out and i should just keep them in the hyper edge i can have yeah exactly but still i still i can do pairs right still i can do pairs and say you know well look now this is equal so i can also just um keep that this is equal right and then in this case we just have well it's a zero and it's a one and then which is like a zero plus one right so now it's the um i mean i i i just don't wanna i wanna think about this twice before i start coding this because last time i realized but i shouldn't have just given up that's the problem and i don't know if i should just do things from scratch or i think i'll just probably do things from scratch it'll be beneficial anyway because you know my my my other alternative is that i kind of kind of trying to keep banging my head against you know the idea of finding a way the finding a way just for me to um just for me to find a way to transform a hypograph that it's already fully you know factored out into another one that's fully factored out directly without coming without doing this kind of like unfolding step and the problem is and the way the gates must be specified because gates are specified at the system level that is that is the is kind of me but this this is sort of maybe i should spend a bit more time on this because this is this would be cool this would just be would just be cool let's maybe let me just get rid of these um and kind of like do again what i was doing here i remember so let's kind of take the case of the ccx right because it's an easy and let's do some examples because this is this is an easy enough game because it actually has the definition where i know that i i just need to replace i just need like two rules to match and replace rules because the rest stays the same but that's the that's the essence of the problem the the essence of the problem is that i i'm being forced to match at a system level and so yeah that's what sucks i'm being forced to match at the system level and and this means i'm in a way i will always be forced to actually transform the system out of like the the the qubit level system i'm not using standard terminology i'm just when i say qubit level what i mean is like from the qubits perspective just not at the system level but but like just with the subsystems in mind so the states of the separate qubits and their entanglements when their correlations um i'm forced to reform these to the full system view so i can do the match replace and that is that is the main that is the main issue right so that's why i was looking for that's what i that's why i'm looking for a way to represent an operation just by talking about the subsystems and not about the system right and that's easy to do if the gate is not entangling because this means that the gate can be broken down right so if gate is not entangling it can be broken down to qubit level operations right which means that i just you know i can i can eventually compress all these and then say if you know just if this is this then replace right you still you still have to it's it's actually an easy example would be the sort of a harmon layer right which we often like write as just one box right but it essentially is just a harmonic in every qubit so if i have a system that is in the state zero plus one zero one you have like three qubits right and i apply a harmonic layer what i'm what i'm actually saying is is apply harmony to each one right so this hardware layer which is basically consists of uh which is basically consists of two of two rules right the rule would be the rule would be find a zero replace it by zero plus one find a one replace it by a zero minus one and so you know for these cases it's easy and in the the case of both uh what what we would do is we would just kind of split the notes right i think that's that's the yeah so would split the notes and then apply these and then we'll need to do this kind of like solving for interference right so i think so you have multiple nodes and you you're going to keep solving for interference that's why this makes sense yeah in a way right so you you know in the case of of zero plus one then you would basically first break into like a zero and a one and then you would do zero plus one zero minus one kind of thing and then you would solve for interference and then uh you would what what am i doing and then you end up with with a zero but it's two simple rules right now to keep it so control controlled operations are easy to do as well because two keyboard operations no control control controlled operations are always you are easy to do right because you kind of have this rule if like you have you have this rule right of say if um if you okay so actually look at this so the way to approach this is by by being able to nest rules right because the splitting happens always so so the the so if if it's just one rule if if the control is one then what then then what you're basically doing is you're nesting the rules so then you're saying um ho ho ho because i'm thinking i don't know if i'm glossing a bit something too much because here here i would say this expansion also should happen if you're more kiwis right well no because again there's no hyper h right because there's no correlation or there's just one hyper edge so um and so there's a one then it's a bit of a it's a bit of a special rule right so there's a one replacing another one so okay so that is that is the difference right so these these these are sort of local replacements and i'm probably abusing language here um just because it doesn't mean they are local necessary but it's just one qubit and so these are um and in these are non-local replacements in in in the sense that local it's a local match replace it's a non-local match replace which means that i'm matching in one qubit but a map and replacing another qubit so so much for q zero and then and then um and then what you're doing is you're saying uh you know run some local match replay so you're saying you know run these right obviously so that's how you would specify the rule for the for the control node so so this expansion it's just this expansion like does not generate new hyper edges but these does generate new hyper engines this does generate new hyper edges even on the if the actual replacements lead to a different value of the second qubit in this case right because what i'm going now is that i think maybe the trick is if every n-qubit operation in general can be broken down into a universal gate set which is just like the hot mark and the control knob for example right then that should be that then that should be what i do then that should be i think the key that is um i think that should be the key maybe that should help me get like a more compact representation in terms of rules because that's the point it's the same with it's the same with um it's same with mattresses right like if you have like a five qubit operation you can yeah sure you can do like a five by five matrix but you can also break it down into smaller matrices which then turn into a tensor network which well it's easy because you have a bunch of smaller mattresses that they kind of connect to each other and then you can do some kind of some tensor tensor um tensor network magic and then you get some you know a bit more efficient uh simulations i think that's in this case same idea it's like break a cc knot like what is the see if i can open chrome and the thing doesn't explode i'm getting a better machine soon that is also good obviously we're gonna go incognito and close this one and so what about the ccx the composition because we could start with this one for today's stream ich shtima ii i should just probably go to pictures what is this beat uh yeah there you go there you go something like this yeah yeah yeah the wikipedia one what is this though polish stuff the main goal of b is to design and implement an algorithm for solving an all relevant np complete class of problems using quantum computers effectively speed is better than quadratic guaranteed by theory are they guaranteed by theory we focus on the hamiltonian cycle problem but since there are known reductions between different problems inside the mp complete class we may say that we're working on all of them at the same time so your quantum your quantum working intelligent knowledge for me who is behind these do i know no never heard of any cool anyway so this is it this is it this is it this is it oh sorry now the question this is so this will be this will be really interesting because that tells you in a way that tells you how the cubiets entangle with each other and so we should just kind of take these breakdown um and and make rules out of these right that's the way maybe that's the way forward so in a way that's like that's a network as i said that um oh man i'm not having a good day today that they this this so if i have a qubit that is in the zero plus one state and then i have another qubit that is in the um zero state right so that's that's that's my first that's the system so then i go here and say okay so cool let's do a match um now yeah so now and and note here difference right so that's important because that's actually a change in the way that i was coding these a how did i call them um a linear a nail comp match so so this so we've got an elcon match here so this means we're going to split the node but we're not going to create we're not going to create two separate hyperedges yet right so so we're gonna have like a we're gonna have a basically zero one so we're gonna split the note like this right i should use different notation for that probably but whatever so yeah so that that's that's what the the elk match does it's just splitting the note into two notes and then i'm gonna say cool so now uh now let's take a look and say so thus running these rules so how okay here's the trick right so if either rule so in this case right i've so in this in this case if a rule effectively does a replacement for every time a rule gets actually run like this means that there's there's a match it it will create a new hyper edge and but of course of course you need to resolve you need a result for these right so that that's that's the thing so you you run the rule you you kind of what you run the rule only if there's like an actual non-elcon match because i think that what you you don't want to have elcom matches right so because that could even work for for ccx gates we'll see now um so if now i would kind of say okay now i so you you you first of all you run this rules as many times as until you stop getting lcom matches so um i know it's really inefficient about whatever then then you basically go and say because the girls just not write the rule but like yeah kind of like do a sort of a swat like a you you you go from you go through all the sort of the nodes that you try to match and every time that you have an outcome match you do a split um and then you kind of do it again and now you have a full match right and so now you're saying cool so now i have a full match um now let me take a look at the so so the first is the netcom match and the outcome match and now you have a full match an f match right and now you go and say now now that i have a full match now um i'm gonna go and run sort of the these other rules in here um for these right but then every time that i actually have a match and and i replace with something that is mathematically different right let's see oh it's just different for now let's let's deal with the let's deal with the um let's deal with the face kickback a bit later we can do this now maybe soon but so so every time you every time a rule hits and there's a replacement that leads into a different node then then that creates a hyper edge so that then now you would have a zero zero and then you kind of would carve this out and say well the ones kind of like that and so and then you're done so what's created the separate separate edge is as a full match remember that we can't have two nodes that's the thing right so we can't have at the end of the at the end of running an operation we kind of have two nodes i mean we can but then that would mean it's it's to go here but okay so let's see how later how will enforce this right but like just because this was split then it means that i have to carve it out because now i've got i've gotten a match okay so it will only actually create a hybrid edge if it doesn't match on a split node yeah so we'll actually create the hyper edge now huh how would it work with the face kickback can we can we make it work with the face kickback without having to do a um sort of a branch merging operation so the face keyboard let's try to follow the same opera the same same logic so to get a face key back um you to get a face key back you do a you can do a um [Music] let me check quickly with quirk this is so slow because this is just slow man this is just so slow 21 cpu okay so so if i do these and then we are here and now i do control x yeah that is it that's it because here the blocks here was plus and it turns minus right so that there is a key key back so we have a plus state and minus state so that's without a plus state we have a minus date let's see what happens now we do a um we have an outcome match so the same happens here [Music] and um and now we do it again and uh and have a full match and so what we do here and now the full match is say okay so now let's let's replace the rules but now cool so but now we have here an a an outcome match right so this means i gotta split the thing so so i split the thing and that means that i end up with with ease right um now that's interesting and now i have a remember i what i want to do is i want to i want to and i'm not creating hyper edges here because if i create hyper edges i know this is not entangled i know the the outcome of these is not entangled at all and i'd like to arrive that without having to create intermediate hyper edges because now now what i'm saying is now run these right we found them so we so what did i do before so before it's like i found a match and then i run these things and the thing is if if i was finding a a full match here um with the mat with the match on a mixed node here then i was creating the hyperedge but now the thing is i found an outcome match so but target qubit outcome match right on control keep it by targeting alcohol match and that's a different case now because and and and now i shouldn't create a high priority i just basically do these so i do the splitting right um and now what now i would run these and then replace the zero zero by one the zero by a one i'm missing something here because if i've replaced the zero simply if i just simply go and now say cool so now i can now i can run now i can run this again because i've split the target qubit so now i'll get full matches so i get a full match here right and uh by the same logic now that i get a full match here you know i should create a hyper edge right but i don't want to do that so i'll just go ahead and replace these and then this goes and it turns into these right so i'm what i'm done [Music] hmm i'm kind of dumb but i'm kind of done but it's not it's not giving me the it's not giving me the face kick back right because i'd say i'm kind of done so so i'd say okay so first of all i should be just writing notes somewhere and i'm ah so an lcom match displays notes um then a full match on a non-split node on a non-local replacement on a full match whatever yeah he then creates a hyper edge something like that i'm gonna write these rules down i think i'm on to something i think i'm onto something a full munch like the thing is here's different case right so i did that and then um yeah and now and now what right you might say i kind of just like plus and in a way minus right i wanted the minus to go up here because that's what the face kickback does because we know this doesn't entangle we know this is the same so we know this is the same like this one here right but it just has this extra kick extra extra phase that's what i'm using but i'll leave it here um but i think i'm onto something so i like these i like this this new approach i like this new approach of kind of like nesting things like that so saying like the non-local microplace that making that distinction is something i was not doing in the original implementation because essentially that is what is missing that is what is what i'm kind of missing from a um yeah to to avoid the kind of full system like hey match is zero zero and replace with that because that's that's not what i want to do i want to do these and i only want to create hyper edges in specific situations and again my guess will be to move to to move to multi cubic gates either there is a way you can do it like that still or then then you're just going to go through the decomposition um uh sort of the the the the see the the c not the composition uh of it c not plus unitaries because then uh basically means yeah and then you can kind of build like a big like non-local match replays for these or a set of rules like that and then that that will act like that will be sort of a parallel or that will act like a tensor network in a way of replacement rules um and and yeah but at least it should leave you that's the goal is that i want to i want these rules to end up like generating the right amount of hyperedges without you having to do some kind of resolving or refactoring because that's the point and and i think that's the same point like what tensor networks isn't it that you don't have you don't end up with big things you just kind of compress small things all the time so you all the time have the smallest the smallest representation possible and everything else stays connected that's really what i want to do sort of uh it's nothing new really it's just um it helps me to think about these kind of things in different ways and and that is quite intuitive because it's a bit of a it's a bit more classical it's like you know if this is one then do these if this is no no no no etc right um then you end up with this kind of hypograph that tells you you know it kind of tells you how the cubes are entangled which yeah essentially i mean i don't really know why would this be useful or not but i mean hopefully to study entanglement in a way uh let's see i i don't i'm just this is just really an exercise for me for myself at the moment and then yeah and then you know i want to go back to the whole quantum mechanics as well but that specific problem buggers me a lot so i really want to get it solved and then kind of get my hands down coding on these while i do the other quantum mechanics stuff again cool let's see how the quality how the stream would look like uh i'm i'm pretty happy with the the way the whole the whole um twitch studio beta looks like and it's fairly easy to use and to set up um so i might use that as well now unless i would change the stream platform but i don't think i would cool see you tomorrow |
okay why why why there you go i don't know why the laptop is going nuts but um uh it's hard it's hard it's hard it's it's it's hard to kind of keep keep working it with such a breaks but it's all i can do i have to live with this and let's just get uh get going so what we're doing we're doing basically uh what do we have here so uncertain systems roadmap and we were stuck with the exercise with the polar coordinates and we yeah yeah so we want to uh do this stuff here so and and kind of like what was the rewriting so this was the solution and so we're going to do uh volume integral three right i think there is the in triple integrals um [Music] that doesn't help me tricks so that is the tricky the trick is going to be this rewriting so what is this person's size i was wondering if anyone can share some tricks instead of going to rewriting a triple integration exercise i'm doing trying to rewrite these two d y d x d z uh i have drawn the picture out and realized it's sort of house cylinder being sliced by the plane if i'm wrong with the description so it's it's just going to change your quantification you can write all the limits as inequalities and then they use the new bounds from them in order for uh in order from outside to inside again what are we doing here so uh what are we doing here so uh shell basically um python python x 2.1 a what are we doing um so i've done these why am i just what am i doing is this really the latest stuff that i did um so let me check in here no here we are finally so this is the one that i did last time wait did i finish times zero so we're current time so it's a function of just of just x so yeah you see but that's what's motoring in here but there's a value involved but if you just give me r i'll give you a value right yeah you see but that's function of so a time is zero so we're here in time so it's a function of just of just x so in this case the way of expressing that is that you're evaluating it's definitely like you know that's my point although it's assumed that it's just two pi right because it's the full three dimensional i'm sure that i'm sure that this is three-dimensional space as we said uh i just had a bit of a lapse when it comes to where's my band this had a bit of a lapse when it comes to and i um find a pan who the stole my pan really there's no pen god anyway um this is like so this is like basically so applaud um [Music] so exponential of like minus um you know minus say one times and then x right and so that's this here um yeah um minus two you know so this is this is basically this alpha and then if i say here like three times something whatever it is but yeah yeah so so this is just um is that kind of exponential like these okay but then x x cannot be x can only be positive what am i doing um so i do basically e to the power uh of minus uh you know one times and then x right times x no i want the oh there you go why isn't why why wasn't why was google why is that that bad what's going on here i don't understand why is this being plotted like these uh anyway this is like this is what we want to do and so so we know so in a way i know that from from one but if i imagine that in 3d it's going to be like a kind of cold it's like a some kind of cone like these right not coma like yeah something like this and then so on one end i won like 360 degrees and on one end i just won like 180 degrees because i know it's never going to be down there right because we know x is always going to be positive uh yeah um you know and even if this is like that then it's just the other side exactly so if off is negative um exactly so i'm pretty much convinced that this is gonna be like that so if i'm assuming i'm just gonna gonna try to understand that too much um upfront but i'm gonna i'm gonna assume that um somehow you know the boundaries so we've got like definitely boundaries um on one end is like from zero to two pi and on the other end is uh from zero to um pi yeah and uh and this guy here to be honest oh and this is where alpha goes i think isn't it let me see no five six no but it's always one is it why is it always one yeah it's always one of course because e to the power of zero so zero to one okay zero to one so zero to one zero to one is sort of the set so this is um this is the z axis this would be the what do we do um what do we do so what do we do what do we do we do what's the convention z-axis i think x and y so the x is going to be this is going to be x and this is going to be the y axis i don't know this is my working assumption and um and so how do we how do we um how do i rewrite these well i have to rewrite so i do have to rewrite the r do i just have to rewrite and say that like r is um do i have to just rewrite and say so 3d radius formula or is it just a or line formula is just this exactly so it's a square root of like pom pom pom squared so do i just have to say it's basically square root of uh you know x squared plus y squared plus z squared we'll just go is this what the function is saying so this is what this is the function that i have and i wanna and now i wanna rewrite it so i can i can do things like these right in uh well in polar coordinates do i have to rewrite anything yes because i have to transform this x and y's and sheets like these with cosigners and craps right because essentially essentially what we have essentially what we have is the following so so we have um [Music] so so we know r and then we have z and we have one component one component so so this z is fine r we know uh or z is fine but then we have so if this is y and this is x right so this is where the two angles come into consideration correct [Music] it's just like sinus or cosines or something some like this i guess um so rewrite sonopolar cylindrical coordinates i just need a formula can academy triple integrals so the volume what there's no soul include the r as expanded as r d theta the r d z oh okay so you only need the one angle radiance yeah volume of a sphere this is the radius okay now we're missing something aren't we i mean this is definitely these is it correct so since the bonsai set will depend on the value of r will let the inner most integral handle z while the outer two integrals take care of r and theta so integral uh okay so are we doing this correctly so hmm so you need sphere i guess you don't oh this is three blue and brown this is just a rotation okay using polar coordinates okay so maybe in our case it is already i mean we already have it in polar coordinates so there's no need to re there's no there's no really need to rewrite anything we already have it in polar coordinates so so then it's just these so um it's r d z d r d theta uh so the main thing to remember about tripling because is in simulation is that db represents a tiny bit of volume expanded like that so it's inside according this can't get similar integral it has some kind of rotational symmetry about the z axis that's exactly what i that's exactly what we want so we wanna so we wanna do these uh okay but they're always oh man i feel i'm one like little very very almost there almost almost almost there um it doesn't matter in which order i do them because python does them for me so that's fine so we have r but what we want to do is um i just don't get um i just don't get how that actually works at all uh with just specifying one angle or how does it work at all if the function is like these you know there's no the theta right so how is that even solved that's interesting i mean the the sphere is just the radius right so if you tell it the radius is just a radius that's an interesting example [Music] so uh of radius one first of all make sure this is just one the function is just one this is hard particularly about the integral that i understand but how is the step by step since the bounces that will depend on the value of r will let the innermost integral handle z while the other two integrals take care of r and theta writing down what we have so far [Music] okay so i mean i have the bounds um it's important to make sure the order of the differential terms designed here and then matches up with the appropriate integral and this is a little trickier it feels like yet another failed video failed session um ethereal function inside the sphere our computer will change okay so the three variable function this might seem out of place in the article but independent cylinder component since everything there is in the collision coordinates indeed the treatment is a bit different if you want it and however in polar coordinates this becomes very simple the angle is pi fourth and radius is zero and two always so what did i say it's wrong that is i mean the re the radius is like between zero uh and infinite right but we know the bounds on z is zero and one so putting this together our triple integral looks like these zero and one zero two y zero two radius oh so it actually steps my step by step so how do they do these first they do set cool then this is done then little r oh so they so here is when you get the the theta angle you get it because of at some point you're yeah because you're it's going to appear in your boundaries uh in this case the boundaries that can be translated to this i mean in a way maybe this is what i'm missing no i mean i know the bounds of so i know it's it's cylindrical uh so i know i know i want to do two pi in my case huh my case is like these right so it's like these and it it's like symmetrical so almost there i just have to find i just have to find the right maybe it's just easy it should be easy because it's just oh i just got alpha this is just a parameter so that's fine it's constant it just got oh it just has r it just has r in it right so it's a bit like the it's a bit like this sphere example in a way it's just got r my function it's just um that one so maybe i just have to indeed have to um you know express these things so as function of r in a way um what i'm not really processing at the moment uh 0 to 2 pi this is this is the same our radius radius radius radius zero to infinity and that how should i express that that is just year one am i just being an idiot am i just like am i just i'm just being too stupid and i'm just saying well like just do just to the triple integral and that's it don't necessarily have to i mean am i sure that zed can be bigger than one no because it would mean negative radius in here right unless if you give it a negative alpha that's not only that so it's like now the radius can be so what is the radius am i being confused i'm being confused i'm being confused it's just it's just a radius right x is the radius so x is the um but he is plotting it and again i kind of um i'm clutching again with this with the same concept with the dimensions and whatnot in my head because i'm i'm confusing myself because i'm plotting it against x and i'm talking about the radius and whatnot and just saying that is how something would look like but we know that x can only be positive um so we know this is a limit x can be positive let's say that's all it can be right but alpha can be negative so so then it means it can also be like that i'm not so sure if that's that's the volume that i'm thinking about but it could well be anyway i have to this is a is a super hard problem without any guidance i mean i have the answer um i'm just i'm just really lost in terms of what is it that i how do i even start like i i know i have to start with something i have to so i know this is the this is the function that is just depends on r right so um but again i'm kind of and i'm kind of circling circling around the same issue of like is it then my function four-dimensional or three-dimensional right because it's like for three-dimension space what the function is telling you is um in a way sort of the probability that it's there right because remember we have to do it square so i have i have that function squared and it's telling me the probability that it's there um and i have to [Music] so in a way this is four-dimensional that's that's what's just that's what's bothering me a lot uh you know what i mean like unless i would say uh boo unless i would you know do something like i just do the one-dimensional uh and and integral of these and then somehow sort of applied in that rotation right so times the the two pi which would kind of tell me why the the pi is here so that's kind of my that's my main problem is that i we we have here's more lack of a four-dimensional thing right so if i try to go on like that when we do exercises one of the exercises i'm like okay so let's just assume x is one-dimensional right so it's then it's just your your wave function actually is two dimensional right so and that's what you're that's what you're then calculating the area off and so in this case if x is three is in 3d space uh then we're talking about a 4d uh four dimensional function um which then all this gibberish that i'm doing does not really apply here um whereas now if i just go and say um you know this is my function uh please integrate it like this right what do i what do i have so i just i have basically what i have here so i have if i say and it's going to be i don't know if n is going to be positive but like a is definitely non-zero and r is positive so if i run this what does it tell me it tells me that i have n squared divided by two alpha okay um but again this is just a slice of it right this is just a slice that i have to somehow steal um i i've just integrated across like r so this is just one um one slice but i still have to apply some sort of you know how is that two pi rotation to these i don't know um it almost is almost feels like it almost feels like this because then i have like a so if i have n squared and then two alpha and then i and i do these two pi then uh then i basically you know the twos cancel out and i have like n squared is basically pi divided by alpha which if i then you know kind of do these it tells me that you know n equals like pi divided by alpha like that and the solution is really close to these it's alpha divided by oh no but that's uh kind of makes sense what am i doing no so n squared divided by 2 alpha and i'm adding 2 pi and so if i wanna say that uh and i won [Music] you know and i think i want this equaling two once it's alpha divided by pi that's correct the twos goes and it's after divided by pi so it's n yeah yeah so it's these and it's alpha divided by pi that's correct that's fine but i have here and the answer it's alphas alpha to the power of three and then this is actually yeah it's almost like these forbidden three times i don't know why but it's it's really similar so i'm like i don't feel like i'm that far away from the answer i i really don't feel i'm far away from the answer but i just don't know what the answer is uh because it doesn't seem like this is something that i need to do necessarily this whole like um yeah i don't know next time i feel i'm almost there i'm just missing a three here and a three almost there because yeah i can't imagine 4d but i can imagine in 3d why wouldn't this work like that you have a sphere and then you just say like look i'm gonna do just that and then rotate it to pi like why doesn't why can't why can't i just integrate a slice to these lines and multiply by two pi if it's a pure sphere why can't i do that um it would seem intuitive but maybe there's just some something something wrong in there anyway see you next time |
okay so this is supposed to be the summary of my analysis of this article published by brian uh and siegel walk so brian basically we've been um discussing about this in twitter for quite a while this is just the latest um article of a series of articles where he discusses these uh kind of classification algorithm where he goes in and calculates some precalculate some angles and then uh we'll go into this right now so builds as a quantum circuit that implements the swap test and um i think my critic to my critique to the article is that you don't need this right because if the way that you encode it is like that so you have this using the same gate and using these angles it's just not if you compare the angles so you don't really need to go through the actual quantum computation for these um and what i did in the previous videos is basically i kind of try to reproduce what he does and he explains in the article so we you know what i did was i took exactly the same data set um and did a bit of a you know the same the same pre-processing he does classically as well so he does and he what he does is he takes the averages of each class so this is the emne's data set or a version of it at least that's already in csv uh and and so um i think each class then or each class can has like 784 pixels so there's the 784 values in here uh for each of the labels and this is sort of these are the averages of these um and then he goes and compresses this a bit more and compresses these uh to eight different uh 16 different um values so he basically does a four by four sort of low resolution image of uh of those numbers so to say and and these numbers is where he then you know those are normalized values between zero and one so what he does is i believe so as he takes these and he turns that uh each pair of the like you know two of those angles puts them into a u3 gate and that's how he encodes the uh the gates that you that you see in here that's how he he creates those gates and what he claims is that well then the subtest kind of tells you uh with you know a certain probability to which class because what he does is the the first you know eight qubits here at the top um represent the the averages of the classes and then at the bottom you have one new instance you know randomly taken from a new data set and he compares it with the averages right um let's put that aside the accuracy of the idea let's put aside you know um whether this makes sense or not whether this is better than a neural network or not i'm going to keep all this discussion aside and just basically try to prove that's what i did in the past two videos that if this is the setup you don't need to um to run the soap test this is totally useless in here so you just can compare the angles and actually it ended up like i really wanted to kind of keep this as a summary if i if i really want to do it so it ends up the proof is actually fairly simple well it's a proof that is simple enough is what i write in the notebook at the end um where i basically say that you know if um the subtest output zero with probability 100 only when the states being compared are equal right and and regardless how you created the states right so basically if the states you're creating are equal then you're going to have 100 probability of classifying you know that new data point to that new class and so you know basically um you know that doesn't really need any formal i think proving there if you are using the same gates here for the data register and for the test register the u3 gate the ry gate whatever you want the same gate and the parameters you're using are the same it's this is the only setup where you would have kind of well the same you know um the same vectors right or high dimensional vectors right because the if the angle of these cubit is the same as the angle of the gate in this cube then effectively the mattresses of this case are the same and so they're going to generate the same state and if you scale that to the eight cubits the same here right so if the angles and the parameters are exactly the same here and here then you're guaranteed to have the same state so and and you know and and it's kind of a known uh unknown fact that like uh this only if you restrict the the parameters to be between zero and pi right so you don't like allow for cycling or having two times pi or three times pi or whatever um between zero and pi then there's only one way to generate one particular state so you know it's not you cannot get to the same state by parameterizing the gate differently so with these two facts in mind you know that's basically the idea uh that's the idea i used to prove this here so by just running this simple proof in here what i also did in the previous videos is that kind of took a look at um a bit more in detail at the function uh of you know what really mathematically the soap test is doing is this part here so the probability of measuring zero is defined by this equation and so you know i kind of developed this idea for one for two qubits so comparing two cubic to one qubit states um and and kind of analyze the way this looks like and you can see here that this proves the point where exactly the diagonal where the angles are the same so you have the maximum probability and then everything else smoothes out gradually um in the notebook i played a bit more with you know more dimensions um and you kind of you know get this pattern within the formula where you have sinuses and cosine is multiplying e you know i i can't plot so many dimensions but i can show that um i forgot about this but i can show here that if you're kind of replacing all the parameters by the same angle um that's where you're getting it and i actually i think i can even go farther than that and say like the angles don't have to be the same like all of them the pairs have to be the same so if you change these angle this angle must be the same as this angle these angles are the same as this angle etc etc but that's that's kind of you see the point here right so this is where you get everything like with a value of one so you see all the ones in here uh one one one one one this all the things in between are the angles that i'm using and i think that's it that covers the proof i mean i don't know maybe it's not super crazy hyper formal but i think just tinkering with numbers and the equations kind of at least gave me confidence enough that uh you don't need a swap test and like it doesn't matter how many qubits you want to put in there it doesn't matter how you want to map things as long as you use as long as you use this encoding strategy you don't need a qpu okay so you just compare the angles and basically here i use the compress so i copy pasted the values from the csv file so i i use the compressed version here but i mean like if you use the compressed version version you kind of see that it literally takes and that's the code that that's a code that calculates the occluding distance okay just the distance between the two let's assume those are points in this high dimensional space um so it basically does this and it classifies this new test which is like a sample taken from the label zero classifies it classifies it as a zero and here you have the distances to the different classes um brian also kind of mentioned that yeah well you know what if you do not compress like then you will not be able to do it classically well then here you go so i actually did it so i actually took the same values but like uncompressed so each class has 784 uh pixels and the test one has also 784 pixels so this is about like yeah so if i do these okay now it's there if i now go and run the calculation because i just replace the variables and in like it's a for loop it's a one-dimensional for loop and then there's like the the the norm uh to calculate like this thing calculates the actual distance so it's just it's just like a subtraction and like off of the array so this is kind of maybe another nested loop right what it does is it compares i mean you're comparing literally like every pixel i don't know does maybe this is like max like 5 million iterations or something like that all together i think if you multiply 784 times 784 times 10 is that it i don't know um but i can so let's check first let's print test to make sure that it's the right one so this is the one with 784 pixels right um [Music] and uh could i plot this to see that it's a zero or whatever and um yeah and if i just now basically run this well it just takes that it takes a ridiculously short amount of time right and it tells me already all the distances to the different classes and it's still a zero although to be honest as you can see the distances are pretty similar so i think we're quite lucky but the point is like [Music] like that beast on a simulator takes like four or five minutes just for the just for like eight cubits or 16 cubits in total right and even if you would have like a million error-corrected qubits that could you know you just compare the angles so yeah that's basically that's basically what i need i'm quite happy the results i did learn some little nuances while building these expressions in here so i'm happy that i learned more about like how to actually build how to actually compose uh things with the tensor product like you know knowing that i want to apply this operation for example to the second qubit not to the first qubit i have to do with the tensor product and i'm going to definitely do another video on these because i've i think that i think there's some kind of there's some kind of interesting points about you know using linear algebra as uh as a modeling tool for for a quantum computation so i think that makes it a good executive summary uh i hope what i said was understandable if not questions below in the comments happy to answer and i'm also going to just push these into github so people can play with this as well um not the csv file though cool see you next time |
You're just offering a classical alternative., That's the point. If there is a classica alternative why use a quantum one? |
i'm like orange can i close this yeah this is the background right i think so okay so um i think the next step is basically to um to go and check the um how to translate the actual problem into a cue ball i think that's what i i think that's what i what i went to today because we've played with the wave um i guess we'll figure it out we'll figure it out as we as we go so um chicago she can't go quantum again so this is the paper um let's click the web page maybe there's some links um so this is some news and this is the medium post right um let's get the pdf and in the medium post maybe there's something there as well so this is just um okay just just a summary of this stuff it's not interesting so the um we construct stock portfolios based on our stock peaking formulations from july 10 2020 data after the u.s market closed market closed we pick from a universe of 60 stocks run them through our classical and quantum algorithms and select to start portfolios from the quantum portfolio results our starting point is 60 us liquid stocks okay so but this is the [Music] so the mathematics in the competition i guess this is the same paper that i'm gonna open up right now yes to find our specific portfolio size in a very few number of tries out of a large number of choices so they we see our quantum values override almost the entire efficient frontier which shows us comparability between our cq and s formulation and the classic sharpie range okay so we want to understand what the cqns is right um the cqs finds high expected returns for certain level risks classical efficient frontier marker multiplier sikiones ckns portfolio secure portfolio ga portfolio universe 60 stocks these are the results finally have no evidence that any of these portfolios are optimal or best or likely even on the top 25 percent of portfolios we could have selected which wasn't based on prior 12 months performance which is no guarantee of future results what we do know is that we use a quantum computer for 20 seconds to look through 1.15 times 10 to the power of 18 portfolios and we still have profiles that will track and measure against the universe of stocks that were selected from so here we actually have the 60 stocks that they get now and this is include okay they also have the indices and then data from yahoo finances of 4 pm ct on july 10 uh 2020. okay so with this we could reproduce at least a 60 stock one we tracked five portfolio for first homework close to july 10 through market close august the second uh let's show the results okay but i'm looking for the cqns [Music] i have no business with any company okay but there's no there's no mention on the what is the particular cq on this right and then i'm assuming that is then what should be explained in the paper let's focus on this today okay um so we've got the paper here let me open let me zoom it in we investigate the uh the user computers this is abstract blah blah blah why is this scrolling so slowly as usual pdfs okay so to find optimal risk versus return portfolio an optimized portfolio based on the markov its formulation and the sharpie ratio a simplified chicago quantum ratio then a new chicago quantum net score um introduction okay so here we have the sharpie ranges right so first let's try to understand these so what is the sharpie right here sopedia shopee rachel was developed by nobel laureate william f sharpie i'm sure i'm pronouncing it wrong and is used to help investors understand the return of an investment compared to its risk the ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk volatility is a measure of the price fluctuation of an asset or portfolio subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking activities the risk-free rate of return is the return of an investment with zero risk meaning it's the return investors could expect for taking no risk the yield for a u.s treasury bond for example could be used as a risk free right okay so you're you're okay so what this is essentially doing is is uh kind of uh removing what you would consider a safe investment from um from whatever you want to take and then you can have a better better understanding of the risk associated with taking that right because you could just as well put the money somewhere else like on a u.s treasury bond and then yeah generally the greater the value of the sharper region the more attractive the risk of just a return the greater the value the more attractive the risk adjusts the return [Music] return of the portfolio minus the risk-free rate divided by the standard deviation of the portfolio success return subtract the risk free rate from the return on the portfolio industry where it could be a u.s treasury rate or yield divide the result by the standard deviation of the portfolio success return um the sun division helps to show how much the portfolio's return deviates from the expected return the standard division also sheds slide on the portfolio's volatility okay okay but i guess so here's an example but i guess the uh so let me understand that so you're have the return of the portfolio and you're removing the risk-free rate so okay so if we ignore the standard deviation for for a second um that alone would be that uh you know it's the the bigger the number the bigger the num the the numerator here the um the better because that means that taking the risk is really worth it because if you know you could if the number is small it means you could as well just put the money into something that's safer that's it right like something that's risk-free um and then the standard deviation that is cringy but like let me recap that so the standard deviation is a statistic that measures the dispersion of a data set relative to its mean okay and it's calculated as the square root okay so this basically is how calm it is like um so it calculates how how much like it tells you how much it varies from from its mean values and so if it's really high it means that like it probably you know um there's like i mean it's what they say here is like there's a lot of volatility right um so the smaller the better the smaller the better yeah so you're correcting you're you're adding this correction of you're saying if it's got if it's quite stable then your number stays because your numerator numerator is big then divided by a small number um it's still a big number uh whereas if your standard deviation is big then you're gonna get like a small number um which is what you want because it means that like there's a lot of volatility and so um that is maybe not attractive right well it's an indicator for risk as well um okay makes sense now why this can't be implemented into a quantum into into a d-wave into a cubo i guess it's because of the why can't this be implemented into a cuba [Music] so what do we have here by the way this work structure as follows it will begin an exploration with a sharpie right here okay so i see here the standard deviation and then i see the ra minus rb plus rb um where beta is the ratio of covariance of a portfolio with a market over the variance of the entire market [Music] r is a return of the collection of assets rb is the risk-free return and is a standard deviation of the collection of assets w is the vector of weights for assets in our portfolio you can also see the sharpie rich in matrix form what is e though what is this e in here so you're taking w is the vector of weights off for assets in our portfolio okay so this is like okay so you're you what you're doing is trying to sharpen ratio off of of a portfolio of particular choice right and w tells you this right like it's it's the weights as in like take you know um invest everything in two of them or invest everything in like i don't know like you know do 50 50 or and then two of them so or do like 30 70 or something like this um beta is the ratio of covariance of a portfolio with a market over the variance of the entire market um i don't know why but um the collection of assets and that will use a vector of weights for assets in our portfolio we can also see the sharpiration matrix [Music] form why is this a matrix form from here develop the chicago quantum ratio which is w times the coefficients where ko i am is the covariance of the i asset against the entire market this is a slight improvement over the sharpie ratio in terms of computation as we need not consider nominal assets um risk-free investments have a near zero covariance with the entire market covariance is like how how they like differ from each other right i keep mixing these concepts up joint variability exactly the joint variability of two random variables if the greater values of one variable mainly correspond with the greater values of the other variable and the same holds for the let's say values the covariance is positive okay so it's negative it's like these positives like this it's zero it's like this no there's no so what they're saying is they're saying that risk-free portfolios have a low covariance right with the market because this means that they're totally not related to the market because they are risk-free and so they okay you could say so um [Music] near zero covariance do they though like a risk-free investment it means is an investment that it's um so where you're always winning right so yeah okay the entire market like i wouldn't i wouldn't say zero right like i really wouldn't say zero i mean it depends on the market it depends on it it really depends on yeah but okay uh we can also reformulate cqr in matrix form it's our matrix form so so but i'm missing something here it's like shouldn't there be like a summer the equation is weird because it says cov covariance i am is a coherence of the i asset against the entire market and okay so the cqr for for one acid is it like the a the sap a here means it's one acid right no it's of a collection of acids it's the the actual peak that you do okay but then there's something missing here because this should be this is a term that's like the ice acid against the entire market let's think about this for a second so so cqr is a ratchet that tells you so w is the vector of weights for assets in our portfolio and i assume the weights mean like what are we picking right so if the weight is zero it means we're not picking this one if the weight is and i guess the sum of weights must be one it's not okay it doesn't have to be one has to be below or equal to one could be bigger as well you can set investment i don't know what the weights are so but this is weight times the covariance and so what you want to do here is um you but isn't that implying that um [Music] you're never gonna peak because the risk-free investments will have a is like close to zero a covariance with the market according to their own statement right and so then you're gonna have like the cqr is gonna be zero because this ratio seems to say you're basically looking for positive covariance or like the higher i mean that's directly proportional to the covariance right it's just corrected by the standard deviation so and um and the original sharpie rachel is basically if no i think it's correct so if an asset is kind of almost risk-free the sharpie ratio will actually be really small exactly and and here and here with the chicago quantum rate is the same because if it's almost risk-free then you're going to have covariance close to zero and so the cqr will be actually close to zero okay i think it makes sense it's still i think still it's a it's a big statement to say though that covariance with the market is close to zero i mean i don't know i'm not intuitively thinking i mean if the market is doing well right then you should have positive covariance shouldn't you not zero covariance because with a risk-free investment you're always winning okay risk-free investments let's see if i can google that actually like zero better portfolio comments definition conference with the market um parents measures how stocks move together positive coverage means stocks tend to move together when the the prices go up or down the negative covariance means the stocks move opposite of each other covariance is used to measure the correlation in price moves towards of of two different stocks isn't it wrong to compare then sort of like an asset in in the whole market like isn't that actually essentially wrong to compare that like are you comparing apples and and pears so to say or apples and oranges i think it's a sentence okay but we'll but we'll now we can also reformulate cqr in matrix form uh we explore these formulations by a variety of classical methods which one we'll find uh in three both formulations are rages and this neither is negative is properly suitable for a quantum annealing solution as the wave requires a linear quadratic form we attempt to rectify this by exploring the natural logarithm what is e what is this that's the capital e have i tried to uh okay uh this however causes a different set of mathematical problems in formulating a consistent quadratic form finally we settle on the chicago quantum net score which is given by these wow okay that was a big step so um so first they attempt to rectify this by exploring uh the natural logarithm set of different mathematical problems in formulating a consistent quadratic form why i know um so what cq s is is basically for w and alpha okay so rw is a weighted portfolio and alpha is a real number the most experiments we choose an equal weighting where n is the number of assets included and which is alpha and year one these are no requirements but they do make the computations in the wave slightly easier there's a wide open question as to finding optimal weighting and optimal alphas we'll explain how to formulate a quadratic form for using d-wave in i guess this is appendix four final independence five and six will give our results visually and mathematically and discuss our future work what is it what is this is this and what like energy like i don't know what is this um so the variance of the portfolio at minus minus e where is this in the portfolio to the power of two plus alpha okay so where are these appendices if this would scroll smoother oh no this is this is not append this is like chapter four i guess the main thrust of this research is in fact how to formulate a cubo which when presented with the similar results to the classical sharpie ratio consider following the sharpie ratio as defined above in one the numerator can be expressed as a simple dot product where mu and w are the expected return and the relative weight of the eyes as respectively you expect a return and relative weight okay but that's not true right because it's not the expected return like you've got to remove you've got to remove the risk-free investment thing the denominator can be expressed as a square root of quadratic form where q sub i is binary classifying whether the eif asset is in the portfolio or not v sub i is the the the variance and co i j is a covariance term between acid i and acid j so i guess that is the formula for formula for standard deviation right like standard deviation formula formula math is fun math is fun well sorry to be late but it's fun yeah that's the square root okay um and so essentially that's the square root and then the uh covariance i always recognize that sharpie which is not a quadratic form and doesn't solve the wave system from the chicago quantum s results this problem can be presented as a quadratic form okay but they've been explained or to explain how to develop the cubo here hmm consider universe u of n assets when dealing with a single asset portfolio we only consider the linear terms in a cuba in particular when we have a lower triangular matrix or zero diagonal matrix products of the form but as we pick off only the linear terms in this case we concisely model the inverse sharpie ratio on qubits and using it and use a penalty on the couplers moving to two or more assets we have substantially more work to do looking at a single asset there are two covariance terms to deal with and we can embed the inverse sharpie ratio directly into the onto the qubits we create a unique cubo for each size portfolio evaluated by applying the weights directly to the matrix so q i and q j can remain binary we divide the linear terms by n and apply the linear f in transformation we divide the variance terms diagonal entries with application and divide the covariance terms of ml inches by n squared over duplication we then apply the quadratic fin transmission then we assemble the matrix and reverse the sine on the linear terms finally we apply a scale factor and write it into our n by n by n matrix for processing by d wave why a metrics first of all matrix oh here we go so what is this why a matrix i need to stop it since there are several types of cube mattresses i will know what i should be used to if mainly in the live hybrid uh [Music] type 2 d wave cube matrix a is upper triangular linear coefficients main diagonal terms quadratic equations upper off diagonal terms which i will try with the other one because there are languages i believe that this quadratic the dictionary must be the type three upper triangular with all diagonal entries equal to zero so i my questions are what are the correct and best types of q matrix to work in the wave applications hell john the answer to your question depends on your workflow it is possible to solve keywords directly with many d-wave tools for example device sampler through the sample curve function in this case the proper formula is upper triangular with linear terms including the diagonal okay so these are the linear terms say you have um it's been a while without using the pen i'm gonna go for some coffee does this work this work this works but i think i can't paint in here no why i can't paint why i can't paint because i can only paint on a tap but it's not this one whatever because i'm recording another browser so um okay but um so basically if we have say um q like sort of q one uh q0 q1 q2 so those are the linear terms and then this is uh so q0 right so this is zero zero um zero zero and then this is like zero one yeah i see two and then one two yeah um the rest are that's why it's just upper chamber because the other values will be the same so there's also more general way to solve i derive both cubes and as in problems if it has a better quadratic model this one separates the linear and quadratic terms in this case you can use a console function sample function ah that's what we used in in in bracket actually right i think that's all we used in bracket [Music] uh okay so this is the this is what i mean by the matrix so each matrix it's each element of the diagonal it's one of the assets right and so basically one asset is just like if you just want to pick one asset you u of n assets when dealing with single asset portfolio we only consider the linear terms in a cubo in particular when we have a lower triangular matrix or a zero diagonal matrix [Music] i'm not so sure that is entirely fair to [Music] um what are we so actually why are we checking the covariance between two assets that we peak [Music] i don't know i might be overthinking this it just goes really fast like um then we're going to express a simple dot product the ice acid is not the is the variance and the corvair the covariance the occurrence between acetone and j which is in case of like one asset that term doesn't exist and so we only care about the variance that's what this is saying so you only care about the variance and so as you're then solving a cubo that's like that your the cuba solution is basically you're trying to find for something that minimizes the the the the the energy of right so it minimizes the actual uh result of the uh of the cubo and so you're gonna go with the asset that has lowest riots are you yeah the lowest ryan's this means highest ratios right because this part here is not going to be there but what is what is cq and s finally i i hate that this goal scroll so slowly so this is synchronized so this is ryan's minus whatever the crap this is what is this e how can i find these horses how are these things called um is it like um math happily no that's not the capitalist i mean math wiki cookies accept the cookies expected value is it expected value applied mathematics oh come on commercials let's just hmm that's what makes the most sense given the context of these being statistics it's the expectation value of but i don't know where they take this from can i can this please scroll okay from here the expectation value ah so this is from here the expectation value being the average okay so that would be then the average um yeah yeah okay i mean i i i think i understand what these mean so it's the average kind of return like risk-related return right and so cqns so what they do here is they take the variance of your portfolio and minus no that wouldn't make sense minus the expectation value or minus the the whole returns like like the returns and and it's even exponential i don't understand how they get here like how do they get to this equation i don't know i don't know what the e means probably expectation but the expectation value of the weighted portfolio still we find that skeleton quadratic form how so consider universe blah blah blah so what are these things eqe [Music] um so we embed the c q and s on the d wave by writing the expected returns onto the linear terms okay so what you're doing finally so we're doing is you're embedding this by putting the expected returns onto the linear terms and then both variance and covariance onto the quadratic terms from here the d-wave inspector shows how the system encodes and embeds assets into physical keywords an attempt at changing the formulation by manually embedding terms to respect the reordering of assets does not yield substantial improvements as we increase our asset size we see that for a fully connected keyboard the wave requires multiple qubits in chains to leverage the available connections in order to crew in in other groups of camera structure increasing our portfolio size above 40 will result in increasing cubic counts utilized to support multiple chains and the potential for increased chain breaks we see consistent results with 40 assets when we tune the chain strength and scale factors so i understand this well uh it's uh basically you put the variance in the [Music] in the terms and the no sorry the expector returns into the terms and then both um uh variants and co varies in covariance of diagonal into the quadratic terms so your so what you're essentially saying is you're essentially you're essentially saying you want to maximize no minimize i mean returns expected returns expect the returns as in like uh the higher the better but you're trying to minimize that right so that's not really clear how this is done like this is super hard super harsh to actually reproduce i think when exploring portfolios of different sizes we presented different metrics to the way for each desired size portfolio we add a penalty for exploring portfolios of different sizes while maintaining accurate values for the desired portfolio size the intuition for this follows closely from converting a cuboid into an izing model in order to convert a cube into an icing model we consider transformation on the binary vector x this transforms x t q x d q x into where c is a vector of matching length and k is a constant which we can remove from consideration since we're only looking for the location of the lowest energy level and that coordinates ah blah blah blah blah definition one visualization results low one year of daily market data for a specific set of n acids i mean this is current as of that moment hold the data for experiments calculate covariance of each asset with calculate covariance of each asset with the market and beta [Music] and then three based on lock results covariance terms between assets underlying and summary values including sharpie range and chicago and escort for an all asset portfolio derive a keyword for each portfolio size visualize minimum sequence values on multiple cube mattresses shift each queue to increase likelihood of choosing portfolio with fixed number of assets dui is using appropriate range of portfolio sizes the waivers also see the genetic algorithm and compare values the following figure figures 34 give some idea how well the quantity performs yeah it almost feels like i haven't moved much forward time is it because i i basically want to try to understand the keyword but like google but i'm like not entirely sure i understand so it's definitely cqns but i don't know how what this e means and i don't know how can this be represented in in the cuba form right so simple.product okay so what they're saying here is they're saying that um yeah you can basically so you can basically the numerator is basically the expected return kind of multiplied by um you're multiplying it by each acid okay yeah mu being more i and wi being expected to and relative weight of the i fasted the denominator can express the square root of of of this quadratic form oh now we oh and i know where the two comes from okay because this is the it's it's um it's one divided by these and so uh okay so it's one divided by these and all can you do that i guess that's where i guess that's where these comes from right like they just go and say um that's where the two comes from in the uh when the formula below where is it i'm so slow i'm so slow where is it where are you yeah i don't know why these things don't scroll properly no no no no no that's not what i want ah there there that's where the two comes from is it is it i think so because or not because you want to have the inverse off but what is this e [Music] um so this is the denominator and then it's basically the this is not the portfolio it's over the the variance and so you're taking the variance and then you're also taking the cover and the cover ions of the elements q i q j [Music] so this is this is what we this one over by like you square root of if you was just like representing like you know just being normalized i guess okay so i'm not sure understand the shifts and stuff that's way more complicated than i okay so uh what i'll try to do next time i need to get some sleep i think how would the would look like for a two asset uh how would the cube would look like for like a two asset portfolio let's say we want to pick two assets okay out of 40. uh let's let's try these and then maybe we'll download the data next time and so we can we can start playing with this um because that's uh shift factor this shift factor is still what the hell what is these when exploring portfolios of different sizes represented by a different matrix to the ui for each desired size of portfolio we add a penalty for exploring portfolios of different sizes while maintaining accurate values for the desired portfolio size so you're adding something to it so you know those values kind of are not definitely close to the lower values of energy you're looking for i understand that but it feels okay because you want to avoid you want to avoid that it peaks more but like if you just want to want to be two then why do you need it why do you even need these why can't you just have like the two terms okay now because you need to have so you need to have all like cues like the key from q 0 to q like 39 because you need to pick two out of all those but you you want to have a constraint that tells you just pick two and so you have to limit so you have to add shifts okay i get it so you have to achieve i think i start to get a picture of these so i have to add something that makes those solutions um more expensive uh okay um [Music] what i don't fully get is the uh i don't forget the uh oh you hate scrolling i should just use a pdf freedom and it's just cool whatever i can't find the i just find this file next time next time i think i have to go sadly yeah it's been a long video um but i think i i i started to get a start to get an idea of the cue of the shape of the cue ball um so you're basically uh you're basically having uh the expected risk the expected returns are the coefficients of the um linear terms and then the quadratic terms or the terms i guess i guess those are the quadratic terms where like you could see the two of those um see i'm not i'm not entirely entirely sure i get the whole cube thing yet but i i i yeah yeah so those are the quadratic terms i guess so in this case the terms is the sum of the variance and the cover and the difference of acid zero wait a second why would it be the variance of acid zero i guess it's the variance of each asset um the variance of each acid plus i guess that's what you want the variance of each acid plus the uh covariance this will mean that you're penalizing by assets that like behave differently and i don't know if that's a good because you have a high covariance it means that they behave really similarly and so if you have a minus it means they behave opposite and and and i don't so what's the standard deviation and it was the volatility right uh of your all of your all ah yeah okay of you of your own portfolio so you wanna prefer that's not really volatile but this means that it always kind of okay so you want you want assets that between them they have like a low covariance right right because it's a high covariance and then you can kind of gonna have like a lot of volatility in there i guess so i guess so but okay i guess next next step will be get the data try to play with it just go hard on these uh because otherwise i can just spend hours trying to understand that but like not make make any progress so um the embedding part then then we'll take a look at the transformations later on okay it i mean it doesn't seem yes to me right it's just a really um at the end of the day it's like a model right so you have a model you're building a model and then you're basically i guess the the what you're doing is you've it's like the the quantum computer is allowing you somewhat take into consideration right like all the 40 different assets in their combinations which is a number that you it's intractable with a classical computer it's it's it's interactable to to brute force but still once you have the cue ball you're looking for the the values like minimize the energy and and you're hoping to achieve this by annealing so basically by naturally evolving that and so because your navy is it's so you're relying on nature to give you that minimal value okay i guess i think i think i should leave it here um it's been a long video and i i don't feel i've moved forward a lot but um maybe |
so now we are in part five is there is no summer in here good ah what I see okay I see I wanna say right so good places here and here I've got minuses okay so those but we have information stored in the Torah code which is a great noisy cubits there are the bad things can happen is if a lot of bits of bit flips happen if it but the girls already the kids get better who have no way of detecting or cracking these so they will build up over time and soon miss our logical qubits okay how can we manage them with me we have a rule for each white square each must have an even number of ones around it by looking at which of these rules are broken we can work out how to get rid of the bit flip errors but let's forget about these rules for a while instead we will come up with some new rules for blue plug ins and States from last lines with this we can declare rule for each blue squares when looked at it in terms of pluses minuses each must have an even number of minuses and therefore whatever number of places there are many possible patterns of positiveness is that obey these rules here if you yeah this should look pretty friendly in there the same ones as before but with zero a place with classes in 1 2 - what squares was Brewin everything turn 90 degrees now we go off and get a cup of tea and agrimony season for some mischief it measures one of the kids to see if it's zero I want every cue it here is plus or minus both of which are superb isn't one so the measure key will randomly choose one of the other let's say for the sake of argument chooses zero when we get back from our tea break we can check whether the blue squares are following the rules okay but what about too near the measured cubed well of course we're asking them to look at how many of the qubits are instead plus and minus and tell us what - this is already even for most of us which is simply a plus 1 this is an easy job but the measured one is in state zero superposition of plus and minuses as to this I was - before the question can be answered if it decides to be plus and is which is what is give it was originally the rules are abating this squares the qubit has also been forced into a state there is nothing like the zero state if the size to be the opposite of what it was originally the effect is similar to a bit flip except this kind of flip turns plus minus minus in the plans we can call this phase flips yeah so this is okay so yeah of course it's obvious well it's not have to seem it's obvious so far because I keep having that problem with the blocks here where the said rotations at the face but it doesn't necessarily surface I get it that it doesn't matter what the global face is and yeah that's rotation that's like between the minus in the plus there is a decrease so there is like the face basically causes the rules to be broken in the two blue squares but just like with me flips and this is to detect and correct the errors also the cubed is again forced into a state that has nothing to do with one of the Kremlin measured this was protecting the code crying the air this was protecting the code for measurements is exactly protecting from bit flips a few measurements here in there won't cause big problem we can just attack them remove their effects and update the state to make sure their results can tell anyone anything about our logical qubit yeah the thing is in I kind of make sense because if you're just prettier than once and you don't care whether they get measured because that's what they are if you put them into pluses or minuses then you do care and the trick is if you're if you see a zero it means it means that from that from the perspective of a plus and minus so the x-axis when you try to measure a dad then you were gonna get a run the result if for the case where there's an error an error which is in this case one anyway yeah exactly combining the rules we now have a set of rules for dealing with beef lips and a set of for dealing with face lips yeah but we really need them to work together so we can deal with both at the same time getting things to work together in cotton because it's not always possible so suppose we have a single chameleon and the only way to make it always give us the result zero four zero one measurement as rule I requires to have it in state zero but that is plastic is the result of the time we do a plus -2 similarly the plus state is only one that always pays to but it breaks wanted the rule I have if but there is a big difference between Rosa and I I and the rules we use enter a code there is no just one big state that satisfies the walls for the squares but there are many patterns of zeros and ones that do it but there is victims using rules there's not just one state that satisfies the rules for the white squares and anything to do this for positions with some possibilities there's hope any state can be thought of either in terms of season ones or in terms of minuses just like a single cubed what about figures where positions so I mean what this is basically telling us is so yeah it's not sighs it's not just so nice both sets of so they're just simple patterns one depicts position to make sure that the rules can work well together in fact for illogical zero Nia's for vision of every possible set of this once this could be used for a logical zero which is a lot in this case making such a complicated set of correlative qubits is going to be quite hard that's why you can't buy toric code from the shops with both sets of rules working well together we just need to keep checking on both types of squares so we see white rules broken we know that the beat flip has happened and we see blue rules broken we detect the correct face flips as long as unlikely things don't happen yeah but you cannot have both both things what is the meaning of having both types of cubed States mixed in one greed if I have zeros ones plusses and minuses because I can still be but it becomes complicated how to map that grid to any possible cubed state that just doesn't seem like a reasonable direction to me so far so obvious oom that the only kind of bad things like having a bee flew in on what a measurement season once but what about face flips or unwanted measurements of pluses and minuses yay our code naturally sorts this out - what about those other crazy week random stuff it turns out there anything that could ever happen can be thought of as some combination of bit and face flips so whatever happens once wishes that rule were being the rules are being obeyed the noise gets for a force to decide what kind of beaten face repiy wants to be then we detect and correct those in knowledge wallet Oracle okay so high level high level makes sense well perhaps it doesn't protect from everything big harbors can still happen to have a blob long time mmm our logical qubit will last a long time and if they are not rare enough we just wake or just make our code bigger so they need to be ever bigger and more horrible before they can mess up so that's the Torah code in five parts next we'll move on to its flat brother the planner code but first in this in this post we take a look at some of the maths behind all these I try to keep it with little maths as possible but it did turn out to be a bit long there's a little bit of repetition there okay let me just check it out quickly fun measurements four pairs of qubits huge measurements see what I'm gonna go through all that I think I'm not gonna go through all that fun okay so that's this one here so is it is it the next part and the planner code so that was the toric code because of the tour the touring geometric figure but the planner code is what we're gonna get to I mean that's what I was what I read somewhere is that the that some feared the planner code works as well and it's it's a simpler model for that but it still that doesn't answer distill dysphoric code doesn't this introduction to the toric code didn't answer how do you perform operations and these are you'd you know in the gate how do you do a set rotation how are you all this kind of stuff and how do you represent a state of a qubit that is neither as you or one or plus or minus so maybe the planar code is easier and then that kind of makes sense let's see this is from 2016 though maybe should find some more reason I mean I don't know how much more has happened in three years since is a research for this but so what is the next one next one next one flat brother planar code is this is this entanglement with a pasta with no with the simplest math possible doesn't look like that's not it's not it I'm just trying to find it before switch okay male just do this out of the video so a couple of lot of things crime interaction the puzzle see if I as if I find this next part okay cool that's it so far |
okay so we're gonna have to be um go back to the old setup for one video because that kind of give up my screen for for tonight um and i hope the audio is gonna be fine as well i'm somewhere else but should be fine um cool and actually i just wanted to use the chance to um explain a little bit what i kind of been thinking about in the past couple hours because i just realized that like what we're doing here really it seems really similar to vqe right because at the end of the day like i was totally lost but i just tweeted this a couple hours ago i kind of realized that uh that really the mps is actually it's actually the quantum state right um where is the where was that that was the part everything so notice how so here here this is introduced as in like um p is the the the level of the system so you can have two levels in the normal quantum computer right like a zero and one two levels and so forth all the speeds right like um n is the number of cubies and so you kind of have really a uh rank and tensor which is what is what i was kind of suspecting last time like i know you you usually write in in the standard that you see in terms of secret circuitry in quantum computing stuff is that you see just like a big array with all like for example you have three qubits like you just see you know a complete array with like all the the eight possibilities um in an array and and you use the column array um or use that like yeah basically that not the cat exactly that can't um as a way to represent the state but like you can essentially do that as well in a like in the form of like an um a rank end tensor where literally like you use let me see if i can open anaconda so actually let's go to let's go to workspace and let's go to powerful optimization and start jupyter because i wanted to kind of play a bit with the code here but essentially each index of the tensor is kind of makes reference to one to a qubit and actually kind of have two components right it will have it will have the not this one it will have a component for because i was playing with these so we'll have a component for um maybe this is what the components mean here so we'll have the component for let's say this is my matrix right like it's going to have if we have three qubits you're going to have um the three indices right so you know these these would be qbh0 right if we index them like that let me comment these right and so and so forth so you know this would be like and i think it's the oh the second is the amplitude of now is it i don't think that's and that's gonna be is this gonna be an amplitude i think so so you have three qubits and then you kind of index them like that i think that's the i think that's the trick right um so that kind of it was confusing because it moves away from the standard way of representing circuits right so that's what you're saying these are the coefficients of each of the you know this i1 i2 until i n those are the kind of computational bases right for for your and cubic state exactly and so it's it's basically saying that it's going to have p to the n numbers and and this is the complex number it's the actual amplitude um describe the wave function psi and then can be understood there's a coefficient of a tensor c with n indices exactly where each of the instances can take up to p different values yeah so a little bit basically because that's kind of it's it's the level right so that makes sense cool and and then i kind of realized that i think what you're doing if i was reading this well because it's like a optimization the position of the first bond so to optimize the first bond tensor a range of procedures can be used but amongst the most effective uh are iterative algorithms such as davidson or lanzo's algorithm so what are these algorithms um client source let's see what is this doing it's a direct algorithm device by criticizing this adaptation of power methods to find the m most useful i convert is either vectors of an n by n have reached a matrix uh maybe i would beat rock i'm not so sure i thought what you're doing is you're because this is some there's i had the impression i read before like yesterday that there was some kind of variational part to these two neighboring mps dances at a time to just perform using the truth if i can solve our approach before the next step single tensors factorize using svt or density metrics composition or to restore the mps form the bond dimension can be adapted okay but but like what's the starting point right like isn't because even these is like for diagram to be efficient h must have some simplifying properties for example h could be the sum of local terms and this is i think the same that you kind of find with vqe where your your you know if your hamiltonian if it can be decomposed like so there's a limitation yeah so if the hamiltonians can be constructed as a sum of poly operators then and and there's tensor products that for example we can have the following hamiltonian right so that's exactly the same idea right you measure for each component and then you add the results up and i think that's exactly the same that is being explained here if your hamiltonian is just the sum of of tensor products of things it's the sum of you know yeah basically of smaller tenses then yeah it's an npo tencent network and so and here's like the mpo is an actual matrix product operator so and that kind of i think confirms what i'm what i was suspecting that this is going to be this is actually going to be like mps is really just a state and that's that's that's an operator and but let's try okay so i i don't know if i explained myself really well to be honest but i think that's that those thoughts clarified a bit the next steps but i want to really get started with these i i don't even know like i was aiming to start with step zero but i got a bit lost and i'm trying to understand before so before you begin with the mrg it's imperative to bring the initial mps into an orthogonal form but what was the initial nps like because that's what you would um that's what you uh you would just start with a random one or something that's what you do with vqe right you start with it you start with a random anzac and then you optimize the answer so that you kind of minimize the eigenvalue you minimize expectation value and so you kind of both find um you both find the uh the eigenvalue and the eigenvector maybe i should just read farther here so can i is this dmrg explained somewhere so the feminine is probably the most famous example of the in states because it is behind some very powerful methods to simulate one dimensional quantum many bodies is most permanently density it's referenced in here it's also behind on the well-known methods okay so where is this mentioned here this is an algorithm from 1992. that's nuts i doubt this is going to be accessible i i i bet that's paid access pdf yeah whatever i really have to figure that out then maybe i'll this will just be the very first easy the accessible tutorial out there um finding ground states version optimization and maybe maybe it's explained here that's relational optimization but it's not the mrg no a few remarks are in there first if we perform the systematization for an mps with open boundary conditions then what we'll recover is nothing but the famous dense engine position group algorithm and the language of mps is particularly appealing since using the npsp pictures okay so basically it is really given a hamilton age generational principle states that for a given quantum state psi it will always be the case that blah blah blah and so so the expectation value basically um therefore if science tensor network belonging to some family of states like an mps or a peps then with a fixed bond dimension we can always approach the ground's energy from above by minimizing this expectation value over the relevant family where by family we mean either nps or p8 or peps or any other family of tn state with bond dimension d and where lambda is a crunch multiplier that introduces the constraint that psi must have known 1. oh this liquid is thinking again ideally the above minimization should not be done simultaneously over all the three parameters of the tensor microstate exactly hence over all the coefficients of all the tenses for all sides however it is normally quite difficult to implement in our particular efficient instead is to proceed tensor by tensor this is to minimize with respect to one tensor while keeping the others fixed then move to another tensor and repeat the minimization and so on in practice one sweeps over all the tenses several times until the desired convergence in expectation values is attained the way to minimize with respect to one tensor works as follow imagine that we fixed all the tensors with the end except one of them which we call a the coefficients and that's what's gonna that's that's going to be the that's your answer um the coefficient i mean that's the way you're yeah so that's the one you're kind of going to be optimizing it's pretty smart so this is so you're going to fix the rest the coefficients of a are now our variational parameters okay so and and yeah because the tensor will be probably just be the small tensor will just be off dimension two so it's going to be it's just going to be like a two by two matrix probably right then the immunization is written like these in the above equation it's just a tensor but it's just a tensor a but with all its coefficients arranged as a vector so this is so this is like yeah so you're reshaping the whole thing so you can calculate these what is hf do i have to have this lagrangian thing in here ah so those are correspondence and networks for these and these but without tensors that is the environment of tensors a but rhythmic matrix form so you oh god this minimization can be done like that so whatever minimizer you put in there which leads to the generalized eigenvalue problem numerically by using starting algebra packages and to calculate both hf and n1 must use the tools explained in the previous section to compute expectation values you can calculate both hf and n one must use the tools explained in the previous section to compute expectation values of physical observables and effective environments in a tn in practice this means that both hf and m can be computed efficiently and exactly for an nps whereas for perhaps this calculation is sufficient but if we perform the optimization for an nps with open boundary conditions then we'll recover is nothing but the famous density matrix normalization group algorithm yeah so this is this is the dmrg essentially a tensor can be transformed into a vector by moving and grouping its indices i'm fancy okay i get it because you get the normalization matrix n so what this is saying literally is like you know take so you're optimizing tensor by tensor like the small tensors uh-huh so these are fixed i guess that's why they are squared or was it the look oh or this is this writer's sub analogy thing i can depart the mps tensor numbers as a change of bases from the basis and the bond index like that this interpretation motivates transforming the matrix h into the following diagram if we take h to be an npr form we can compute the transmission efficiently defining the rj tenses along the way for efficiency this question that the edge tensors be created by contracting each amp as an npo tensor one at a time in a certain order as follows next combine the first two nps tensors by contracting over the shared bond index so you have these at this point it's have to observe that the bond tensor does not contain just partial information but defense represented by the mps rather than is it is the entire tensor eigenvector of h just written in the bases if the alpha 2 base expands sub space after whatever vector of h we seek them to measure the vector i don't understand this okay to optimize the first bond tensor a range of procedures can be used for example gradient descent ah or conjugate gradient we're going to stick with great in descent but among the most effective or iterative algorithms which is davidson alliance algorithm so okay we'll just stick with gradient descent the key step of each of these is the multiplication of b12 by the transform matrix h using the projected form of age in terms of rj tensors defined above i'm getting lost even with the diagrams i think so what this means is we should what i guess is we have to come up like with a we should try a small example right so let's try a small example insert cell below we want to find the lowest eigenvalue of like a simple matrix um ah damn it i mean we could even use these okay we could even use these so let's let's use masty thoughts example so let's imagine this is the hamiltonian there's only one cubit okay so the hamilton is two times that plus x plus i it is with the following matrix so this is our hamiltonian cool let's let's uh let's do this so uh how do i even start i got numpy so our hamiltonian is basically array of like what did i say was three one y minus one okay three one one minus one is it not defined um i'm gonna keep these things just in case and now what still running yeah that's nothing uh it's just like that no what is the syntax why isn't this working oh there you go so this is this is h now um it's just one qubit right so uh basically but it's just one cubit that should be actually fairly simple i wanted to kind of have too many cubes um well one could be illustrative enough so if i take one qubit and that's the that's the hamiltonian and now i want to have so sort of basically the mps right so the nps is just really in this case one tensor isn't it i want to have two key beats i really want to have to keep it so how can i make up an example can i just make up an example i just want to make sure that i don't uh let's do a cubic handle because i i don't want to get like i don't want to get something wrong in here or or a matrix where this wouldn't make any sense i think the vqls hamiltonians i said there are some lectures actually i just want to have an example i should read this probably that actually seems quite useful let me bookmark this well let's just i don't know let's just make up like or what if i just kind of like have uh basically the same but like somehow is that a valid hamiltonian and now i basically uh kind of copy the same but like with the zeros here i guess that should be a valid one because this time taking the same thing here and then just having uh zeros in the other places not at all uh zero zero minus one and then one minus one okay so you can have these now whatever whether this is correct or not we'll see so we have two qubits and now basically would have like we kind of should have uh the the mps kind of basically have to be two tensors so what i should what i could do is i could actually use these functions right because this is okay so this create tensor of dimensions and rank what is the difference dimensions one of the dimensions then here great tensor create a block because i have new metrics for the mps with random numbers from 0 to 1. i know that just creates tensor with random integers what is dimension though okay i think we don't need these so we can i think what does that mention is for here dimensions dimensions for dimensions 2 to 40 so it creates a temple dimensions least ah it's just the name of the dimensions right but it's kind of i don't really memory cost uh whatever uh cell i feel like i should understand this but i also feel like i don't want to waste time with these dimension and then sample dimensions list what is this actually two by two three by three four by four five by five so the rank all the dimensions is the p i guess so that's rank two but then it's like the actual dimensions okay i get it yeah but this is not needed so cell edit delete sounds cool so all we do here is create a block so this is dimensions would be two create an nps of rank two i guess that's what you're doing and so it creates a block of dimension two with bond dimension what happens so dimensions 292 ranks 2 950. that's all i want to know that's all i need to know this is all i need to know this is gonna basically delete this stuff because i've retrieved i'm gonna keep this for a second so let's say i want to build like an mps of rank so we have two qubits right so we have two qubits that give us two two bits okay so we have basically a rank so team is the kind of the dimensions right so this is two as well we're working with cubiets what is the bond dimension i don't know so what is the pawn dimension um where did i take this from actually i took it from so let's maybe one dimension what's the first time it appeared where is the first time it appears it's since there is called bond okay so now we're in different values that h1 is different measure of the amount of quantum correlations in the wave function um these new degrees of freedom like i said you're creating the tensor network different dna blocks these new degrees of freedom are represented by the connecting indices amongst the tensors it's a network the connecting this is turned out to have an important dimension an important physical meaning they represent the structure of the many body entanglement in the quantum state and the number of different values that each one of these indices can take is a quantity quantitative the number of different values that each of these indices can take is a complete measure of the amount of quantum correlations in the wave function this is called bond or insulin indices and the number of possible values that refers to the bond dimensions the maximum of these values which we called above these also called the one-dimensional tensor network terms in better and such that okay well what should be the bond dimension for a cubiet there is no tanglement is present in the wave function then no entanglement is present well that's interesting so if the bond dimension is simple okay this is of the tensor so actually it seems like by design you're already kind of saying well the way you're going to connect these and this is you're already going to be fine with this some sort of entanglement there or not it's not some dt overall tenses where i assume that the rank of instance is bounded by a constant the d is the bond dimension i think here they assumed because this is carrying the external indices are all right or floating on this the pawn foreign can be adapted this adaptation is optimal i said the bond dimension so that's something else you can adapt the one dimension or tensor train rank of the mps can be adapted this adaptation is optimal in the sense of preserving the distance between the network just after the optimization step and then i work with the restored nps form so this is also something else you can actually parametrize where you're parameterizing this means you're basically defining the entanglement of your anzad's this way it's a bit mind it's a bit it's a bit difficult to grasp i think oh okay this is the typical set type of answers that is used yeah this is so heavy that i'm just like too lazy to read through all of these if d equals one then the upper bound says sl equals zero so no matter the size of the block that is no entanglement is present in the wave function this is a genetic resolve for any tn if the bond dimensions are trivial the entanglement is present in the wave function the end state is just a product state uh this is the type of this is the type of answers that is used in mean field theory second for any d or bigger than one we have that the answers can already handle any area of the entanglement entropy changing the one dimension d modifies only the multiplicative factor of the area law therefore in order to modify the scaling with l one should change the geometric pattern of the tensor network this is just way deeper than i thought this means that the entanglement in the tn is a consequence of both d but also the geometric pattern oh wow okay really okay let's take a look at what happens if i just say pawn dimension one no entanglement between the qubits and so i create the mps and now i say nps nodes what do i get so i get this node the dangling edge it's it's distancer so can i just say so for tensor in for node in nps nodes print node tensor note uh so api docs so i get the tensor network so intense network how do i get the actual tensor tn probably i have to actually do nothing connecting what's the what's the second i don't want the right one to reference okay that was the introduction i want i want to be able to paint these okay so that's this bonds the bond stuff in here so and so blah blah physical indices the new horizon lines are called ancillary indices with physical answering this isn't sign being labeled respectively um in the product of nps's i just wanted to not want these uh basically pair tutorial no i want to get the reference documentation oh that's it here actually tensor i want to find a tensor and i want to find so get a note and get the tensor parameter tensor get tensor let's get tensor okay so this is then two tens just look like that if you drag the mps like these because when i uh if i take a look at for edges for edge in mps edges print edge what does the edge look like actually upon dimension see our contracting if i i don't just to do that what i want to do is i want to actually doesn't print so what's the actual length of these print oh actually doesn't go in here so basically there's no edges right probably zero why is that possible rank two so i have two qubits or isn't this the the way you do it the rank of the yeah well of course that will not create any note in between because it would just be a y minus two creating new metrics for the mps numbers build the nps sensor i'm confused now so what i should create if i'm working with two cubits so if i work with three cubics maybe that's just like dimension two and then you know whatever we'll just fix that and now this one edge so what if i print this edge it's a named edge obviously if i print the tensors now we have so the last tensor always looks like these because what you want to do is if you contract the if you do like if you contract the edges this edge and then you print an a it's got two dangling edges so two dimensions now what is the result of it's 2x2 so it's a 2kb is it i'm just struggling with the mapping you know that's that's kind of what i'm what i'm realizing but i'm struggling with the mapping i don't really understand how to map these so you have bring the weapon into small pieces maybe maybe this will be explained i mean this is the part that i write yeah that's the part i've already read so figure empty is the number of parameters it tends here this is just talking about the parameters i don't really care right now i just want to have the replacement of tensor c by t involves the appearance of extra degrees of freedom in the system which are responsible for gluing the different dna blocks together these new degrees of freedom represent by the connecting indices most tenses in the tensor network yeah that's also what i just read okay so here they're gonna try and understand me better what is this entanglement how you understand this entire one that's definitely worth taking a look at um i just wanted to move forward with this small example but i'm not really understanding what i have here so because i want to have three cubies right okay but by their but their rank of the tensor one dimension is one dimensions is two and if when i have two qubits the rank of mps should be what should it be maybe the rank of the end maybe it should be eight because at the end of the day you wanna have one two three four five six seven eight now then you're gonna have three there's two tenses and there's no edge connecting them that just makes absolutely no sense what is the rank of the nps state awesome mps one two three four four side nps with open boundary conditions if i want to have so build an nps tensor so you start with the node in here with two dimensions and then one dimension then you have an actual node in between oh look at these okay so you're going to create a node for e this is where rank is used so if you want rank to be that's the price your rank to be the rank to be two then just know but in the rank to be one there's gonna be one thing here and then to build the mps so okay but there should be an edge right because you go from one until rank minus one okay probably not connected edges so rank just this this just tells you so with two sides i'm really lost that's not what i'm looking for it's definitely what i'm looking for so this gives me two notes but no edge and that that's confusing two notes but no edge why why there's no edge let's give you one that's q2 whereas if i say rank 3 it gives me three notes and one edge that is kind of confusing is the edge between these two nodes and what is this thing here that was this basic tutorial here right so so we begin by creating directly the node structure of the api so let's define functions to build each block and then the mps itself create fps uh physical dimension 2 and rank 20 with 20 cubits so if i give it rank 20 dimension two bond dimension two ah so what if i play let me play with the pawn dimension rank two one dimension two oh that gives me two notes first of all that gives me print length of mps nodes and the length of mps edges steals your edges upon dimension 2 will go so i really have to have if i have three qubits then i get two why you get one edge i don't understand that i should be getting two in what is the mps edges first of all actually connected edges so you create a node then you're creating sort of rank minus two nodes and then the final node and then you take connected edges and you're connecting the zero one dimensions with the one zero dimension so you're literally doing at least you're adding one connection right ah there's a bag in here it's not adding it it's not adding it like it's definitely not adding it like i think that i think that's the way this should look like uh what am i doing c is not defined what have i added here the c that's not what i want okay i don't care about these so what do you mean this is for array split cell delete cells yeah that makes sense okay that makes more sense because now if i say rank two dimension tool yeah that makes sense okay leave it here i think there's a body okay so i don't know how can i how can i report that maybe twitter or is there stay away from you how to contribute so it should open a ticket or something code reviews country real license i have no idea i've never contributed to this but i think that is definitely a bug because you're creating the connection but you're not you're not adding it to the connected edges yeah that's definitely back okay but now that makes sense and so now i have these edges and these connections okay so if i take a look at the if i now do four nps f4 node in mps notes print node get tenses get tensor sorry then i have these two tensors and then if i do the same for the edges i'll get the edges right edge and mps edges print the edge h okay tenseness can i do this no get whatever but there's one edge right so that makes sense and what do i do next so now i have the tensor network what is the next step of the algorithm i got lost i got lost so now i have the hamiltonian which i should kind of be able to have in these form of an npo how do i get so i have my hamiltonian i i want to have my hamiltonian now in an mpo form can i can i get that that doesn't even appear here but i think that if i do matrix product states i would kind of have to now so by some local terms or means suggest like you know these so as an npo how do i so here's an example maybe is this even is there anything in here about mpos no i don't think so matrix product operator example whatever so this would be something expressed like that expectation value that's another library though so you should the next step is basically get the empties in mpo form if i do i guess if i just doing like what was the example here i think there was an example quick start was here in github if i do these these basically so i basically do these which i have done it i had done here right fine get things blah blah blah left edges uh-huh so that's what i that's that's the way that i was doing node 0.1 i don't know so if i so this has got two dimensions if i split these no that's not like if i do the the sve whatever if i split this with the singular value of the composition svd i'm not going to get an npo sorry the knot again an npr um that's kind of my next step and then i can get on with the algorithm actually cool for today |
horrifying example and some um uh but i don't think i'm gonna find anything norma normalization uh so normalization wave function 3d space oh okay look at this maybe maybe normalization of wave function in 1d and 3d maybe that helps actually that one of the most is thing is how we're going to be able to discover our particles in a specific space drawn as this over here where we have a particular curve over here and we're going to target this possible region where position is flavored as x however at a region now the probability of finding a particle because function was this is for all space over there yeah now let's look at how we'll be able to start the normalization process over here now for normalization the probability of finding a particle in a region which is dx in this case in one dimension is actually equal to the product of a normalized constant multiplied by a complex conjugate alongside the normalization normalized constant multiplied by our wave function all with respect to dx since you are all involved or this is applied to a one dimensional system now the sum of the individual probabilities must be equal to one so much must be speaking what we are trying to say is that so probably he's gonna do the three dimensions separately which is the square of your wave function your square of the wave function must be equal to the normalization of your complex conjugate okay that's so we've done writing in this three-dimensional system is written as this long given i think quantum mechanics terminal plus conjugate and the wave function with respect to x or square root over there so the end will be determined in order to normalize how we remember okay so it does so so it's just dimensional system over here now this particular segment of coordinates over here are all equal to r over there and once you're able to solve this for our three dimension then you can be able to apply that for three dimensions while on the other side okay i mean yeah why not we could try it i mean we i don't know i don't know if i thought i had tried that in the past um instead of just playing with with polar coordinates and what not i thought i had tried that in the past i mean i i assumed in three so three this space this uh distance is like the the distance it's just the square root of and then uh yeah exactly so that's the origin so it's it's this thing here okay so that's r basically i mean yeah why not we can try that and and so we can do this basically cartesian um and so basically you know our s1 is so this is basically square square root of x squared plus y squared plus z squared okay so here we actually define basically let's see let's just define these things again n a uh we don't define define r and all these status and stuff like that so we just define um we just defined can we do x y z uh and then define x y is that i think you can do this they're real and they're real but they don't have to be positive um so and i hope that is yeah and then what we're doing is we're integrating from minus infinite to plus infinite and so we're we're yeah okay but the thing is probably this is i don't know so this guy just literally does this like that yeah and so basically uh what we're doing is s1 so s1 is the integration of s01 and then first we we do x and we do minus infinity to infinite okay uh what the is shouting like this man then we do first x and white and z for example so i mean you know that would that would have been quite quite stupid if that works out then this is why i think this might just be complicated three and this is z and s1 s1 and then s2 then s3 so once i have everything then i solve for then i uh then we solve for you know this and then times n and the conjugate of n minus 1 so i print s4 will that work out i don't know at this point i'm already just desperate enough that python exercise 2 [Music] this might just take a long time what is these or do i just define them separately maybe um although i'm sure that that's not necessarily half or or you know the solution is really alien because it has to because it has this pie i forgot almost i think i must have tried that at some point yeah it must have tried that at some point the thing is i don't know why it doesn't work um i don't know why it doesn't work but i i actually now realize the the reason i was i abandoned that idea is because of pi being in the solution which kind of i was like where do they even take that from um and so and so i'm basically stuck at the same uh same at the same step i'm very stuck at the same step what have we tried we've tried the we've tried to this space we've tried to displace and that almost gives me what i want because the solution is uh uh and again i'll just probably have like 10 more minutes i just wanted to try this one because i i thought maybe i hadn't tried that before but i did i did have i think the solution is this dish a t0 um i'm sure i'm sure it's uh i'm sure it should be it should be this but then this also doesn't give me exactly what i want what's more it's kind of even worse in a way it's like what are the boundaries and this constants in the middle r is definitely bounded from zero to infinite hmm cylindrical okay so cylindrical coordinates okay that's just cylindrical coordinates right no n-dimensional syndrome coordinates on my contribution routine especially in finding narrow peaks in small the integrands this was using multiples of the monte carlo normalized function and constructing a normalized function r3 to r which speaks in cartesian uh polar and some difference of coordinates it consists of lines and peaks not normalized as they cannot be normalized in higher dimensions for radio components if requirements regardless what the hell um question for n dimensions n terms with learning but i also need n dimensional cylindrical coordinates in 3d okay so is there anything that's what i'm interested in generalized cylindrical coordinates for n dimensions this is possible to just use spherical kernels and dimensions and just attach an orthogonal cartesian coordinates to it like in 4d that seems to be like something that i should they may try to understand super useful from an abstract perspective but i'm also like it's too much time what if it's a halocrine assistant great there's no answer how do you mean with this very electronic system that's almost this almost seems like too crazy for that exercise that's my problem 2d space is so much more comfortable and it makes much so much more sense but it doesn't give me the exact it doesn't give me the solution that i that i want it tells me just spherical spherical cylindrical integration what's more cylindrical integration would be even easier right into the because then you're saying it's r so you start with uh integrating ah r that goes from zero to s one my integral is z sorry then you integrate or and what was that giving me yeah i have to go ah that's that's far from far from optimal far from good actually um uh sparkle coordinates and that's cool petra thing khan academy so this is basically the bottom of this yeah and so essentially this thing is like r squared times sine is a sine is a phi that's why r squared times sinus times this and so this is what you would do the r ether and this that is what finally here in this picture is this because the other one is psy correct yeah i'm sorry oh no this guy the long is this one and we first do this because we are this so if we do it this way then it's okay uh phi and theta okay so phi and so here we do the function times this with the radius then we do five and yeah that is it actually is not a fight five house it's really fine right okay because i know because i know that it can't be i can yeah i can i can i can yeah that is fine divided by two because you just go because you know that you're not gonna get negative values that is never gonna have negative values and so you do these and then and then two pi for theta yeah that is that's that's what it seems like it should be but that's what's giving me this crown man and if i uh oh that's giving me a better result okay yes sorry let's give me a good result i mean if i if i think if i do that then i'm gonna get something bigger than that okay but look i'm not getting the coefficient so that's actually better in a way ah almost there somehow almost there or isn't it the [Music] saw where's the thing here so this is uh am i stupid where am i almost there it's kind of almost there a positive true if we don't say what happens if we change some of these preconditions and i'll have to drop guys i'm sorry does it change anything okay okay okay there's something there there's something there there's definitely something there yeah i mean i don't know it sucks without there's a lot of stuff missing here but anyway anyway see you next time |
okay so actually i i um i just had a chat with uh uh with couple people like with someone in in my discord about this mathematical proof and like um you know sort of good practices in general around proving things like these and and we kind of basically he pointed out to a very simple thing and a way to prove this which is even simpler than what i'm trying to do here and i just wanted to kind of um you know uh put it get it in video right away because i will still play with this a little bit now but i think i'm gonna leave it here uh basically uh i don't even know if it's yeah i'll do i'll do an executive summary video because brian wanted that so i'll do i'll do a quick one explaining um instead of what's in the paper what's in the article and like uh and and why uh kind of this makes no sense so there's a simple way to prove this basically and and you can see so i can write this here right uh so the the essence comes to basically establishing the fact that uh can i write with like can i do something like uh insert i want i want a text cell cell type 7 markdown now okay cool so it basically comes down to saying you know so this is the simplest proof to be honest simplest proof so um this is working simple as proof i mean simplest arnold will be the simplest but um oh call it simple proof basically the uh so the swap test outputs zero with probability hundred percent only when the states being compared are equal regardless of how the states were created and regardless of the states um and regardless of the of the number of qubits and and we talked a little bit about as well the complexity like what could be a complexity for these right and um the swap test i my guess i'm i'm sharing the thoughts with the people in the discord here's my my uh the swap test has a complexity of like o to the n um in terms so i'm saying this because you have um like you you gotta do like one comparison per each right so at least gotta have like these n steps um but you should add someone pointed out in the discord p in here like p being so p being [Music] related to the amount of iterations you you run to basically in order to establish the probability because because that's what you want to do right you want to you want to see what what the you need to like akka amount of shots basically so now but the important part is the first one the slope test outputs 0 with probability 100 only when the state being compared the states we converted equal now this means if we would if if we would know up front that the states are equal then why do we why do we even want to run a swap test right um so and and what's more we know that that the swoop test tells us kind of kind of a little bit of a how equal are they right and so well basically yeah if you construct the states like that like you use the same types of gates and you're kind of you're kind of in this case putting the angles in the proper like places right so data zero the angles you're putting in here are the same ones that you would put for the test zero and in terms of not the same angles but like the same um feature right here in this case each qubit was i think encoding two of these average values that we talked about because i think that he was using brian was using two angles well then yeah i mean you're just using u3 gates it's the same type of gate so if the angles are equal if the angles you're using here are equal then you're kind of going to have like 100 because you're you're basically creating the same like two states right so the two states are the same it's just by definition of the fact that these gates will have the same matrix right so it's trivial i mean i i kind of feel like stupid a bit because i spent a lot of time digging into these but it was helpful but i think it's i think that that makes sense i mean it's just it's just kind of trivial if we have the same the same angles then we have the same gates assuming we are using same gates to construct the states being compared therefore this means therefore this means that um so if we have the same angles then we have the same gates if this means before this means that if therefore this means that if we compare the angles we can well what's more like and and you know that that particular gate like the u3 gate or the r like ry gate this it's just a one to one there's no way if you assume the parameters can go from 0 to pi there's no way that you can get the same state using two different angles so this means that like this must be the same angle it's not determined it's enough to determine uh you know how close the state will be right or something like this um yeah to be really strict i don't know if i would have to prove that like the other way around also that when the angles are yeah but i don't really care right like what happens when there's like the orthogonality i just really care about the equality of the states um and [Music] you could try to do induction like would you need to prove like do you would i need to prove that like would i need to prove that that this hum what like a relationship between between the angles and the and the and how equals and how equal the states are right um i don't think so i don't really think that this is something you need to prove because um because this means that uh like you know what i mean i'm trying to see if if you need to prove somewhat that the smoothness of these is there in higher dimensions as well that as you as you go away and say okay i'm at that point where one priority is three or pi and the other one is zero then we're like at the farthest distance right um because that would probably basically mean that's kind of then you know sort of equivalent but this seems to be this seems to be obvious i don't know if there will be a formal way to do it to prove that i don't think i can but that kind of makes sense right so if the gates are the same then the overlap is the same and so just comparing the angles is an auth or should be enough so that would be a simpler proof now let us play with this a little bit i i was just curious because i like you you see that the terms follow a pattern right so you have the so you have the thing here uh plus zero five and this is like time zero five right uh [Music] so the only way that you get a one is when this whole norm here like this whole squared thing is a one right because otherwise you would you know multiply whatever by 0.5 and it will be something smaller and so you will never get one so this is for sure something that uh yeah so so this means that that kind of we only have to focus focus on these being to be honest on just these being one right because if it's bigger than one then the norm square will be bigger than one and we don't want that um and if it's smaller than one then it's not going to be one either so yeah only thing we need to prove is so let me see insert cell below so um let's check that like these or when is these equaling one that's kind of another thing that we could do right um so i'll turn that into my expression and then i'll print the expression let's see what we get yeah so we get these basically now so those are four components right and it's basically multiplication of sinus of the sinus and the cosinus function so the parting that we see here is that um first for all the angles you have everything multiplied like it's a sinus multiplied uh sinus multiplication the the cosine is multiplication and then you have like a mixture so what you have is two two more components let's see if i can i can't paint actually here i can paint if i open the pin whatever i'm not gonna discuss gonna mess up with stuff but these two other components you have sinus times sinus and times cosinus times cosine is and then sinus sine is cosinus so you're going to have like the um yeah so you you you basically have and the angles change so the sine is of the x and the y variables and the cosine is of the p and the q um this pattern is something that will i guess as we add variables it will just scale right if i was if i were to add like say uh now i don't know r and v right r and v so i just need to kind of do these right r and v and so you kind of have r and v r v r and v i like the way that i coded this because it makes it easy to extend now i could probably do this somehow iteratively and i guess what i would have to do is the tensor product of r y r times distance product so to say right let's x [Music] hmm one second what am i doing i know the right is it's p [Music] i will write it here just this would be the gates applying to the first cuban to the second qubit so px r [Music] and then you have uh tensor product r y v this whole thing right so if we do these then we get like okay maybe i how can this be a where is the the size mismatch oh yeah well because now we have uh yeah because now basically have like eight possible states so this needs to grow we've got now eight and valid matrix well well well there you go no okay zero oh i'm stupid i'm not using it here and i should use it here okay zero okay zero okay that works so now you have this monster one here and you kind of see the same oh sorry now i need to do these cool so you see many more terms now but the first one is all sinuses the last one is all cosinuses and then you have like sinus sinus sinus sinus cosine sinus sinus sine is something that's expensive to sinus okay so two cosines with three cosines is really like all the possible combinations right and so i i think there's just a pattern there that i mean there's definitely a part in there that's um it's it's a you know a function of uh the number of qubits or the number of angles that you'll have and so my intuition tells me that if we take a closer look at how the these functions can you know look like [Music] um so let's say sinus is basically uh sorry x symbol x and now you have sinus is basically the sinus of x right and i say plot sinus this is work what is the sinus sp sinus yeah so this is the plot of the sinus then the cosine cosinus can i go like this i doubt you can plot like these though oh yeah that's awesome okay cool so um i'm not so sure but yeah but the point is yeah so one one is zero the other one is one right and vice versa and they all go from one to minus one that's important so this means that all these multiplications right they won't ever be over one that's good now we want to make sure that they don't go r1 because this you know more than one here activated somehow now what happens what happens to what happens to these right let's define another symbol y and then let's um let's basically define the sinus of x and the cosine is f x or like this and the sinus of y cosines of y i basically like these and now i say well please multiply or show me the show me the plot of these now how i did the uh how did i do the 3d plot under the fancy 3d one like that okay actually i'll just copy that sinus of x sines of y times sentence of y and a limit also between these angles see what happens okay cool okay so what does this mean this means this function this product will only be one i guess that's that top here when so if p and q let's see let's let's actually take a look at the diagonal so if x and y are the same it doesn't always equal to one right but what happens if we plot the the cosine as well yeah i think yeah it kind of feels like it's the it's sort of the inverse and not not of the inverse not the inverse but like if you would superpose them you would actually get oh you get though because you start off a one in here at zero right so this means that you know this term here where it's all sinuses and the last term which occur sinuses kind of enforces the fact that it's you know um one right now what's interesting will be interesting to notice what happens with the sinuses and the cosine is like the terms in the middle where there's like for example for the case of for the case of you know just two angles not three so three two qubits whether these terms are basically kind of adding up that's interesting so when it's in the middle for example oh actually we can do that right i'm stupid well we can actually plot plot the sum that's what we want to do so we can do is we can actually plot this function stupid that's the first one that's the second one yeah here you have it bam compare these with these there you go there you go so plot how can i call this one plot uh probability of zero but that's one cubit right so if i run this ah i gotta run everything because i'm reusing variable names all good if i now to be fair right let's kind of try to yeah so if i if i if i do now what one cubit i think that's how i called it right give me a second this glasses are just turning me making me nuts oh come on like you're not printing it whatever but you can see it's the same right so what if i what if i plot the because wait a second because that was these basically yeah that's these that's what i'm kind of having here uh it sucks that i can plot a four dimensional or three dimensional one or four dimensional would be the case right because these are the terms yeah but i can actually plot the values i can i cannot put the values so if i let's take this expression right now expression is this big thing so can i just insert cell below i really wanted to do for this video to be just like under right under 30 minutes which i think i'm gonna make it because i think i'm gonna stop it here i kind of see these i mean it's what i wanted to do to do quickly is if i print this expression it's these so if i basically expression uh how can i senpai senpai senpai replace symbol uh which multiple values i want to kind of have like a um substitutions uh i wanna evaluate i i kind of wanted to evaluate uh you know multiple times i probably have to loop i just probably have to look and that's it um yeah i just probably have to look it'll be faster so if i just say for example right i say uh four oh no no what i'm doing so for i in uh say wrench i don't know yeah actually basically zero to pi or 0 3 yeah but i i want to have 0.14 like how can i like in python for with um decimal steps python range with decimal steps that's what i want to have um i saw the lead with that with the lean space okay how many values i want to have okay cool that makes sense so so what i can do is i can save in for i in basically uh between 0 3 15 and i want to have i don't know 20 values right um then the angle is basically length of steps right so steps will be basically this array [Music] and steps and then i just get steps i it's basically the angle and then i want to do expression substitute because expression is these right and i have p q r and so i basically have like yeah x y p q r v x y p q r v so i kind of have x y b q r v so we're just gonna we're just gonna you know print some of the values right to see how how this works since it's going to be this is going to be angle so it's going to be angle so i'm basically replacing um so print angle right and uh yeah let's see what happens okay interesting oh yeah that printed the angle but if i have to print this yeah so you see it's all once it's all ones it's just one it's exactly when the angles are the same right yeah now i don't know how these will vary yeah because that's kind of the shape of these right that's that's that's the whole point the thing is with with like already you know six dimensions it's just not easy to imagine but like that's that's the idea right yeah i don't know at this point i think i'm just trying to i don't know i'm trying to prove something stupid i think um essentially if the gates are the same then the states are the same right so if the angles are the same the gates are the same and therefore comparing the angles you know is is equivalent to uh running the swap test and therefore you don't need to run the swap test um another story another proof i think i'm gonna stop this upload it and then gonna shoot the executive summary right away basically that was it i guess |
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okay so what are we going to be doing today uh more exercises so far this has helped me a lot with um exploring uh so far so far the only thing I can do is really uh Firefox is Firefox qm exercises so the only thing that I've been able to do so far or the the thing that I've managed to learn out of these is from these first exercises I did explore the concept of you know sort of R like 3D space and and then sort of doing the uh this kind of circular integrals uh or polar integrals um uh then we learned about uh [Music] the thing with momentum right so that was actually this is something that I will definitely be digging into a bit more um soon at some point but other piece what I what I kind of understood is while the white function is not what I thought it was right so this was kind of the hammer when it was like really I always thought and I always understood the wave function as uh you know it's the equivalent of whatever variable um whatever Dimension you're looking at like classically right so classic looking one things like position right so so the wave function off Position will be in position um but it turns out the wave function of position encodes also momentum um so it's really more it it's not just you you can't make that like analogy of like position equals you know it's not the same kind of level of citizenship sort of thing um okay it's a short probability density of finding the particle with momentum and I assume these in two pinch width show the probability density of finding the particle with momentum h-bar K is isotropic so it does not depend on momentum um I'll work on the details later uh I'm just trying to find what what is going to be so we're just going to maximize my learning so again doing just the 80 um surely the spatial probability density um uh this is this just uh I I don't understand uh uh okay it's also isotropic so it's just calculating that and then show that it's isotropic um this leads me to the question of like is is there but that's more of a technicality right so how do I uh um login How can I check that a OE function is isotropic is there like a how how can I check the function I don't know if there's like uh I guess it's just a set of like a like a recipe for these guys all right I have actually approach I guess it's a combination of just doing an analysis it's just doing analysis so that's just all fine um okay if I calculate the expectation values so this is um let's dive into expectation values okay um off position and uh and and no matter what time t and then show that the expectation value of r squared so the question is what does this mean right increases with time and it satisfies this inequality first of all let's let's just dive into um expectation value so uh your channel so let's ask GPD what is the it's like tension rally off it's really it's an expectation of the expectation of expectation value is the observable right that's the right way to ask the question um so essentially you have the position operator the momentum operator instead of what are the expectation values let's just do the classical also what is uh off on the position let's ask K to be more creative position operator hyper physics God we're you know it's kind of like all right I like approaching it like this because it helps me kind of pin all these different um all these different key elements right in Quantum constitutionalism observable is the average value that that observable when the system is in a particular Quantum is the average value of that observable when the system is in a particular Quantum state the expectation value is calculated by taking the inner product of the observous corresponding operator um where the state Vector of the system and then taking the real part of the resulting complex number for example consequently I'm not sure that's correct okay so that's important um let's double Let's cross check that with the Google search but the protection value of a in the state PSI is given by these now I guess I'm also looking for an interpretation of these so what is the uh uh okay in in the continuous setup it seems like we're doing an integral of two requirement kind of calculations is something you can observe in the laboratory the expectation value of the measurable primary is calculated for the position X Exposition value defined whatever X is right so is this the position operator this interval can be interpreted as the average value of x that would be expected or expect to obtain from a large number of measurements okay so the expectation value is just an average of what you expect from a measurement perspective a lot of these Concepts what's interesting is that I'm not so sure what you know how well they map to a super super deterministic view on quantum mechanics because when you get into measurement that's what and measurements being correlated with the detector settings alternatively could be viewed as the average value of position for a large number of particles which are described by the same wave function for example the expectation value of the radius of the electron in the ground state of the hydrogen atom is the average value expect of the information well there's similar functional business the appearance of an average of the function the expectation value of momentum involves the representation of momentum as a quantum mechanical operator while the expectation value of a function of position has the appearance of an average the expectation value of momentum involves the representation of momentum as a quantum mechanical operator as an equal position where this is the operator for the X component of momentum since the energy of a free particle is given by this then the expectation of value for energy huh okay it's like that okay for particle in one dimension so you do multiply by here it's like you do the operator now what uh why why does this make sense why mathematically speaking is this intuitive what's the again in a way it's like what's the what's the meaning of um is that the current transpose conjugate or or what is that the the uh uh all right why does this make sense right so multiplying the operator with the wave function makes sense because you you're you're basically because you're basically saying you want to get you know for every X you you basically apply and and we want to get the uh or you just apply the operator right and you want to get the so you're basically what is the result I mean what is the result really that the thing is mathematically speaking like what is the result of applying an operator right um what is the result of applying an operator to a wave function uh because it's still a function right or breakers like the remote operators triangle's equation we'll begin with a reminder that the safe access are written in terms of the basis vectors of each space is four hours position space uh although in most cases uh but I mean you know if that's the if that's the position operator like what the hell what the heck does these what what is the you know what is it that it that it actually gives you right you have a wave function um you apply this thing and that's kind of like another function um you know what I mean it's like what I'm trying to see is uh what is the so we're trying to break down basically uh intuitively what that means right so you know if you take like an average classically you just what's an average right like what's really the definition of an average another expression is actually the same but like what's the the average is like I mean what's what's how do you interpret an average right that's that's crazy I mean in the median is the mean the same as the average I think so yeah I am blue so you have the mean 100 uh indicate where the center and of the data is located and and what the typical daily number of newspapers hold these yeah that's kind of like uh yeah okay it's in a way indicates centricity it indicates like it's just there I mean how how do you interpret it depends on the context I guess but it's like this the center of things right um so it's it's a kind of a center so essentially what the Expedition value will tell you is the same right so if you but but that's what I'm thinking is like you you have a wave function and then uh you when you apply the uh position operator the idea is that applying the operator is like measuring right is applying an operator to a wave function same nice measuring I mean that is what I assumed that up to this point right so um this is a good I think it's a function over a space of physical States onto another space of physical States so on the formulation mechanics operator operating quantum mechanics um of operators the wave function um okay so if PSI is an eigen function of the operator a uh that then it will really give you so that is the item value of the operator value corresponding to the measured value of the observable if something like a function of a given operator and that's in it then a definite quantity will be observed even measurement of the observable a is made of um the state of PSI conversative side is not a magnet function of of the operator then it has no eigenvalue for any um and the result does not have a single definite value in that case since the measurements of the reserve layer will yield each eigenvalue with a certain probability so essentially um but what is it that you're expecting to so what you're expecting to have after the operator is applied um and that's actually an interesting topic because this is like I I like breaking this down um breaking this down you know kind of getting at the bottom of the definition is is kind of um that's what's complex right so we're based like a huge component we'll hear the corresponding every value I think these underway functions so if you multiply the eigen the the operator by the by the wave function then it's like you give it a foreign [Music] uh so position yeah it's funny right because position operator is just it's just these that's the general definition so so it's it's basically why so why why is that the case if I have a wave so can you can you give me give me an example of a wave function and 3D space suction space foreign yeah let's take that so so you have meaning that it is equal to zero at the walls of the box and non-zero elsewhere oh my God stop for this answer okay so that's an example of a wave function okay um can you Express the wave function above and polar coordinates viewing only variable r can you do that certainly okay that's all fine and then the question would be now uh foreign the position operator hmm I'm confused I don't know if I'm explaining myself why right so what I'm trying to understand here is what does it even mean to apply the the if that's the operator okay um another kennergy total energy momentum Total Line development how how let's let's uh one dimensional let's start with an easier case can you give me a simple one-dimensional uh wave function and position space in a one-dimensional box let's plot these ah come on this Ghost Rider gonna eat that up so if I say Main uh what am I using me here so let's say Main and let's say generate code ask it like uh can you plot these foreign foreign how do I solve this I think I did that before already copy how can I solve this warning so we've got a name if I run this it should work now oh yeah we're a functional program state of a particle in the Box uh one dimensional so the thing is what does it mean to just multiply that by X uh foreign [Music] times x how how do I get Ghost Rider engaged again that's what I've done it just um generate code algae okay applaud uh what PSI times x now well whatever just uh X nah yeah but it should remember I guess I just I guess I just do um my case is plot X PSI and then PSI what if I just do PSI times x how different are these right so so now we have this shape uh and what and now that's the shape what does this even tell me foreign plot can I plot um PSI and like so How can I overlay two ports very much and Michael leap foreign so stop okay so if I just copy this foreign just say well essentially supports one two so so this is uh yeah so so this is one plot and then and then this is times x yeah all that work kind of already here so I think that should be that's that's where we're going so I'm trying to understand to try to understand uh very slowly right so I'm trying to understand what is the meaning of taking a very simple uh wave function multiplying it by the position Operator just being these why what is these and what is this giving me at all okay so it does not overlay them but comparison all right I assume I'm doing that correctly so uh Okay so um is that the way function though uh is that correct like the wave function that it gave me here at the beginning uh square root of yeah yeah and then pi times x divided by divided by Alpha exactly and so uh okay so why I mean this is supposed to give me what what is it supposed to give me um I don't know is this supposed to give me um it's another function right but it doesn't really I don't know what it is that it's telling me like what what have I transformed it into right now by applying the operator um I guess it's gonna be for uh next video oh |
okay so we left it at the expectation values I was checking now so expectation value expectation value formula [Music] um basically uh what I'm trying to understand right so uh quantum mechanics it's really from I'm just trying to understand why the formula makes uh intuitively sense okay um cause from a uh from a purely positioned perspective let me come and do an expression expectation value May then be stated like these General formulation because the expectation value is in in the end some sort of average right petition value expectation value quotation value operational definition consider a the expectation value of the operator right so the thing is it's telling you it's a property it's it's a property of the operator um the operator being thing that you're doing sort of the the thing you're measuring right so if you're measuring is observable I'm measuring position they have the the position operator in direct notation um incredible experimental setup is described by the observable a do we measured and the state of the system the expectation value of a in the in this specific state is you know like these [Music] um automatically A's itself I joined operating a little bit space pure stayed and then um this is a formula for that [Music] um if Dynamics is considered either the vector PSI or the operator a is taken to be time dependent depending on what is wrong with the picture and you just use the evolution of the expectation value does not depend on this choice however if a has a complete set of eigenvectors with certain eigenvalues then the expectation value turns to be the sum of these eigenvalues okay this expression itself to the Earth arithmetic mean exactly so it illustrates the physical meaning of the mathematical formalism values are possible outcomes of the experiment and their corresponding coefficient is the probability and that this outcome will occur it's often called a transition probability the probabilities seem simply simple case Rises when a is a projection but a particular simple case always when a is a projection um but let's dive more into the why does this work right or even the um in the continuous uh in the continuous Spectrum so in quantities they probably have a non-discrete spectrum such as the position operator this operator has completely continuous Spectrum with eigenvalues angle vectors depending on the continuous parameter X especially specifically they're pretty X acts on a special vector spatial Vector in this case the vector PSI can be written as a complex valid function on the spectrum of X usually the real line This is formally achieved by projecting the state Vector onto the eigenvalues of the operator and as in the descript case it happens that the eigenvectors are the position operator from a complete basis will be a closure relation um the above notes used to derive the common integral expression of the expected value so that's the integral expression by inserting ideas into the identities into the vector expression [Music] um nintendent expanding in the position basis so this is specifically this is what I was looking at before right so um to things to things to basically unpack here right so why why is the transpose used in in this context mathematically speaking um vector single integral to the composition when mechanical integrals um all the above formulas are valid for peer States for Pure state only interesting prominently in certain thermodynamic thermodynamics and Quantum Optics also mixed states are of importance these are described by positive trace class operator The Cisco operator or density Matrix Expedition value can then be obtained okay okay so what is the what is this here so what is example in configuration space as an example consider a quantum mechanical particle in one spatial dimension in the conviction space and plus Defender configuration of system are called generalization space or physical system okay um here's Gilbert space the space of square integral functions on the real line vectors functions call wave functions this color product is given by this um the wave function the wave functions have a direct interpretation as probability distribution okay so you are doing that because essentially [Music] essentially that's how you calculate the that's how you calculate the probability distribution okay it could be that's as simple as that um it's a probability of finding the particle in Infinity as an observable consider the position operator Q which acts on Wave functions these okay the expectation value or mean value of measurements of Q performed on a very large number of identical and many systems will be given by these so yeah okay now this makes this make sense now this the the reason we're doing the product like that is because um we want to calculate the we we want to calculate the probability um the probability density it's the sum of all the different probabilities probability densities I think that's a nice and then the probability of each X I would say that's kind of um that's that's kind of how I would release that that makes it that really makes it look like um an average right in average you'll find it there like there's 50 chances of finding it here 50 chance of finding there so you'll find it like in the middle that will be the expectation value um I'm not so sure I uh uh although from a sequential perspective that's that's a bit odd right it's like you have the state and you apply the operator and then you so you evolve the state it's not that you're evolving it is that you're this is easy to understand than the direct one right because this is just this is the wave function then you have the operator so you're applying the operator you're making the measurement and then you're multiplying by the by the the uh transpose um the Expedition value is not the case for all factors unbounded and stitches in front of this domain of definition up to the average momentum one is the minimum operator and configuration space okay [Music] viral theorem serum since it's good particles about financial forces in mechanics the viral theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system Bound by potential forces that was the total potential energy okay [Music] still I'm not so sure I understand the con I I understand fully um how to read the result of I have an operator and I apply the wave function and what does this tell me or give me right like because if I have a wife a wave function and then I multiply by X like what does this tell me you know what I mean that that's that's definitely um it's kind of in a way here it's like you're applying the you're applying an operator but you're not like applying applying an observable applying an observable is because an observer is a special kind of operator that that probably then actually does this um it's protection um but not entirely this this makes sense okay so this there is an explanation for these which is you want to calculate the probability it's like the the sum of like an average right so you're integrating over and then it's the operator um right at times the probability still not so shy I I don't feel 100 comfortable with this I understand yeah I understand the expression but it's like no yeah that's the point that's it's an integral right so you're doing this for every single point for every single for every single um DX it's a sum of every single DX that's what you're doing so yeah essentially that's that's what the expression means here though um it's a bit trickier because as I said I would typically read these right to left and I would say boom then we apply an observable and why do we need to do this so it's like what's the maybe the next thing to check I'll leave it here for now and we'll do another session later another brief session later but this this makes this makes more sense I'm not sure why X makes sense so or in a way why is the operator shouldn't these be shouldn't these really be I know this should be the eigenvalues that's true these are the eigenvalues it's not the operator that's true right these are the eigenvalues correct foreign ly achieved by projecting the state Vector onto the eigenvalues of the operator as in the discrete case is the projection it happens that the eigenvectors of the position pretty form a complete basis um ah there you go if a has a complete set of eigenvectors and eigenvalues and that's what you're doing these are the eigenvalues and I think that's and then in that case X is also the eigenvalue this is also the eigenvalues of the moment of the position operator I guess eigen it's just X right think so to each observing Quantum cases in a product correspond to it I don't understand what's the meaning of their values of the extra person's X is for me let's correspond to real numbers exactly so uh values that are measured in the experiment a values are values in the position obtained when measured every measurement will produce different result we're unwrapping some of this stuff we're moving a little forward step at a time we're unwrapping some of this stuff just to feel just feel more feeling more comfortable with this um notation uh with notation and elements and stuff like these yep |
okay so this is progress update for i don't know what week um i'm not going to be doing those regularly as you can imagine already by now but i think every four or five videos i'm going to be doing one so it's uh um i'm going to be just numbering them differently but the progress log for these wikis or since i keep the last progress update is it's pretty it's pretty simple so i'm still i'm heavily focused on the multiverse computing paper just because it has so many different things to play with inside that i'm just having a lot of fun and and i kind of like the way that it's kind of gotten me to explore um tensor networks to explore and to you know to explore d-wave in the past couple weeks and to explore also um vqe in a bit of a different way than i had done before right like i'm actually really like digging deep into the the whole expectation value and and why you know why these type of cost functions are differentiable and why why we can calculate gradients at all um so so this is what we probably you probably can see this week like if you take a look at some of the videos um so i did a couple of long sessions on on uh on matrix product states and matrix uh product operators this is tensor network stuff um and i'll go back to these right i in the previous week i did about a bunch of d-wave stuff which i also have to go back to because i really didn't completely finish the whole thing and but but i kind of wanted to move forward to the qe this week quickly and vqe constrained and so i got got hooked into plotting like it's the third video already plotting plotting expectation values so what i'm doing is i'm playing with vqe i'm making small circuits with one or two parameters and then just plotting the expectation value and i'm basically trying to see whether you can differentiate that and trying to see whether you can find the param plateaus that people talk about where you're like okay so you get to a point in your expectation value land where you cannot like move anywhere because you've basically converged somewhere which is just like a local minima or something like that and that's what i'm trying to play with so basically not like on the other side of things i i keep i i keep thinking a lot about like another way to represent quantum states not just like having a system having like the wave function that shows you you know what the different states are at the system level but like the zero the one or like if you have two qubit zero zero one or one one you know one zero etc but having a different way to represent a state where you have um you you keep track of the single qubit states and then you can keep track of entanglement between those states using um hyper edges so edges connecting multiple like more that one two or more uh of those nodes um i promise i'll get on with the videos in this direction as soon as i'm done with the multiverse computing paper but i'm just like it is so much to do with the multiverse paper that i'm i think and because those things really neatly come together really well um kind of the thinking about the hyper edges for the entanglement together with the the tensor network stuff um and i think the variational part is something that i want the vqe thing is something that i wanted to explore anyway because it's a direction that i definitely want to dig into as well in the next in the coming months so yeah plenty of stuff to uh to watch this week plenty of stuff to take a look at in this direction so uh yeah let me know i hope you're enjoying the videos uh as much as i am um yeah i think that's it for the for the update basically it's hardcore multiverse computing still just because it touches everything such as d-wave system it touches vqe vq constraint it touches like tensor networks it even touches classical staff like classical optimizers which i haven't really done yet but i've um yeah basically that's it that's it really |
Quanah teleportation comparison so that's where I want to do today to explore a bit farther sort of you know the continuous verses of this create quantum computing implementation so what-what Senator dies and I am still unsure about the premise that correctly but does nobody has complained so far I think that's that's the correct way of pronouncing it and then the the sort of the discrete version of Quanah I so this is the IBM experience in this disk Senate though I'm gonna try a key I found the quantity repetition here I probably need to find it I think was in Strawberry Fields documentation mmm so algorithm somewhere let me see blah blah blah blah blah blah operational circuits kind of know it's introduction maybe should you search teleportation docks it's not that's no but it's and it's not Strawberry Fields Penny Lane so software Strawberry Fields okay now this is Strawberry Fields straw refills to commit yeah exactly the right one yeah cool stuff by the way here with this simulating instance could use very well chrome so I haven't tried the editor yet them I probably play with this as well Zhou mmm teleportation I just keep like I'm getting distracted dilapidation yeah Conant repetition tutorial but is this or is this because here we solution lost Direction wish for chronic quantum algorithms no quantum algorithms state teleportation I think that's what we want to do what else was there state gate this is something that I'm I'm guessing the equivalent in this great quantum computing would be sort of the matrix dilapidation so sort of your encoding something in again I don't know encoding a matrix in a gate and then having the same effect on the other one maybe something like that cows in cloning Boston sampling quanta neural network and this is something that I also worked on that I also have done a discrete quantum computing version of it and okay but let's take a look at let's take a look at that and then next we'll take a look at and beam splitting and then probably we'll just go back to the actual Yellow Submarine project because I think I think I'm diverging too much of that and then probably I'll try to understand that based on what I understand here and then I might make a separate series really dive into strawberry fields or exon ado because I just gonna be I'm I think I'm losing the focus here a little bit but let's just just get onto it so States repetition I took a look at this briefly in the other and the other video as well so because meme splitting is the way you create entanglement in continuous quantum computing but let's refresh our mind my mind actually not yours so was the teleportation here kind of thing right so Allison Boclair entangled Bell they hold one of the qubits each of them exactly today by using her Margate and then applies in gate okay so the curry Bell state basically and then they have each of them one cubed Alice applies it in form of gay to q1 controlled by this next Alice applies a harem arcade and applies a measurement to both qubits that she owns then it's time to for phone to call Bob she tells Bob the outcome of her 2qb measurement depending on what she says Bob applies like a different gate and exactly how old how will the test result the real computer blah blah blah blah blah here we had here we had the this circuit so this is the Q this see if I can zoom in this over the way it seems I can maybe I can open the image in a new tab that makes more sense [Music] yay so this this basically the the idea the basic idea here so if you've got an entangled pair of qubits exactly hmm you've got an entangled pair of cubits and so alleys does this on her anzhela cubed which basically I'll just I'll just try to fire up just try to open up a circuit so we can take a look at it I just don't want to repeat the whole thing again but actually I realized that I kind of forgot a little bit about it so but you know one since using the actual such three three so Alice does this and this which basically gives her a superposition [Music] no no no no no no no anna-san skew took a mr. Churchill one of them to the murmur getting the place in and get into Cuba controlled by the one bells so I don't I don't know why the Zen is here flying okay but the idea the idea here is you you apply a Haram art here and and then you do a control so you see entangles Allison tangles her control qubit perm and silica bit with her own cubed in a way that they are let me see correlated right so if I now do this they're always gonna increase this year or one one these two here so and then she goes ahead and she and that's this operation just let me just I'm I'm a bit confused by this ed here right now I think it's not needed so then she applies a measurement here that's not what I'm looking for that's what I'm looking for a measurement to both qubits okay so on mmm I mean of course these two qubits have been entangled before so there's also these and these entanglement and now al is basically so Alice basically applies a Hatem art here why was this Haram art needed she applies a hot iron because then it's funny because it's like it's one of the first things I did and I kind of realized it's hard to come back to those things it's pretty hard to come back to those things and it's pretty hard to read the circuits like that I remember that I did this like really with his ears and once on a notepad and it worked fairly well but I thought by now I'd be able to actually Stanford to the data at least applies how am I getting appliance measurement both qubits and she ohms so if she misses she measures zero zero then Bob shouldn't do anything if she measures your 1 1 0 1 1 10 Bob should apply to X gate Z gain or Z and X Cade [Music] so basically jumping a lot you okay put that that kind of works so that kind of works with so basically because these measurements with their dis partial mesh measurements what they're doing is they're kind of clack collapsing the rest of the state right so so if she had a I don't want to use the notepad because I know that because that's not scalable and I just know that it's gonna be easy to understand like that but I try and understand that just by reading the circuit kind of but it's yeah I feel I'm missing that intuition again that sucks so this is entangled exactly and then so when we measure these and we measure that basically this is gonna make this is what's gonna make the difference right so if you if you measure zero zero if you measure zero zero it means that was a zero it means that never happened that control nod so it means that was this hero so it means that also never so you know it's kind of like feels awkward but it's like so let's let's do the case zero zero right so bye-bye meaning that that this is the zero and this is the zero so if you if those are if those measure zero it means that that this was zero right it means that this control not never happened which in turns mean interns intern here means that this was also no I cannot say that this was zero right because this was zero and then it was in a harem iron so do still zero and one yeah okay but this is still zero and one this is still zero and one right exactly so so this means that because this could be zero one and if it's zero then Bob's keep it stays zero and if it's one Bob's keep it stays one we've effectively you know kind of teleported this right so this means that Bob needs to do nothing and just measure that's what they're saying here exactly so Bob does nothing just rescue bit and in the same so exactly so you see you see how this this is the if feels like you you gotta go down to the measurement and that kind of go back and and see and understand why this works that kind of it's one way to read the circuits but it's not trivial so if it's zero one right then it says applied the X K so then if it's 0 1 means we've measured one here which in turn means that I'm not so sure if this is the correct read but it means that this was a 1 which means that this sign here was flipped here the sign was flipped the sign the the amplitude so the controller was applied and if the control not was applied and we measured a zero it means it was a 1 here it means that because this was one the control here this controller was applied but because exactly this controller was applied so basically this qubit was then kind of moved to one so kind of I kind of yeah I think that's I think that's that's what's happening here and then which then means that in this case because this is a concrete example right where this is a zero in this case then we got apply a next gate to turn that into you know and to its original thing into the zero so the concept here is that the idea of the discrete and I'm probably missing some stuff so I definitely should do a sort of a refresh and this um I mean there's so much to explore it's just it's it's a lot of fun but this basically means the concept here is you say you have and then you use partial measurements so alleys Allison tangles and in silico bit with her own - then - then measure like she does she does she does the the harm our operations or not and then she measures and then backtrack sight so because of the measurement that she's done then Bob knows what transformations he has to make in order to get Alice's original stain and I'm guessing guessing this is just one implementation so but this is the this is the basic idea whereas here it in in Xanadu it's sort of a similar concept because you've got conditional actions based on measurements right so they say is while originally designed for the screen variable bottom okay the results showing in the following circuit so I the thing is I don't know why they use the X in the set gate which are not that are they I don't think they are continuous quantum computing variables operators operations so but it's the same concept right so you see here that you've got in Silla bead is the other one I think because well you're trans you're what you're teleporting is the momentum and the position so and and so you're entangling alice's entangling that and then making a measurement and then based on the measurement so a measurement and and here the thing is because you have different types of measurement if I remember well so you've got measurements and position measurements on momentum and some other fancy measurements like counting stuff because this is really tied to the implementation right which is the photonic type of implementation so this is not like I think the measurements here really do not map into to the measurements in discrete quantum computing but it's more like okay how do we measure our beam aligned right how to measure the fall in sight we can count them we can check their momentum you know stuff like that I think so basically that's the idea because but I'm curious about what is the entanglement really looking like inside because the beam is it's the same concept that then the phase and the moment the player that the momentum and the position the position are always agreeing for example is this what it beam splitting does maybe that's what I should take a look at now quickly but that's the concept is roughly the same so you measure position and measure momentum and then if measure position then you go sort of an I assume the X is a displacement and I don't know what the Zed is if it's the same thing here chemos are spatially separated teleport her known state to Bob Alice now performs a projective measurement of her entire system now this is the one that shows a teleport is the ancillary thing it's different it's a little bit different there's no insular cubed here a commode is there I don't think so there are three because they're three because one is used for position and another one for one for momentum one for position right cameras are now especially separated teleporter non-state alleys now performs a projective measurement so the first one is the actual key mode to teleport the state to teleport protective measurement offer anti system into the maximally entangled basis States this is llama entangling this is this be a 50-50 beam splitter homodyne measurement in the X and the P quadratures respectively the results of these measurements are then transmitted to Bob who performs both the position displacement exactly I know momentum displacement okay conditional to the P measurement yeah but took over the exactly transmitted stayed but those displacements should be permit rised if I understood well the whole concept here so how does Bob know the parameters for those displacements uh-huh I don't know this can be implemented using the Blackbird quantum circuit language ooh Blackbird sounds cool okay that's it let's take a look at the beam splitting to understand a bit more in the anatomy of this the anatomy of this entanglement then splitting beam splitter okay now I won I won I won gates I want gates quantum programs I want kids introduction states gates do we have here yes yeah beam splitter here you have beam splitter so what is it doing what is it doing I have no idea absolutely what does this here mean Medina measurement heterodyne measurement photon counting yeah yeah yeah so you see the homodyne measurement measures the deposition the heterodyne measurement measures position and momentum then photon counting okay whatever but I what is the beam splitter to win game splitter beams later entanglement beam splitter entanglement it's a paper which quarter our physics forums this is this benefit or conditions different to phonons to be entangled by beam splitter lots of references given by the forum users and conclusions massive entangled pairs must already be part of the of entangled pairs and that's the beam splitter just swap the entanglement between the members can the beam splitter be used alone two entangled photons or can it entangle them only in the presence of many other elements like polarizes wave plates prisms and tangle man I mean I don't want to go so much into details I just wanna know what's the entanglement that this creates like seems clear beam splitter gates because it also has parameters applied means for operations with specific modes it is assumed that this is real parameters transmitted amplitude reflected amplitude first mode second mode what is it transmitted amplitude and glossary references and further reading beam splitter nope nope nope okay here we've got some definition for the annihilation and creation operators of to me olds denoted and the beam splitter is defined by okay they will transform the operators according to the Phillipines respond to input coherent States to input coherent States to to output coherent States where by substituting the by substituting the in the definition of the creation and annihilation operators in terms of the position momentum operators it is possible to derive an expression for how to beam splitter this forms the quadrature of the quadrature operators okay so it seems like it seems like it is correlating it seems like it's correlating the position of one with the position and the momentum of the other I'm just guessing right this is kind of correlating these with these and these okay action on the position and momentum eigenstates controlled face da-da-da-da-da-da-da cubic face this is some pretty dense stuff but it seems like the beam splitter is correlating position or position and momentum so the position of of the one key mode with the position and the momentum of the other one and momentum of the one key mode with the momentum and the position this sort of like a difference here with the sign is up here the minus sign is up here and here is up here whatever that means really so it's kind of where am I I wanted to go to circuits quantum algorithms state teleportation but I was as I as I was saying um as I was saying so it correlates those things and then it would make sense that you measure the position and the momentum and then you can probably derive what are the parameters for the displacement you must do here correct I don't know so let's see what's happening here so let's take a look at the code a little bit if I understand if it can be understood PCI Ellison Alice is one bob is two so there's beam splitter this beam splitter then there's squeezed why what was squeezing I don't know squeezing squeezing then beam splitter Alice Bob Alice performs the joint measurement X Kate scale psi some important notes I infinite squeezed vacuum stains are not physically real realizable the function scale whatever that means the function scale can be imported from this the BS gate accepts two arguments set and Phi a variable storing the value of pi is used for setting these parameters in Python this can be imported from Nampa what a sigh so you do so here the the ex Kade because it seems like it's not a conditional application it means it's applied yes oh yes right it's applied yes oh yes it's just somehow permit rised so that's kind of the difference here because we're probably dealing with because of the continuum thing I guess I mean intuitively that seems to me what's going on said good Alice but measure X measure P okay so you measure X into into upside and the measure P into Alice okay okay okay so you see okay so then those that I was confused by the variables and there's some sort of scaling going on which I don't know why but the idea is you use the exact results of your measurements as input for the x case and ZEB gates which DX get and ZEB gate which is then I guess like parameter izing the DES displacement it's permit rising the displacements okay this is really at an intuitive level fairly more simple to understand because it's kind of it feels more logic it's like it's like you entangle you entangle this with you entangle this with your channel or whatever right if you don't call it this way or you entangle this one in the cubed so so they said so the positions and and it seems like really what it's doing it's just it it really entangles them in the sense that they get the same position in the same momentum I mean there's a scaling factor here but it's what it this seems to say it's pretty fairly simple is here or seems the code seems pretty straight like fairly straightforward so and then literally the measurement and then literally you've also entangled that with this one so you need so you kind of create you entangle the three of them really and it's just you need the three of them because you need to destroy those because you measure them right so you destroyed this one in the middle and then you literally apply whatever you measure position wise and whatever you measure momentum wise as display displacements to to Bob's cue mode and and then you have it it's funny because it's much more complex from a theoretical perspective like what those things are kind of looking like but at the intuition level seems more obvious at least to me that seems fairly obvious and fairly similar to what you would usually do with a traditional classical computing thing you would basically kind of copy stuff and then you know and basically use the result of your operation as a parameter of another operation etc so this feels quite quite intuitive whereas that the discrete one and I might be missing done don't get me wrong gamer I might be missing here some of the details so I might just dis might be PS not as in beam splitting bus in the other meaning of PS and and whereas here is complicated because you got a backtrack from measurements you don't understand the circuit because of the entanglements I think the entanglement here is fairly complex because of the controlled operations intuitively it's just a bit more mind-bending I think this is just a bit more complicated but you you can grasp it easy okay cool I hope that was helpful I'll try in my next video to really go back to the yellows to the Yellow Submarine project and then actually try to dig into that a little bit more um and then kind of jump back and forth the Strawberry Fields documentation and the code and then see see if we can grasp that it might be too early but let's give it a try |
okay so we left it at the expectation values I was checking now so expectation value expectation value formula [Music] um basically uh what I'm trying to understand right so uh quantum mechanics it's really from I'm just trying to understand why the formula makes uh intuitively sense okay um cause from a uh from a purely positioned perspective let me come and do an expression expectation value May then be stated like these General formulation because the expectation value is in in the end some sort of average right petition value expectation value quotation value operational definition consider a the expectation value of the operator right so the thing is it's telling you it's a property it's it's a property of the operator um the operator being thing that you're doing sort of the the thing you're measuring right so if you're measuring is observable I'm measuring position they have the the position operator in direct notation um incredible experimental setup is described by the observable a do we measured and the state of the system the expectation value of a in the in this specific state is you know like these [Music] um automatically A's itself I joined operating a little bit space pure stayed and then um this is a formula for that [Music] um if Dynamics is considered either the vector PSI or the operator a is taken to be time dependent depending on what is wrong with the picture and you just use the evolution of the expectation value does not depend on this choice however if a has a complete set of eigenvectors with certain eigenvalues then the expectation value turns to be the sum of these eigenvalues okay this expression itself to the Earth arithmetic mean exactly so it illustrates the physical meaning of the mathematical formalism values are possible outcomes of the experiment and their corresponding coefficient is the probability and that this outcome will occur it's often called a transition probability the probabilities seem simply simple case Rises when a is a projection but a particular simple case always when a is a projection um but let's dive more into the why does this work right or even the um in the continuous uh in the continuous Spectrum so in quantities they probably have a non-discrete spectrum such as the position operator this operator has completely continuous Spectrum with eigenvalues angle vectors depending on the continuous parameter X especially specifically they're pretty X acts on a special vector spatial Vector in this case the vector PSI can be written as a complex valid function on the spectrum of X usually the real line This is formally achieved by projecting the state Vector onto the eigenvalues of the operator and as in the descript case it happens that the eigenvectors are the position operator from a complete basis will be a closure relation um the above notes used to derive the common integral expression of the expected value so that's the integral expression by inserting ideas into the identities into the vector expression [Music] um nintendent expanding in the position basis so this is specifically this is what I was looking at before right so um to things to things to basically unpack here right so why why is the transpose used in in this context mathematically speaking um vector single integral to the composition when mechanical integrals um all the above formulas are valid for peer States for Pure state only interesting prominently in certain thermodynamic thermodynamics and Quantum Optics also mixed states are of importance these are described by positive trace class operator The Cisco operator or density Matrix Expedition value can then be obtained okay okay so what is the what is this here so what is example in configuration space as an example consider a quantum mechanical particle in one spatial dimension in the conviction space and plus Defender configuration of system are called generalization space or physical system okay um here's Gilbert space the space of square integral functions on the real line vectors functions call wave functions this color product is given by this um the wave function the wave functions have a direct interpretation as probability distribution okay so you are doing that because essentially [Music] essentially that's how you calculate the that's how you calculate the probability distribution okay it could be that's as simple as that um it's a probability of finding the particle in Infinity as an observable consider the position operator Q which acts on Wave functions these okay the expectation value or mean value of measurements of Q performed on a very large number of identical and many systems will be given by these so yeah okay now this makes this make sense now this the the reason we're doing the product like that is because um we want to calculate the we we want to calculate the probability um the probability density it's the sum of all the different probabilities probability densities I think that's a nice and then the probability of each X I would say that's kind of um that's that's kind of how I would release that that makes it that really makes it look like um an average right in average you'll find it there like there's 50 chances of finding it here 50 chance of finding there so you'll find it like in the middle that will be the expectation value um I'm not so sure I uh uh although from a sequential perspective that's that's a bit odd right it's like you have the state and you apply the operator and then you so you evolve the state it's not that you're evolving it is that you're this is easy to understand than the direct one right because this is just this is the wave function then you have the operator so you're applying the operator you're making the measurement and then you're multiplying by the by the the uh transpose um the Expedition value is not the case for all factors unbounded and stitches in front of this domain of definition up to the average momentum one is the minimum operator and configuration space okay [Music] viral theorem serum since it's good particles about financial forces in mechanics the viral theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system Bound by potential forces that was the total potential energy okay [Music] still I'm not so sure I understand the con I I understand fully um how to read the result of I have an operator and I apply the wave function and what does this tell me or give me right like because if I have a wife a wave function and then I multiply by X like what does this tell me you know what I mean that that's that's definitely um it's kind of in a way here it's like you're applying the you're applying an operator but you're not like applying applying an observable applying an observable is because an observer is a special kind of operator that that probably then actually does this um it's protection um but not entirely this this makes sense okay so this there is an explanation for these which is you want to calculate the probability it's like the the sum of like an average right so you're integrating over and then it's the operator um right at times the probability still not so shy I I don't feel 100 comfortable with this I understand yeah I understand the expression but it's like no yeah that's the point that's it's an integral right so you're doing this for every single point for every single for every single um DX it's a sum of every single DX that's what you're doing so yeah essentially that's that's what the expression means here though um it's a bit trickier because as I said I would typically read these right to left and I would say boom then we apply an observable and why do we need to do this so it's like what's the maybe the next thing to check I'll leave it here for now and we'll do another session later another brief session later but this this makes this makes more sense I'm not sure why X makes sense so or in a way why is the operator shouldn't these be shouldn't these really be I know this should be the eigenvalues that's true these are the eigenvalues it's not the operator that's true right these are the eigenvalues correct foreign ly achieved by projecting the state Vector onto the eigenvalues of the operator as in the discrete case is the projection it happens that the eigenvectors of the position pretty form a complete basis um ah there you go if a has a complete set of eigenvectors and eigenvalues and that's what you're doing these are the eigenvalues and I think that's and then in that case X is also the eigenvalue this is also the eigenvalues of the moment of the position operator I guess eigen it's just X right think so to each observing Quantum cases in a product correspond to it I don't understand what's the meaning of their values of the extra person's X is for me let's correspond to real numbers exactly so uh values that are measured in the experiment a values are values in the position obtained when measured every measurement will produce different result we're unwrapping some of this stuff we're moving a little forward step at a time we're unwrapping some of this stuff just to feel just feel more feeling more comfortable with this um notation uh with notation and elements and stuff like these yep |
kind of reconfigure the whole twitch integration cool um so what are we doing today so we're now basically going to take a look at um so we did the whole thing with the particle in the box last week and um i'm kind of trying to figure out how can i kind of at least carve more time to just do more um drilling on these or grinding on like actual exercises but i'm looking into you know kind of to to take a step forward and and maybe not a step forward but kind of like approach the whole thing from another angle right now and see okay so we've seen how you solve the schrodinger equation at least the very simple versions of it how how do we get to the path integral right because my my ultimate goal is just as a reminder for myself as well um to try go try and um understand entanglement from a path interval perspective that is the ultimate set of goal in here um so what do i even start how so let's just how do we go from short and go to path into girl like or something like you know should i get a path integral okay so there is a wikipedia page which is going to be definitely too complex to go through that would be my guess god but we can try to do that and then see where that's where this takes us in terms of maybe there's a nicer front path integral [Music] this is from mit path integral method maybe okay then maybe that's what we should maybe that that's a better one to go through oh and they go through the harmonic oscillator awesome cool let's do that let's let's bookmark these let's just bookmark these here and um cool so let's just go through that so let me just double check we're live aren't we i'm not so sure i have to log into it yeah yeah yeah whatever i'll set this up later i just want to make sure that on my life let's just go check to each quickly because now i change the certain c's systems and certain systems yeah yeah life okay cool that looks good um path integrals in quantum mechanics we present the path into a formulation of quantum mechanics and demonstrates equivalence of stronger feature we apply the method of the free particle and quantum harmonic oscillator investigate the euclidean path integral and discuss other applications so a fundamental question at quantum mechanics is how does the state of a particle evolve with time that is the determination that is the determination um of the time evolution of some initial state quantum mechanics is fully predictive in the sense that initial conditions and knowledge are the potential occupied by the particles enough to fully specify the state of the particle at all future times um so uh in the early 20th century derived an equation um [Music] specifies how this instantaneous change in the wave function depends on the system inhabited by the state of depends on the system inhabited by the state in the form of the hamiltonian in this formulation the eigen states that hamiltonian play an important role since their time evolution is easy to calculate um a well established method of solution after the entire eigen spectrum is known is to compose the initial state into the eigen basis apply time evolution to each and then reassemble the eigenstates okay so that's the okay so that is the this hamiltonian formulation works in many cases in classical mechanics however the lagrangian formulation is known to be equivalent to the hamiltonian one thus we seek an answer to the above question that relies on some analog of the lagrangian action um so direct made a mysterious comment to this effect which later inspired richard feynman considered trajectory xd between an initial point and a possible future point let the transition probability amp transition probability amplitude be the inner product of the wave function the transition probability amplitude and these be the inner product of the wave function in the schrodinger picture of the particle evaluated at these two points what does this mean so the probability of transitioning to to that state i have still a hard time understanding sort of the physicality of of inner products in general and things like these like that's still it's still a bit of a mystery to me mathematically why that why does this make sense um at all but that's something that i definitely like to dig into at some point feynman hinted at the equivalence of the probability amplitude and the exponent of the classical action of the trajectory where equivalence is not yet well defined it was not until 1948 that feynman as a postdoc student at princeton formalized his connection so five percent of formulation quantum mechanics based on this principle so let a given trajectory x t be associated with the transition probability amplitude with the same form as that given by drag um of course by quantum mechanics we cannot speak of the particle taking out any well-defined trajectory between two points instead we can only speak of the probability of finding the particle at these locations which is related to wave functions that is all that can be determined is the relative probability of the particle taking one path or another okay i mean that kind of makes sense to me just formally speaking i'm not so sure i fully appreciate it's not about understanding necessarily but it's appreciating that that way of formulating a transition probability that it transitions from one state to another it was the probability that it transitions from that state to that state right and and y inner product like that that's kind of something that so does the inner product have any physical meaning that'll be an interesting so there are so the inner product has the meaning related to a projection of one vector into another vector for true projection the wave functions needed to be normalized it's the integral let's interest this is defined as an integral summary book for beginners are the quantum mechanics in one section the author shows the inner product of two-way functions i am wondering what's the significance of that product i googled that it's not called probability amplitude but that product could be complex uh does it tell any physical significance so suppose you have some linear algebra background um the most important thing you need to know is that the inner product has the same meaning of what you have learned in linear algebra class the inner product has the meaning related to a projection of one vector onto another vector so it is similar to the projection of a three-dimensional vector into another unit vector which gives the result a so the inner product can give you the length square of the wave function so you can normalize your y function by the condition this second allows you to show that two wave functions are orthogonal to each other given by the conditional that the inner product evaluated to zero okay so maybe that's got something to do with the fact that so you take the wave function so you take the wave function at each of these two part dot dots right so if you do that what you're essentially calculating is if if that is if that gives you zero and you're probably saying because these wave functions are orthogonal right because that's the meaning of a zero there um then it means that it's impossible for this transition to happen but then wave functions being orthogonal this just god states right so it's the state that it's a two states these two states are orthogonal this means they why why so that would mean you wouldn't be ever able to transition to these states okay that's interesting so if we write that wave functions only combination of orthonormal wave functions similar to a general vector in linear algebra then we'll have the inner product yeah the meaning of c is the probability absolute of and it is a complex number in general so probability p end of the wave function is given by diminishing is very important when you learn how to perform measurement lastly you should add two wave functions amplitudes together before you take the square similar to adding the amplitude of two water wave functions more precisely if the new wave function is these then the probability density of a position x is this now that a is a normalization constant given the condition that okay so the top product is as you mentioned is the probability amplitude of one state transforming into another that's interesting right so that that is okay so that's definitely clarifying so this is but this is something that um and the possible future point let's just probably have to be better in a product but who defined that like who so is it i understand that's something that feinman kind of hinted at um these two points you know let the transition probability amplitude be the inner product of the wave function in the showing a picture [Music] of the particle evaluated at these points finally an equivalence of the probability amplitude and the exponent of the classical you know what he that has the there's an equivalence between these and then the exponent of the classical action of the trajectory okay so there's this there's this equivalence that that it's probably what we're going to get into at some point right this i'm kind of happy with at the moment but i definitely want to dig deeper into that but okay i'm happy with the projection being sort of the interpretation of these in a prediction but the way that you model this kind of stuff so that that kind of makes a bit more sense um represent information quantum mechanics based on this principle so there's a there's a bunch of trajectories and then they associate with the transition probability amplitude with the same form as that given by drag of course by quantum mechanics we cannot speak of the particle taking any well-defined trajectory between two points instead we can only speak of the probability of finding the particle at these locations which is related to wave functions that is all i can determine is the relative probability of the particle taking one path or another feynman's um inside was these the total transition probability amplitude can be obtained by summing the amplitudes of the particle having taken any individual path [Music] okay so ah i see so what feynman's claiming is in here was claiming is that the the transition probability right to go from this state to this state can be can be calculated by adding all the amplitudes of the different of every single different given path okay so if you sum the probabilities like that seems to make sense but it doesn't seem to be that obvious like so the probability that i go from a to b is the sum of the probabilities that i go through each of the different paths right so so if there's no path i can take if all the the probability that i take any possible path is just zero for all the paths then kind of that makes sense right so it means that they have no chances to go from from that point to another but then but that what's awkward in this definition is that it seems like what you're saying is that the addition of all the different paths doesn't have to be a total of one which is what's confusing here right because you imagine that if there is any change that you go through any of the paths that that is going to be one at some point right um but maybe i'm reading this wrong because you kind of have to go one level up speaking in terms of like sort of at the meta level and say like well the probability that i'm at that point in time it's something you're comparing against all the other points that could be in that time right uh okay yeah i get it so i guess so in a way let me try to draw a picture just see if that becomes a bit more clear let's open our friend paint so you know kind of that's my initial state right and then i guess there's many points i could be in the future right i mean it's almost really a continuum right if you just think about these as space and then like you know maybe that's probably the wrong way to draw it right like just have to have them all at the same um point in time so that's like a slice it's a point in time so i could be in multiple places right that's that's the point so maybe you know it's like there's just a lot of paths that go out of out of these kind of way right so and you know i guess they could just go yeah so there's this point in here where there's just no path going do it so that's why this will be zero yeah so it's the sum of all the path the probabilities of all the paths that that actually lead to that right if if i've got like an equal amount of probabilities then you know those will distribute here i guess that's what that's what these this is saying so which which pretty much kind of reminds me of really um yeah that is that is really similar to um to warframe's feature right of of you've got your your multi-way hype graph that tells you from that state you transition that state and should that stay to that state from you know that state tradition to that state to that state to that state right so you kind of have you you have your and if you take these as what they you know called it the foliations right and then you consider those being sort of the time slices basically yeah that's basically that's basically analogous to these it's just discrete right it's not continuous um so it's pretty yeah that that seems to be quite this there's really an obvious at least at an informal level not an obvious equivalence to these right um so i wanna spare you my sneezing i almost feel like i'm gonna sneeze but i can't anyway that is all that can be determined is the relative probability of particle taking one path or another okay but so we're we're talking about we're talking about paths i'm going slow on purpose right because it just i just want to make sure i process these like um so what's different between the path integral and the schrodinger picture is so in the schrodinger picture you have you have a hamiltonian that kind of encodes how the system how the state is evolving that's why what you what you're doing is if you have your initial state and you apply this hamiltonian operator now you are at the next state right so now what's the probability of you measuring a specific outcome here is you and now that's when you've got to project these uh that's why the inner product works now you do the projection of like i want to know i want to know you know what's the probability of being in the state 0 for example so i do that projection yeah and that's why the inner product works well because the inner product is going to tell me for example if it's zero it's going to tell me that i'm there's no way that i'm in that state right so so that's the shoring away but then that and that's what we're getting rid of that the beat here instead of that we're doing some sort of sum or integral right across these paths but i don't know what is it that we're like shouldn't one just like slice that into into smaller steps and then kind of i don't know i don't know let's see yeah sometimes inside was that the total transition probability amplitude can be obtained by summing the amplitudes of the particles having taken an individual path if the quantity can be calculated in the method suggested the time evolution of the state can be determined by considering contributions from all possible future states exactly yeah yeah so so that's so so the conclusion here is then you can define your own kind of hamiltonian by just kind of adding up all the different contributions to like yeah so you would say the probability that it's in this state in this state in this state in this state and then you kind of sum up all the things together and then you kind of have your time evolution right and below the kids are eigenstates of the position operator such that integration over all x spans the entire basis okay so what is it so okay so what what what is this telling us so the state so that the the wave function at a point of of position at a point t prime d t tilde whatever equals the integral of what what are we integrating here okay so for a small change of position i don't know why this has the tilde here oh yeah because it's that's that's this here so we have uh okay so so what this is saying is actually do that projection yeah that's that's what i that's probably that's kind of what i mean here so so do that projection for like it a very small interval of x and remember this oh sorry no what am i doing that's time shouldn't we just be doing this through time why are we integrating over position i mean okay yeah so we're calculating a time prime so we're calculating let me try to draw a better picture just kind of more focus on i'm here instead of if this is position no i i know physicists represent time vertically i've learned that recently so we'll do time vertically so this is time this is position right and so i know i know i'm here let's say i'm in a well-defined state right and at time like t prime my position is going to change and it can be like that i'm here here or here kind of just i don't know i'm really guessing right so you have these these potential different paths i don't know there's no real your path because that that that that is so small but that's the thing we're integrating over small small position intervals that's what i'm that's what it's confusing me a little bit right now because you're rather you're rather talking about like when i'll be there you know what i mean yeah which kind of makes sense because you want to know will i ever be there right will i ever be there so you're saying uh yeah so you're you're going and saying for each for each little little change in here in position and i'm just imagining position is just one dimensional in this case for for that for for the the the dx right and and wrong because the thing is because the thing is you might have you might have so this is this is x at t zero x at t prime is this one like x like say t prime prime is like maybe you know what i'm saying is like the position you know you might have a superposition at least of of your position right of your state and so so each of these points is going to have a sort of a dx like associated to them right so you're saying for each and so you're taking that final state and you're doing a projection on no the integral is because the integral is just like until here right it's confusing me the formulation is confusing me in terms of is this bounded on on position so are we doing like why why are we using the same x prime notation here where we ah so you're taking a look i'm just having just trouble reading these right it's just i'm not used to the um so so we're looking at a state we're looking at the wave function we know the wave function at time t prime right and we know that this can be yeah so we know that a time okay so another time t like say like or this is t zero t one i try t1 t2 when there's going to be d3 right let's say we're taking a look at t2 at t2 so we're going to take a t2 then we have multiple possible x primes right for example let me change color so so we have this guy here and this guy here of course these these have an associated interval right so what we're doing is or yeah you're projecting each of these and basically but how how is that uh i don't i'm not so i know the integral in this case is is like you can read that as the sum right i'm just um it's another kind of mental blockage that i have that i'm not so sure how like what's the role the integral is playing here um i'm just basically kind of pretty much blocked um just by by understanding the exact like what what is what is that that the this is telling me right and just i get it conceptually that what we're doing is we're calculating that we're adding these projection things right because essentially when we're when you're doing integration i might just have to go back to the basics right but that's what you're doing integration of a normal function of classical it's just like say you have a constant function right x equals 5. oh um f of x equals five right uh like the integral of these is is equivalent to calculating the area under the function right so you're saying what is the yeah i think that's that's very basic but that's that's what that's what's that's what's confusing me here is this integral known as the pathing of militia quantum mechanics method gives the same results as those dictated by showing a picture but also eliminates some of the deeper aspects of quantum mechanics in this paper we will present the method used by feynman though it is pedagogically backward we will then demonstrate the use of method before showing its equivalence so showing the feature we will then investigate the method as applied to the harmonic oscillator following these will introduce the concept of clearly impact girls and discuss further easily okay okay so that's the propagator thing but let's let's stick to this a little bit more i i know because i also don't have you know more and i i'm gonna try to schedule longer sessions because now it's getting a bit more i want to be able to have longer time spent while i focus on these things um but that's let's start be boring i just want to make sure i understand i understand this a bit more in terms of what is it that we're doing what is it that we're integrating because what is it you're integrating that's not a how do you integrate a wave function like what does it mean to integrate a wave function well especially what is like you're not integrating a wave function you're you're integrating the inner product of a wave function and the inner product of wave function this is color isn't it integrate in a product like inner product integral so there is something the integral form of inner product i wouldn't know how scientists know that the inner product of f and g equals to the integration i like that it's a definition shut up and the guy's like oh there must be a way to derive that right yeah okay cool let's let's see because that that seems to be what i'm lacking right in terms of how do we know or not that's not maybe like it's what's called the inner product integral it's what's called the inner product integral and it works like this its definition for periodic signals is that a equals uh 1 divided by t times the integral over one complete period and i'm just going to do integral that's not that's not that's definitely not a the stuff that i'm looking for how do you integrate a vector product okay so i'm given the book says that integrated this with regards to time gives so i assume that you here is cross product which is right if you have two vector functions what is it what are vector functions so the vowel is in r3 then you can form the cross product this is again a vector function maybe i'm having a hard time understanding what this is it's an inner product of two way functions that's going to give me a scalar isn't it it's a transition probability i don't get it why it's an integral right that is or is it that the integral is the whole the whole thing ah maybe the integral is the whole i don't get it maybe i just have to go ahead but that's what's that's confusing me that is the unless unless what that's telling me is do the inner product confusing this is confusing me as in like is this part of the integral or not like wikipedia path integral formulation oh actually i think i had it open already okay that goes too much directly into the exponential notation which that's not that's not what's gonna so one way to read that integral would be you calculate you calculate the probability so you calculate the probability of each of these points the transition here this is each of these points for like a small delta the small delta x or the whatever the the dx that actually represents that and what you're integrating is then that so you do that and you're doing the and that is this color right times because that's then the probability okay of being in here right so yeah so indeed actually that must be part of these it's just weirdly formulated i thought i don't know why is this something you put in here and not at the end for example right because then you you you multiply that by this wave function and so and then you're doing the integral across time position position i guess you do that for all the different positions it's very confusing it's indeed a bit confusing let's see let's see if we get a bit more maybe practical down here right so we get the path integral method and then you define the propagator of a quantum system between two space-time points okay so we're we're working at in space time here but be careful with this to be the probability transition amplitude between the wave function evaluated evaluated at these points what is it the wave function evaluated at these points okay so what the value is so you so so this is the propagator and you give it an x prime and t prime and an x zero and a t zero and that is that is exactly that is the shorting away right so you do the inner product so you kind of project one one to another um so you know what the probabilities that you transition there you know you know the points and the space time and so if the inner product gives you zero it means that you're not you're not going to be able to project you're not projecting so they can project on onto each other so so there's no chance yeah i still have like a hard time with this projection thing but okay let's assume that it's let's assume it true and then if the hamiltonian carries no explicit time dependence we can relabel the first time value to zero and work only with elapsed time t prime minus t zero what so we will often write three as just that to illustrate these the propaganda valve along with an initial state cad fully describes the evolution of a system over time it is also cast custom customary as is done in sakurai to to use here the simple k instead of you and refer to as the kernel or the faint the feynman kernel the path integral method as we are about to see is an explicit way to construct this propagator we we can see the possible trajectories xt of a particle moving forward a through a time independent potential with n points fixed at x zero to zero and x prime t prime an infinite continuum of such trajectories is possible each with classical action i gotta revisit the action stuff yeah the firemen points poses that the contribution to the propagator from a particular trajectory is these that is every possible path contributes equal with equal amplitude to the propagator with the face related to the classical action okay that is getting interesting summing over all possible trajectories we arrived at the propagator the renormalization constant is independent of any individual path and therefore depends on time okay so this is where we kind of move from these thing to like these exponential thing here and we don't understand again what the action is i addicted to this before so the action is kind of is what your system will naturally try to minimize right something like that so of course each path has an action associated to it and like minimal actions will have higher probabilities to be to be the ones that are taken i think something like that okay so actually we're going to the classical action here a bit more how can it be that infinite some above does not diverge the different phases are keep these okay so that's going to be have to do with interference i guess but like if some above does not diverge let's try to understand why you would even ask this question to yourself right uh step back a little bit try to understand the propagator more there's a probable transition amplitude between the way the propagator is the probability transition amplitude between the wave function evaluated at those points so the propagate is defined as in something that is associated to two specific points i kind of keep keep coming back to these to this way expressing that below the cancer eigenstates of the position operator such that integration all over x spans the entire basis so you're integrating over x let's remember what is psi of x t is okay so that's a wave that that's that's how is the wave function i mean the wave function is these right that's the wave function it's telling you kind of for each x and t what is the th is it because i'm coming i'm coming from a pure quantum computing background perspective you just say that's psi right but it's not as it's not a function of anything that's just sine then you have your base states right here the basis is the position opposite the version of expands entire basis i guess the way of reading psy of xt prime is uh that that this is a wave function whose components are the eigen states right so the the possible definite values this could take and then they have that um complex amplitude associated to it um i see i'm kind of like burning my fingers a little bit probably with these um at the moment uh i'm kind of lacking a solid understanding of of how you how you go and read these just coming from a you know a more strictly kind of pure quantum computing perspective where you just have more abstract you know you have a system of you know n qubits and then you have like 2 to the power of n basis states and then talking about a psi of x it's what's confusing me a little bit here doesn't allow me to understand this better and that kind of confuses me as well so i need to stick into these a bit more work i guess so but i guess i'll leave it here so what have i it it seems to it kind of makes sense but i can't really make i can't really appreciate the details these the projection kind of thing i understand and this is confusing me a little bit i mean you're kind of reconstructing if i would understand these as a sum right you're kind of reconstructing you're reconstructing the wave function at like a given point you know x prime t prime if this is like t prime and this is like x prime right by saying that um that you're you're integrating over all x right now that's that's that's that's wrong that's not where that's not what you're doing that is just not that that's just a wave function so how is the wave function at t prime so at this point in here so what is the well you know what is the wave function you're defining t prime right right but you're saying what is the wave function here so how you know how does it look like across the different x points and then so we're doing is you're you're calculating the amplitude the amplitude that you're in each coin uh and then and then multiplying it as you know that that's the that's that would be then an amplitude for for these as these being the basis state right so if you if each position is part of the bases then you're calculating the probability that you're here no that what is the complex coefficient that you're here the complex coefficient here the complex coefficient you're here and that gives you that wave function but i don't know why what is the integration here doing like mathematically speaking i understand that it means just consider every single pawn in this continuum right because position is continued so it's understood as continuous so that's where you're doing like from plus minus infinite plus infinite um and that's why it's an integral not a sum right because it's uh and that's of course a really small interval and if you do that then you know yeah but essentially that's just saying it's just formalizing what is being said here but it's not clear how how do you kind of go about calculating these like for some reason the equivalence between you know the sum and the integral makes sense but it's still it's just i know it's probably just notation but it's just it's just still coming hard on me like as in like what is the integration part what does it mean if i want to do these like what what what's this how do i calculate the integral of how do i integrate yeah integral sums integrals like i got a dick into this it's just basic math i guess but yeah and then you do this propagated thing where the propagator is just a very specific so the propagator is just taking down a specific point in time right two specific points and so what feynman's saying is that that is trajectories and then a trajectory has an action associated to it and then each action contributes to the propagator in that form so that that was fireman's claiming right so it's like the action being s so this so so this is the way that these contributes okay so now sort of the next challenge is how that how the hell is that how do you you know derive that right so how do you how do you how do you get here all right here we get a sum of all the trajectories why he get a summon and like not an integral right cool i mean i get i i kind of get that a little bit more let's try to understand a bit more how does the action play into these we'll do that in the next video bye bye |
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reading day um and i'll try to yeah i mean this will take me this will definitely pissing me for a couple streams uh i just wanted to take a look at these which came across my twitter um you know kind of i don't know how you call that dashboard or twitter you know whatever the recommended things kind of like just because i'm following jen's isaac and then i think he shared these and a bunch of other people commented on that and i just i i kind of went through the abstract which i forgot i have to go through this again and i thought i'll do that because i i definitely want to read papers from time to time i think it's interesting especially when it comes to kind of resource theories and things like these and um there's gonna hopefully gonna do something in there about entanglement as well and you know um the whole thing about quantum complexity is something that i've recently stumbled upon and that it's not i guess it does not refer to the computational complexity like you know the old big o notation and stuff but it's rather something about the complexity of a state so how complex or how complicated it is to build a state maybe or maybe this complexity is going to do with complex numbers i don't know um we'll we'll we'll figure it out i guess but i so let's um let's see i mean what do i expect about this paper to be honest i don't know um i just want to see if that you know it's always one of the things where i think reading this paper sometimes kind of uncovers little little aspects and little corners of the quantum mechanics and quantum computing world that once didn't know before and so you know you can then either turn them into rabbit holes or you know they enrich at least sort of the the knowledge about the landscape in this case so um [Music] let's go straight into these and let's kind of do the typical abstract conclusions and figures because i did scan this through and to be honest um you know for my taste for my taste as a non-expert and and the subject matter this seems to be quite some i guess but that's the appendix so let's see how much we really want to get you know deep into the appendix actually because it's it's it's a proof and complexity starts to prove i mean eventually i would be cool to go through these things and and kind of double check them just for you know for the sake of just practicing that as well but this is quite complicated this is this is definitely something that i wouldn't be able to approach right now i'm not i'm not at least not just like that i guess if i spend time enough i'd eventually go through something oh let's see okay so that's that's actually this is a long appendix and then uh but the paper goes down until here okay some conjectures let's see um let's see there's not so many figures and whatnot but i guess from the title that's to expect to be expected i guess it's more about describing a resource theory um because so let's kind of recap what it says quantum complexity is emerging as a key property of many body systems i like that it's about many body systems including black holes topological materials and early quantum computers so what are topological materials though actually i've seen that term come up a couple of times before topological materials uh you know one i'd like to actually see that topological insulators are a new state of quantum matter uh topological materials this is probably two what is this interpolational materials topological insulators are a new state of quantum matter with a bulk cap and odd number of relativistic direct fermions on the surface the bulk of such materials is insulating but the surface can conduct electric electric current and well-defined spin texture okay well what are these useful for these exotic materials could find applications in electronics catalysts and quantum computing just trying to figure out a way to build even smaller transistors that already existed making nanowires of bismuth and nickel that were only four to ten items thick he still doesn't precisely know why the wires had such thing uh high resistance in the four low conductance but unfortunately getting around the contacts okay so are these just i just i guess i don't know where the topological comes from the idea of topological materials grew out of work in the early 70s when physicist michael crossed her leads of partners and actually used the concept of topology to explain why super super conductivity superconductivity happens in certain materials at extremely low temperatures but disappears at higher ones topology but topology that's just that's a my thing isn't it maybe physics there's maybe a physics concept to it attached the manga so in mathematics topology do this concern the properties of a geometric object okay but i'm interested in the context of physics topology topologies relevant to physics in areas such as condensed metaphysics quantum field theory and physical cosmology the topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and material science electrical and okay so i guess it's just the study of the um like sort of the surface of the objects right of the materials okay the arrangements of how the the the molecules are structured and and things like that okay topological materials so i guess this is maybe when you take a look at materials from a topological perspective maybe rather than just like pure i don't know chemical composition or something like that uh a state's complexity quantifies the number of computational gates required to prepare the state from a simple tensor product i guess a simple tensor product would be just the basic say initial starting state like the all zero state for example right the greater states distance from maximal complexity or and complexity the more useful the state is as input to a quantum computation okay so that is that is that's interesting right so i i'm curious to know and what terms is more useful in in terms of like being more resistant to the coherence being more um i don't know resistant to noise i i don't know maybe highly complex states well of course right because highly complex like to prepare highly to prepare complex states you could use a lot of gates and so that means you better have like really good error correction maybe i don't know maybe naive but separately resources resource theories simple models for agents subject to constraints are um are pronounced at birth journaling and quantum information theory english english why doesn't this catch the thing in the first place burgeoning i'm not germinated i don't know what the alph caiman means i want a definition you know never search for translation just search for definition of a word growing spending developing rapidly okay we unite the two domains confirming brown and saskins conjecture that a reserves theory of uncomplexity can be defined with this theory of and complexity so i guess by resource because this the resource theory of entanglement as in like how useful is entanglement right and i guess i guess a resource theory [Music] dictionary of psychology i don't know that's a good source i don't know what's going on with my internet a theory of interpersonal relationships holding that the amount of resources a lot yeah okay but i guess you can translate that right in terms of like see you have different agents you have different things you want to do with and then you've got like resources and so that in a way constrains what you can do the more resources you have the more you can do things like this like in in terms of entanglement i think it's it's something along the lines of the the highest amount of entanglement from a resource theory perspective is the entanglement that allows you to kind of from that place it allows you to move to it to any other possible state both entangled and not entangled right and the minimal resources when there's no entanglement because because you can't get to the whole entangled state space only with like local operations right um yeah i think something like that the allowed operations fuzzy operations are slightly random implementations of okay so the allowed operations fuzzy operations are slightly random implementations of two cubic gates chosen by an agent we formalize two operational tasks and complexity extraction and expenditure their optimal efficiencies depend on an entropy that we engineer to reflect complexity the entropy being the measure they're gonna define here to reflect that complexity we also present two monotones and complexity measures that decline monotonically under fuzzy operations in certain yeah okay so i get i guess i guess the idea here is you you've got a state right and so the more you apply this op the more oppressions you apply to it the less resources you want to have right because i guess that's the point it's like that's how you define this complexity or and complexity right it's how what's the minimal uh what was that the the amount of gates required to prepare state from a simple tensor product i forgot what a monotone is in mathematics it's some kind of type of god can't i just get this stuff in english because i'm in cognitive mode right english yeah but is it gonna remember that i guess a monotonic function is probably a function that just kind of a monotone is a function between ordered signs that preserves or reverses the given order these constitution calculus or the theory okay this is not monotonic okay i guess what it means is that it preserves the order or reverses it from with respect to the dimensions right so if this is the important thing now this is the input this is the output or the other way around like you can see that for higher so the more you go here the the the farther along this axis you go the farther down along this axis you go as well from a from an output perspective whereas here it's not like it just messes up the order right which is which is what you want when you define this kind of stuff because you want to you want to make sure that the more you know the more of course the more operations you use the less resources you have that's because the more complex than the state becomes um unleashes on many body complexity the resource theory toolkit from quantum information theory cool so let's see let's take a look at just quickly the sections how this is structured so this seems to be a bit of an intro resource theory thingy and then we have a figure here it says state complexity and it's geometry so a a pure and keep it state psi has an exact complexity c of psi of psi equal to the least number of gates required to prepare psi from zero and here what's interesting is that they pick this set of they pick the set of gates to be random two cubic gates i don't know why because maybe okay because i mean this definitely allows you to include entangling gates and maybe maybe the thing is this kind of allows you to also include things where you apply the identity in one qubit and then another operation at the bottom keyword you know what i mean so kind of covers all the cases in general thank you um the state's exact and complexity so equal to the least number of gates required to prepare psi from zero from the zero state the um the state's exact time complexity is a distance c max minus c but uh i'm curious why this is defined in this way because i would intuitively say that c-max doesn't have to be unbound like it doesn't have to be bounded well like you can just you you can't just you can go nuts if you want like you can have like okay unless you unless you'd be saying that you know you would obviously not consider circuits that are redundant or circuits where it's like okay so if you combine these gates but it's this is equivalent to do these other gates uh which is like less amount of gates and that defines and so there's a maximal complexity so you want this difference to be as as big as possible to the maximum incubate state complexity so why is this interesting why is this e to the power of n what okay in a revision of nielsen's geometry complexity curves the state space negatively complexity curves the state space negatively applying a gate corresponds to moving a unit length along a direction in this geometry yeah most directions point toward higher complexity states to transform into a generic state psi prime psi likely must pass through the zero state no significantly shorter path exists that's interesting so what this is saying is that that that likely to go from side to side prime what you have to do is then compute psi and then compute psi prime so kind of like there's no that there's not always right like a path that goes directly from side to side pi directly that is shorter than i'm computing and computing b i don't quite understand why the complexity curves the state space negatively what is reference 15 good luck understanding any of these i'll just open it and see what happens it should always be a way to go back to the reference uh applying a gate corresponds to a unit length along the direction in this geometry i mean yeah you kind of have so you're in you're in in a geometry right like in a space that is like the complexity space and so you're like at a given point say and then um of course applying gates will always move you somewhere right but it almost feels like why why would that space okay yeah because depends on the depending on the gate you apply right like the the thing is thinking like why isn't that just kind of like a one-dimensional space right but it's more like but that's not true right because depends on the gate you're applying that might limit the choices you have for later gates to apply and so kind of each step in the process constrains sort of the next steps right and and and if you make the wrong decision then you might end up with a much more complicated path um yeah uh [Music] consequently and complexity is desirable in quantum computation given a complex psi to prepare a desired output psi prime one must uncompute that that's what i said side to this and incurring an overhead so that's like yeah you kind of have the overhead cost of doing that which is basically you know the cost of is the cost of psi prime to be honest our resource introduces randomness green shaded regions into the gates such gates tend to increase the state's complexity okay using the resource theory okay no sorry that's already the figure so what else do we have here so we have definition of the resource theory of uncomplex so here's the definition and i see already some kind of su4 blah blah blah so things which i'm not familiar with but i'll definitely have to get into definition one every fuzzy gate is defined in terms of an arbitrarily chosen two qb gate u the fuzzy gate u is selected uniformly randomly according to distribution uh of these that satisfies conditions i and i i above one into above every composition of fuzzy gates is the fuzzy operation okay so these are just definitions what else we have complexity entropy the complexity entropy of an incubated state i guess this row is uh the minimum [Music] of the trace of q oh god that's complicated okay so here they go and explain that a little bit take a look at these complexity entropy suppose that oh god of course you might still have like mixed states in the whole thing right let the number of r of performable gates be high compared to the number of gates needed to prepare psi so you are approximately a dust gate psi it's a mos tensor factor [Music] cube projects into a lower dimensional subspace so very small as it requires in this extreme case i don't know why that what is a trace here what's got the trace to do here but that might be interesting because at least we'll kind of maybe give me some insights in some of the some unknown things yeah it's good okay so this is about the actual theory then there's a theorem theorem theorem theory theorem so some protocol that extracts okay so figure two let's see what figure says operational tasks of uncomplexity extraction and expenditure uh operational tasks of uncomplexity extraction and expenditure in the resource theory since gates are fuzzy the agent can perform only less or equal to r gates less the state grow too noisy to be useful yeah a extracting on complexity from a state okay so the extracting and expanding extracting is like the inc like i'm computing and uh extracting and complexity from a state row one applies and fast against the number of qubits left in the state zero is the extractable and complexity which equals which equals the complexity entropy of rho yeah given in all the zeros an agent can spend on complexity to imitate raw the agent performs are fascinates preparing the state belief by a computationally bounded referee to be a row okay yeah are the fuzzy gates there just to kind of account for errors i don't know maybe um it's still it's not obvious to me right that you have to pass through zeros what the is this palm in the chat man bye viewers and big fellows that come you're a guy you're getting a band give me a second how can i ban people can i ban people yeah ban ban time out mod ban you are sir our band that's not the kind of messages that we want in the chat um cool so uh that is not so obvious something to go back to that you actually have to go through the all zero state i mean let's think about this for a second so if yeah so actually it it might make sense because if you define so if you know that you have a state but that's because of the way it's defined right so the way the whole theory is defined because if you're saying look the resource theory this whole research theory thing is like um i'm gonna define complexity as in the amount of um gates it takes to move from an all zero state to my desired state right and the complexity of a state psi is the minimal amount of gates that that take you from zeros to to that state so that's the minimum so yeah so actually you know if you reverse the thing like if you have a stair state you know psi you want to go to psi prime like it's it's almost like saying actually the path from side to zeros and and zeros and psi are like both minimal in terms of you're adding two minimal quantities and so probably that is the minimal path probably but i don't see why that has to always be true like i can have a state i can have a state right even just like say the bell state and and i can say look like i only need to apply one operation to get to this other psi prime right uh but if i would uncompute and compute i would actually have more so it probably is one of the things where it's true for most of the cases but it's not generally true um but i don't know we'll i'd like to see if the paper gives a bit more details into these in the text a theorem three uh so this is monotony city whatever however that's pronounced um i guess you won't know this function is always it's it's monotone conjectures theory two theorem two let p let row denote and every triangulate state like r and uh i'm so bad at greek letters um as both and assume that let's be a two times r and this is i guess an error or something for every and for every n qubit state can be imitated with worldwide okay gotta unpack that conjecture one they cannot increase under fuzzy operations okay so and already conclusions are here already i can go through these now and then there's the and complexity extraction proof that's for the details okay so what do the conclusions say uh we have confirmed brown and syskins conjecture that a research theory of complexity can be defined the the research theory is allowed operations balance random evolutions which model chaotic systems with the agency in resource theories the agent choices operations to perform using the resource theory of formalize and complexity expenditure and extraction the tasks optimal efficiencies we quantify with the complexity entropy that we introduce finally we identify two fuzzy operations monotonous and regimes those are introduces into quantum complexity the resistor into blocks as garnet successive cross-conformation theory our resource theory deviates superficially from two holographic conventions what the hell like holographic conventions we invoke the circuit complexity instead of nielsen's geometric distance correspondingly gates act in discrete time steps whereas hamiltonians act continuously our model is motivated by quantum information theory where discrete dates from circuits the closeness of circuit complexity to nielsen's complexity and uh random circuits modeling the chaotic evolutions uh reasons suggest that our results might extend from fancy gates protecting for turkmenistan evolutions what is this nielsen's nielsen's geometric distance what is this this work establishes several opportunities for future research first the complexity entropy um evades the shortcomings described below definition two this entropy can quantify the efficiencies of computationally restricted tasks such as data compression with few gates and beyond and complexity extraction and expenditure computing the complexity entropy may be difficult typically however the complexity entropy properties and applications to information tasks and one-shot thermodynamics will be explored elsewhere with one shot thermodynamics second the complexity entropy suggests an operational answer to the question undergoing active research how to define mixed state complexity oh yeah that's interesting that's what i said right it's because in theory that includes mixed states as well quantifies the difficulty of extracting uncomplex zero cubes more precisely the complexity entropy can anchor a version of the strong complexity introduced in 73 would evaluate the complexity entropy similarly three and four may generalize to by the regime regime fourth the resource theory framework suggests many questions locked in 16 for example can allow operations in interconvert any two states asymptotically if every time many copies are available furthermore we anticipate connections with other resource theories for focus on the difficulty of implementing unitaries the registry of the resource theory of magic what what is this that's cool fifth the resource theory can impact holography many body physics and quantum computation one might reframe black hole paradoxes quantitatively in terms of resource extraction and expenditure also though fuzzy gates involve little randomness they can oh give me a second thumbs up the door there you go back cool um yeah we'll have to do it here soon but basically what else we're doing here one might version black hole practice completely that's it i'm missing i'm kind of missing i'm missing in a way to um i think i graph the general structure an idea obviously then you just have to go through the way things are defined and understand that but maybe that's less relevant for me right now although i i i will do that in the next in the next um string this week um i like to understand a bit more the whole thing with the going through zeros and whatnot um [Music] but again like i i don't fully understand i i don't fully understand or can take from what i've read at least sort of how that maps to the usefulness in quantum computing so to say right uh application of resistor of magic states i forgot what magic states was where uh uh one shot through one and one shot now good luck finding anything here the geometry of quantum computation is this nielsen like the nielsen from from from uh nielsen and tron the book [Music] yeah i think so uh you know your quantum processes but i don't know really what the what the usefulness applied to quantum computing you know obviously resources for magic stays two faults and the taller quantum computer is where was i let me see congratulations geometry come on i forgot the paper i was in there though uh resource theory of uncomplexity we can see a system of n qubits the node by the powers that appeared in an event eigenvalue and vector denote the one qubit identity faster gates form the building blocks of the allowed operations the agent can attempt to perform any two cubic gates and su-4 this is something that i've also i've also have to at some point get into in terms of the mathematical background of all these you know is it category theory i guess the groups and whatnot right and it took on any two qubits our results generalized to two scenarios in holographic literature first each gate can couple a second the target gates can form a discrete set uh the gate implementation suffers from noise as follows fix an error parameter denoted by there's no direction it's not zero uh i'm getting lost in here i'm getting definitely lost um uh and try to understand where is the usefulness of the resource theory of uncomplexity right like it allows you it would allow me to sort of order and sort states by their complexity right so you kind of can like yeah yeah i guess it's more like a theoretical application i mean it's practical as well right in terms of to be honest i don't know i mean resource theories are useful i guess to discuss about like optimal transactions optimal things to deal with optimal ways to deal with transformations and things like this or even like decidability at all right like can we do something with that state that can can something be done with these states like no we it's too complex like or or yeah uh i guess that that's that's what this theories are uh are for or what is the resource theory for i mean that that is i otherwise resource theory quantum resource theories oh there's a there's a our off uh hyperscience powerful framework for studying different phenomena in quantum physics from quantum entanglement to quantum computation research theories can be used to quantify a desirable quantum effect develop new particles for its detection and identify processes that optimize it yeah okay cool so i i okay so that that answers my question i guess hey different phenomenon in quantum physics from current time into quantum computation resource theories can be used to 25 example quantum effect that develop new protocols for its detection and entire process optimize its use for a given application particularly accuracies are have revolutionized the way we think about familiar properties of physical systems like entanglement elevating them for being just interesting fundamental phenomena to being useful in performing practical tasks the basic methodology because for the the entanglement reserves one it kind of makes sense because this this whole thing of how useful is this resource because i know i know at least i've read that certain states will help you will will actually act as catalysts right so we'll help you in conjunction with another state to reach another state so that's this is something that that's why that that that seeing entanglement or building a resource theory around entanglement kind of makes sense but building a resource theory around just the complexity of a state uh like i'm missing the point there i think in a way so you have free state a free quantum operations and then yeah you have restrictions placed over the place and the studies how different processing tests become possible using these research operations [Music] reference frames appear to have great similarity or deeper structure levels it's already original framework for quantum resource theory focusing common structural features okay that's just like a meta thing about quantum resource theories um [Music] which still like that might be an interesting thing to take a look at to be honest like it's probably emotionally how you go about defining a resource theory for these stuff cool okay so but it's still i've got two open questions one is sort of uh uh like a concrete application of these which i'll i'll see if i can dig one out and then some details right so i can dive into the technical details uh probably spend a couple more couple more streams on these anyway it was fun uh reading through the general structure of the paper cool see you next time soon any of these days i haven't really figured out a schedule yet because i can't with the babies at home i can't i tried but i'll try to do like i'm trying to do two to three streams a week that's my that's my goal and it seems to be working so far so um cross fingers that's that's gonna that's gonna work and stick for a while bye bye |
One day i have to make a list of all your papers you read and send you back to upload to your github to create an awesome quantum computing papers list updated + twitter accounts, That sounds like a great idea also just to keep tabs on the stuff I read for myself. Some of them even with code and maybe attempting to replicate some results :) I'll eventually get there that I'll have more time for this so stay tuned :) |
correctly and what we're going to do is what are we going to do what are we going to do we are going to basically um we're still we're still doing the we're still doing the uh fourier transform stuff so i want i want to kind of do a simple example um i want to try to do a simple example back and forth and see see why and how that works so for your transform i mean first of all what's the difference between um something like the fourier transform in its pure form versus the fourier transform something like and the algorithm like the fft fast fourier transform right so that's kind of like another question that i have i guess the algorithm is sort of this specific list of steps to to do that given a specific function or something um the descriptor transform yeah uh but i'm like mathematically like in a more puristic way like what or how do they how is this defined no way we don't want i should change that i should just change that how can i change that so it always speaks english by default maybe somewhere in the browser settings i think um a finite sequence of equally spaced samples of a function into a same length sequence of equal space samples of discrete time for a transform which is a composite function of frequency i mean that is a very specific algorithm this i i'm for sure fourier transform let's just stick to the fourier transform that's the integral right so um definition there are several common conventions for the definitive free transform of an integral function one of them is these the transformers denoted here by adding a circumflex yeah that's it when the independent variable represents time t instead of x this transform variable weird thing represents frequency so that is a function of frequencies uh i guess what this means is that if with a given frequency you have a specific function i think that is what it means right maybe i'm i'm starting to understand it needs to be better so it's like because you always have to think about the the underlying so so what that kind of thing here means is that it's basically a map between you know for all frequencies that are real right because i think that's what it definitely means here so say frequency one you map it to a specific function right um we'll use the hat or something like that um frequency two and then you have another function like like two one right and so each of these right is you can obtain that by doing the by doing the integral so essentially what that means is it's basically a function that each input is a function so it's a function that outputs functions right if i understand that correctly like let's do let's do like could we do a simple example where it's like the fourier trans like let's pick a function let's just say let's just actually take something that is basic right although that might be difficult because it's a bit of an edge case but let's pick something like i don't know um again what was that app thing desmos this moss i think is desmos decimals graph calculator so let's pick something like three times sine of x plus 2 times cosine of x times 3 or something like that right so let's speak up this function right so what what is the what will the friction form give us in here right so basically what the free transform will do here is i mean okay so f of x right so so f of x because sometimes this comp these concepts are so simple it's more like um trying to understand them at the meta level and and rather than just the technicalities of it so the f axis is like three sine of x plus two times cosine of 3x that's f of x right and so and now what i don't quite get is how do we approach like that right so so what is it what is this supposed to like how do i populate these shitty poo thing here so this kind of weird e like is these um something that you you just say cool so it depends on that belongs to the reals um and then it just depends on how granular you want to go like you wanna that's what i don't really get so much so you go from f of x with the free transform to this thing here right so that is known as the inversion theorem that that's the the the other thing um angular frequency instead of ordinary frequency the other convention conv conventions um the phrase clearly in space history differently it was a very exciting business position aha so that is treated differently harmonic analysis i will never know what the these things are okay so let's let's look at these this is what we're interested in probably the first transform can be defined in any arbitrary number of dimensions as with the one-dimensional case there are many conventions for an integrable function f x this article takes the definition um so x and this thing here are n-dimensional vectors and it this is the dot product of the vectors alternatively this can be as belonging to the dual vector space whatever that is in which case the dot product becomes the contraction of x uh all of the basic properties listed above hold for the n-dimensional free transform as do uh plateaus and vermont's theorem for the integral the freezing term is still uniformly continuous and the riemann levesque lemon holds laplace transform complex domain eigenfunctions okay so we are here in where are we here for your transform why does this ultimately have this e component to it this is what still confuses me i mean that is definitely the component that is used to kind of create that um the sort of the uh the repeatable pattern right because that's the point of a complex exponential but why is it an integral that's the fourier series yeah that's that's these anyway so there's no no no special case in here number eight functions synthesize see i understand i understand the concept of the phrase series but i don't understand the integral part of these an example okay so there's an actual example here that's actually uncommon for the wikipedia stuff the following figures part of the visual distortion however measures whether frequency is present in a particular function the depicted function oscillates at three hertz if t missing seconds and ten squared to zero the second factor in this equation is an envelope function that shapes the continuous sinusoid into short poles its general form is gaussian function this function was especially chosen to have a real fruit transform that can be easily plotted the first image contains its graph in order to calculate okay so here there's like you know calculate like the first transform of three it's like how do we know that how do we even like the second image shows the plot of the real and imaginary parts of this function the real part of the integrand is almost always positive because when f is negative their part is negative as well uh the result is that when you integrate the real part of the integrand you get a relatively large number on the other hand when you're trying to measure a frequency that is not present um okay so you know on the other hand ah okay i get it so so what this means is so this is just general recipe right so what this will will ensure is that when if if you're trying to measure frequency that is not um that's not present then you're getting something like zero on the other hand when you try to measure a frequency that is not present as in the case when we look at f5 you see that both real and imaginary components of this function vary rapidly between positive and negative values as plotted in the third image okay so why five hertz therefore in this case the integrand oscillates fast enough so that the integral is very small and the value of transform for the frequency is near zero ah there is that that's okay so so now i understand what this is telling us so what this is telling us is this is just a formula that you can use to measure or kind of kind of that's why you have where was i here right so that's telling you i get it okay so that's actually building a function that is why okay i get it so that is why you because i was always thinking like why why is it that like how do you how do you make the free transform give you actually the frequencies right and then i would realize that in a lot of the examples you see online you kind of see like okay so if your function is something like that then what the free transform does is kind of squish that into a function that it's telling you the frequencies right so it's like it's a function that for example like like this right and so we'll have peaks telling you kind of what is the combination of the different frequencies and you have to interpret um these values here as the frequencies that make up these functions right that that's why that's why you have that funky thing here because essentially what this is telling you is like cool so if the frequency is not going to be there the integral will basically the integral will basically evaluate to to zero right cool now the interesting thing still to figure out the the the thing that's still important to figure out is that why why does this work like that right so you have to function like why does this tell you that why does this work so you have a function and you multiply it by these right like this this the the exponential part is chosen like that for very specific reason which i don't know which one is it but it it's i guess what will well let's analyze that right so let's try to uh let's do it with our example i like let's do it with our example so but that's going to be yeah that's going to be impossible to plot um well let's simplify the example let's just so the fourier transform of the sinus function so just something very simple right wall from alpha let's go awesome here you have okay yeah yeah that's the thing the frequencies okay now this starts to make more sense so if the frequency but i i won i don't want w the angular thing i won is there a way that i can uh normalization oscillation factor is there a way i don't want that i just want the uh well i mean what's the frequency right the frequency is a the frequency is like because that's just the amplitude so the the frequency in this case it's just like pi something right or is is a factor of pi frequency of this sinus function r1 okay okay so one cycle occurs in two pi oh yeah okay so you're measuring it against two pi that's here right so that's one cycle that's the frequency so what would happen in this case how would we how would we approach the let's say f of x is just yeah if yeah that just that that only changes like the amplitude right so that shouldn't have any effect so so let's just say that our example is uh is just like that and so now we are basically well basically what will we do now so we will um like how okay so you have you can carry that so you probably carry that see yeah okay so let's try to carry that i don't know how because i've had to do a complex how to do a complex integral i don't know but let's forget about the integral for a second so how does the f of x times this thing look like so i've have like 3 times sine sine x e to the power of minus 2 pi i x and then this thing sh frequency here okay that's interesting though because this means that what will this function depend on this this pulling e is just something that it's part of the exponential and i can't can i pull i can't plot a complex exponential can i yeah while i can it's just going to be a circle right the question is how how can we visualize that so e to the power of like pi i is how do we integrate that how do you integrate that you can't just integrate that that's that's what's complicated mood a complex exponential complex exponential but i guess the fact that it's complex is the trick here right because you want things to cancel out probably whenever the frequency you choose is not the right one but it's it it seems to me that it's pretty [Music] in a way crazy that that these factors here would have the impact that the only impact you need but let's let's try to do the following so okay so we know that we can expand that right into um and do i forget so um complex exponential euler's identity alice formula there you go so e and then factor it's then it's these it's the complex sinus and real cosines so so this means that in this case right this is basically uh so minus 2 pi x and that and so [Music] e to the power of i and this thing is basically the cosine cosine of all these of minus 2 times x and that uh plus i times the sinus of all that yeah cool right so this is the whole thing and in this case we're just multiplying the whole thing by sinus again right so it's like sinus times these and then sine squared as the complex part sinuses the complex part sinus sine of x which is kind of like cosine of minus 2 pi x and that thing here right so if this is one if this was one how does this look like because the real so so the complex part so we can plot them separately the complex part is i times the sine squared of no sine squared no sine x times sine of minus 2 pi x and that here right so god how do i do that um and this thing here yeah excuse me a second i'm gonna take a quick call yeah that was not the route anyway so what do we do what do we do what do we do so so what if i multiply let's just get rid of this sine sine times cosine of like and we're just doing now for this e little e crop thing being one um we know that's somehow going to be a peak in the in the outcome in the sort of in the result function right and so the idea is minus 2 times pi i like that how the input works here times x with xuminia so what does what is this doing so this looks like that and then the complex part looks like -2 no minus what that thing over here all right so it's okay so it's the same just a little fake a little bit faced interesting um but it's it's just what if you i don't know what if you just say times five what happens okay huh so these integrals cancel or what in the case of one in the case of one i mean i know when it comes to the integral like so the integral like how does this work like the it's minus cosine is right yeah okay so but ah i don't even know what i'm thinking what i'm thinking is how do i how do i how do i if i'll evaluate the integrals in this case right what does that happen cancel out some of these things should cancel out but i don't know exactly why i guess i guess it's got to do with these maybe like if the blue thing is the complex part so you imagine it it's like 90 degrees rotated or something do things cancel out i don't know so this part cancels out or one um complex function so it's a contour integration yeah baby i don't know i don't want to go to quora i might i really might have to get into that have i do i have to so this is a complex value function ft of a real variable t so ft is like something like that okay so that's our case which is i seem to be a piecewise continuous function define a closed interval so the integral is the same like just simply that okay where is the common integral with every value of integration so in this case we will just integrate the two things i guess the point is if one of these integrals i don't know why would things cancel out i just didn't know enough of that so i might have to i might have to dive into that a bit more but that's my intuition something's got to cancel out here if i just do times five and then it's like times five they're getting uh even like the case of two right it's just i don't know man what was that saying that was saying that basically where was i the example now that was not here here not here here the real is like that and the imaging imaginaries like that integrates to zero so there's the original function the real and imaginary parts of the equivalent transform three hertz okay so you do actually when you take that integral you do actually count that as negative oh ah sorry yeah that those are negative areas of course that because of course because the sine of x is not it's just no what am i doing that's no that's also negative that should just be oh man what is the integral the integral is should be negative here right between 0 and and one and and two pi yeah i keep having those basic building blocks of this stuff but simply it's just like they i keep having blanks like that intuitively i thought that was the case but then it confused me that the integral yeah because if i just make a general integral it's like yeah sure it's not going to be zero it's just like in that in that frame is zero zero two pi right because the the you you cancel things like the stuff that's above zero is plus the stuff that's below zero right so the point here is in this case if you just have one like okay but what's the period that's funny like how's that change so that basically this is what it's doing it's just translating by multiplying a function for by this thing here is it that you're able to why two pi x y y it's like why these y minus if it'd be plus then what would happen nothing god but it's got to do with the behavior of of of this multiplication by something simple some not something simple something which is like it's a pure wave thingy right so i'm trying to try to think about um what's the what's the what's the intuition about this it definitely behaves so this is definitely one thing where you'd say like for example take and it doesn't have an easy thing like you can't yes there's a pattern right so but for some reason you're really here taking the two pie thing maybe because that's the example of when things vanish right in this case it's maybe a different way to practice but it's it's it seems that seems to make sense so the real part here is it's it's positive right whereas in this case the imaginary part cancels while here things five hertz don't count uh no four five hertz things cancel so they both get to zero so that's why you have the free transform here okay so that's the the result of the fury transform is that at these values i get the highest and it's really crazy that i think that actually tells you the proportion the value corresponds to the integrals yeah this is the integrals exactly i get the intuition i get i i definitely get the intuition but i'm not so sure i work out i can work out all the mechanics of these and well or maybe i get the technique but i don't get i don't think i understand why it works as in like why why multiplying by this factor has this effect and you just forget about the complex part we can just keep keep just the real part right just stay with the real part like that's interesting three four five six seven eight nine like the higher the frequency you know the more you kind of get this pattern where things just obviously cancel out in the one is not that obvious because these are two negatives so one of these cancels out oh yeah that's a bit dangerous right because here you have like a minus 0.5 minus 0.6 and you have like a minus 0.9 so for your maybe that's just not that obvious the first is from the sine and cosine functions like maybe i'm just taking an example that i think it's simple and it's actually not simple this is definitely not so whoa no it's not that simple look at that that is definitely not that simple i thought it would be something simple though why isn't it why isn't it simple i thought it would be just something simple i'll leave it here guys i don't have time i thought it would be something simple i just picked the wrong example i think like why why like what would the integral give me in this case right it's something complicated i get the mechanism but i don't really fully grasp why it works think about this way i'll think about this a bit more bye bye |
so I got the coat I got the code from from the team from the authors of the paper and I wanted to basically use the time now to take a look at it and then see how much this differs from what I was doing right so and basically I think this is I think this is my this is my implementation yeah exactly Kyle have to it's been a it's some days now I don't have to refresh my mind on this and I think this is the one recent for the for the paper in our chief okay for the Mexican problem so and this is also for the the fork is it the fork you yes that's also the k14 I guess and it seems like if I just you know i I've taken a quick look at these but it is so much in here but it seems those are the results okay it's gonna there's going to be quite some stuff to go through here I think that's it right because they also shared this one but this is a what is these code I think that's a cleaner one is it well that's a more like a tutorial one I can't see I see the there's the the surtitles and here so they show the transpiled circuit okay it's not that it's useful to us I guess right now but with our Toph IX I guess it's the same is it so anyway this is k14 so I don't know reach which one I should choose cubed mapping I said how do they do this so keep it mapping kids and physical kidnapping without them babbling good cowl mapping physically mapping on IP MQ Palencia okay rich if survey this looks like this is some heavy heavy executions in here because they do a lot of jobs in there then pulling all the jobs somehow plotting the results okay so here we got some poor son results I see that all set hasn't looked so okay so it also kind of doesn't look so so good on D like you see the the red ones or pink ones or whatever the color disease it's a real machine that's also because this is my and I was wondering why this turns not this looks like to me I mean you look at the green ones like look at the green ones really okay but that's what this babbling machine read are aromatic alien KL divergence I don't know okay so with this and today I hear palencia okay let's take a look at Valencia so with babbling as you can see doesn't really work well and like like not at all the two solutions of these are like the solution to this would be one like all ones in here and okay and so in with Valencia let's see let's try to work for the details nothingjust was there for the results similarly results of up there so with Valencia looks better now I can't access Valencia if I remember well if I go can I just go can I just I can zoom in and out reset if I go back to I need to get the providers provide it back hands if I execute this point I just get me the back ends I ain't got a good connection today so nurse that but it doesn't doesn't prove me that I I see but I think I can get so I am cute I'm kidding I just wanted to at least try with the same machine but this is probably not available for me so so we're we're bad luck here we could try the other ones and but I see already that it doesn't have to really work Ryan I mean you see the machine the machine here doesn't really now I just want to talk let's let's learn this lesson spend the rest of the video taking a look at the coat and comparing sort of comparing notes see if theif my see if the stuff that I've been the way that I've been building it is is anyhow similar to the way this has been building here and I think I should I think there should be they followed it they followed a bit of a different approach I see because they follow an approach where they created functions that basically generate portions of the circuit whereas I have whereas I have kind of like created the created the different parts of the circuit and then turn them into instructions which kind of leads to kind of leads to this ugly thing here right which kind of kind of doesn't allow you to see the full picture together because I could turn those things into functions that they just returned that portion of the circuit and I think that's that's that but yeah well let's let some I don't know if I with which file should ice I'll just stay with that what this seems what can take a look at Falls doesn't really matter so let's take a look at different the different gates see subdivided Oracle so this divided Oracle are tough Lee's are tough for me stuff legate with one swap device topology optimal motor control X so they do I got a see an X I got a car I got to see what that generates but it seems like it's the algorithm in the paper where they actually and they do actually on compute this is the under computing point that's what's that's something that I don't do and then optimal diffusion operator circuit Haram are then multi control x CN X and then out of mine so this is building the diffusion operator and then the subdivide a Kroger method basically does and it initializes a circuit it adds the Oracle and adds diffusion but it does this only once and do we do it once no we do it multiple times okay okay but let's take a look at it subdivided face I guess this is the part that says so this is say if angle equals zero fruity in range number of data okay circuit RZ so it adds so it this is the this is the part that we're gonna try to do it this way it's not that's not that's not gonna be that's not good let's go back here so let's use the full screen data is the first that is the part that initializes the circuit that's the that is my this one's yeah so there's this seta times pi and then they go and they say if angle is zero then you do they do are that rotations of pi over the number of data and I guess what they if they give it an angle they just do the angle times pi this is for the for setting up the optimal art hopefully em art of em and art of IX art of em so not in the art of em in my case I use you to gates they use our i gates I don't know why I went with the UI or the u2 and the you won games honestly but it's the same idea Ric xri cxr eyes and then this is these are both positive PI over 4 and this is negative PI over 4 yeah and then the T is the tartness the target so here they pass the Cupid's and um tougher eix I guess that's my art off I think this isn't this is the one yeah ah but because wait a second I have Haram arts here and they don't why don't they have Harlemites case instead of for Haram oh and a tea instead of Haram arts auntie's they've replaced that with a u2 rotation let me check the paper was over music so if you place the harem in the TK by this you to rotation which should be equivalent what I just want to see if this is something that's mentioned in here you know but it's instead of this HD this H harem warrant and its harm or implicates and the T dagger in Hanover gates okay and here dated the other one that's equivalent to the T dagger in the harem arcade so this would be something to take a look at whether this is equivalent but it it problems see I'm working some nice a how could i how could I see this so what if I create a new it's a new - 3d notebook I just gotta get gotta get better at in general so what if I is all basically I'll create like one circuit circuit with one qubit and I'll create another another quantum circuit giving and to see I will add basically a Hummer gate and followed a foggy gate and q1 is gonna be a YouTube rotation that it's gonna be this one but he is gonna be one and now I want to basically [Music] keep watch here mmm I just plot blur effect or something like that yeah not block sector I were title science now that's the block nectar I want to plot the plot I want to plot a cubit I did that and I forgot how to kiss get a kiss key not do that I first need to actually run both through a state vector simulator so basically get them and I haven't I haven't touched that for a couple days and I'm already lost I'm gonna get this seen how do I get to see you later yeah I do getting the simulator stay well look at the stake like the simulator mm-hmm and then was it execute see simulator saw sometimes it's a cute cue see me later thousand times something like these see shots act like I've gotta say shots in here shots shots [Music] out of French oh sorry cute Europe keep it zero mmm that's of course not hi x/np got hi ice not defined that's why I say n P dot pi has PI not defined I was fine not defined and now I wanted to kiss kids stay back the simulator what the box here that's what I want all right I find it difficult to remember this but it's me Angus what mrs. source code I know in the source code elements [Music] documentation oh oh okay here measure no backhand get uh-huh okay should get the results actually and then you get it the state vector and that should okay that should just plot block multi-vector let's just do it basically so Topsy this is gonna be C and this is gonna be me top Q so we're gonna get rid of these and then we're gonna do a ski and we're gonna do another plot Q so so from the job so from the job you get the state vector of a given circus you have to give you this circle what you have to give it this circuit okay so that's Q did it just get overwritten do I just have to let me see yeah so that's the same that seems to be roughly the same okay and if I now if I now add two to the to the circuits the was done t dagger and Hatem art or versus these so now I will say what or today did I for example Q is why using YouTube right so Q for Q at these and now I kind of and for C for she basically a tacky dagger and a hot iron he's not defined yeah cuz thats Sarah so oh but I hope I know I don't have twice the t dagger let me just let me just do that again so just that cuz I think that might have added the t Daria the harm arcades twice okay now and now I basically go through these again we're back at zero and throw this again where is here kind of lost on Excel and they do this and we're stuck at 0 so those things those things work there I I gotta get used to I mean it's not that I gotta get used to it but yeah so whatever that's okay those are equivalent thoughtfully swap so I was so okay so this is this is basically these are tough VIX you too so what I did here is for the art half which is the the IX is haram RT and then did our harem artists are the two things in here and then cxt TV g CX r and c x TC x TC x yeah it's the same to do now the quantum so okay so now let's take a look at the top i swap so now I did a trick here I did a trick here where I I added that's that second so that M to correct that already we see me was in the operation so I don't have to change whatever comes next in the circuit tougher we swapped off we swap so c1 c2 t that he you know takes it to c 1 t c2 so now that we complicated to check because the spin States basically swapped the rest of the circuit after that but that's why I don't see that's why I'm thinking why why can't we just swap it back right after these you know I just then it leaves the contract of that function sort of sort of the same device topology okay so but this is the second look at this so we've got a heart mark c CX n TD g hamas listen to gee I'd love to compare the side by side on this way so we've got this seems to be the same since you the same I had a career at the rear here okay this doesn't seem to be the same let me paint that circuit that's kind of easier to compare so it's gonna be easy to compare if I take a look at these right if I do this and I go so this is just my I'm playing here with stuff so I'm gonna create circle equals quantum circuit so this is gonna have three key weights right three cubits and I'm gonna basically all call it basically and easier I just copied me I copied it as a function of easier fitness I copy this function and now I do you see equals quantum quantum circuits circuit three and I do basically what I will do now is I'll just call the function with this circuit and I will say control one I guess this zero one two right okay and now if I do you see draw out two MPL should tell me something nice now let's compare okay so mm okay some some actually quite some differences in here and it could be on my side so this could be where where things go south this part should be equivalent so let me just zoom in here zoom in here so no also my house because I cannot use the drawing tools so up to this part it should be equivalent then we've got these key now we've got this team here which Wow which I don't have okay is okay so now is is it's this the is this the could it be that there is a mistake here on the paper that this is the same so if I take this to the other side and because I mentioned that I thought this has a prior no this is because I mentioned I I thought there was a problem but I essentially have stuff here I also have this thing it's partly my circuit flipped which is not good now but how could we test these so how could we test these circuits so basically the way we should test the circuits is by maybe running them on a simulator right so so now I got I've got QC that generates this function I'm gonna now create a so I'm gonna add another swap at the end just to keep the consistency just to keep the just to keep the the controls and Hargett sin in order so we've got these last swap at the end now I will basically remove that okay now I'll basically take my what did I do cancel I'll take my stuff in here now make another function I'll call it so call it the same I will call it hopefully swap intuition and I'm we're gonna copy that over because i'm lazy i'm gonna call this i'm going to rename it here and uh now I gotta now I gotta basically and so everywhere that says too should say target target target target target good target target and they invent the rest should you say control so it's control I would control one sorry so once are basically the zeros are basically control one on control one control one and ones are basically control choose talk to control choose what I want to do is I wanna basically now that I've got back when I do you see you see intuition isn't equal quantum circuit circuit three now I will basically find nice to QC intuition you want to what's wrong that I get perfect I just called all my targets packets target target target target and I can also now take a look what I've done here I think I can do draw I'll do it equals so here it here we have it and it's exactly the same circuit so now we've got you see intuition and QC right as if I now do QC optical equals now we're going to test these by so did it wrong one second first all do so what I want to apply with a wet door wait I want to taste is I wanna apply a QC how to more your one two I got the three hundreds I got a starting superposition and and and then I'm testing the toughly on the superposition and see you see and and we should get then see what outputs we get and then here here I wanted to do the same so I'm going to do part so you want to and clean okay so we've got this now and then I wanna run roaster circuits in a simulator and output the and plot the histogram plot the histograms and we should only get results which are valid subset of you know inputs and outputs for that for the toss the game so we've got the simulator so I guess we're just is we're just gonna copy that the one is going to be QC easier on it's going to be QC in creation so I'm gonna call this like that you see questioned and jobs kissy intuition is not defined in touch intuition of course that's the name of the circuit [Music] you know we should try using patient kissing pictures not the papers pick somewhere they are okay so we got this and now I basically want to plot histogram what Instagram off cue see how does a plot the histogram work not this way [Music] yeah what a histogram which I have you probably here somewhere but but it but it but it but uh get counts I need a I need to get a histogram maybe the counts first first I get the results first thing I guess I got the results already and then I just need to do pop to see it it can't think that's that's the way it works and get counted and that's the yeah and I can plot them all together that would be nice okay so comes up to see intuition house let's see what happens okay they both work interesting they both give me the same result tonight so I've still got the nice way to compare that though but that's not well maybe I should measure you know maybe I should add a measurement information Oh God yeah that's what happens so basically same measure support like that as well or do I have to specify you see measure yeah okay and here kind of basically the same and now basically I measure at the end without you seems possible and circuits not condom circuit and intuition oh come on measure just fine my dimensions the measures drops results and I plot the stuff and essentially that's weird that's that's weird hmm because I should be getting I should be getting so I should be getting all the possible combinations right like if I have a layers of layer of Hana Mart's here oh sorry I know it happens that shouldn't be here it shouldn't he just was for the inputs just for the inputs okay so this we go no still he just won Hanuman in cubed 0 and 1 oh yeah ok so this is the first hard work in there yeah yeah so this is correct and then here also I want to get rid of this hot Abarth yeah so I need to get rid of 100 because I want to do a hot amides on the inputs only I didn't see what's the outcome so alright it doesn't work either why that's what I expected you know what I want to do is these right so if I start here and I go into super position and I do it damn thoughtfully you see that the probabilities of I want to get those probabilities just different altitudes right but I'm not getting them somehow that's what maybe because I just running maybe I should just shots sounds come before one of the shots thinks one of the shots in towns so if I do it is really the only stuff that's in the I think it counts [Music] that is awkward and and for this as well right why those things special and a simulator should work fairly well actually hmm okay so I'll stop here for now but basically I have I have to figure this out I don't have to figure this out it's there there is a difference of implementation in here so this might be one of the reasons things don't work I just just want to make sure that I can compare them even if I compare them like that which they should give me the same result it dawned so but it's not giving me the result that I expect that's basically that kind of distribution right I wouldn't get all these different elements in there with a go probability that's what I mean that's what I'm working for basically this is a simple as this cool but I'll save that and then I'll go back to these later because now it's basically time for the live session already right in five minutes in school um come on couple minutes why doesn't this work why doesn't this work I just don't get it I start with the hannam arts kissy-kissy intuition wanna see me later maybe I mean that's not to be eventually okay why doesn't this work let's do let's do a stupid ik let's do a one last stupid try okay basically I'll call the quantum circuit called like be I'm I'm circuit three three I'm gonna start with a Hana Mart and keep it 0 the 1 and 50 I'm gonna add the control X is your off one tool I'm gonna go off sorry the major make sure one to your to and I'm gonna draw up good MPL have it and now I'm gonna basically run run a job in the simulator and drop the I'm gonna product what is the ground 1:1 histogram off job easier one so enough of these awesome really what am I missing in here what am I missing why I don't see if this is the I should be seeing I should be seeing these 25% no yeah should be seeing each of these possibilities 25% of the time I'm doing something wrong cool I'll stop here cuz I got much time now but I'm definitely doing something wrong but I think the idea of testing it like that it's not that safe as awfully test tougher test all cuckoo cuckoo so so Twitter studio |
second pass through so I guess for number one we know what is for we know what it's vertical coordinates are right so it's the radius and the uh and the angle so those two components that are like 3D space so it's probably about building the operator right instead of maybe the metrics I'd say um in such a way that it's the thing that when you multiply it when you're going to apply it to a state to any generic state it I guess it will just respect the balance of The Young and the exercise asking average is saying this fear of abuse a so we'd have to start thinking about it myself and operator that passes Behavior Maybe you know do some trial and error and see if we can find what shape this thing could have um and then determine the energy levels and wave functions uh I guess that is then some of what related to the Schrodinger equation um but I'm not so sure about the second part |
so I'm trying to basically use the staff in here see if I can fix the qls the Vicki Ellis issue so that's not related to it's not related to the vqs but the Vicki OLS if you remember well there was a so I was playing with a cost function that is instead of using the expectation value that we use alone the overlap the harm overlap test because then the circuit gets I wanted to try a circuit that simplified so a circuit that has kind of that double amount of key weeds but but it's it's not using you know all the hard tests and has the control inverse off of the a components and all this kind of stuff so I wanted to but I was not working so if I take a look at the at the solution from the Kiska tutorial then I plug it to my cost function it still gives me a really high cost and interfere and what I would I what I'm doing wrong and so this is the solution this is the solution right yeah should I run to rewrite everything probably was processing and so I okay because that was some douse and it sells this only is there if I've executed the the actual message but we're not so I'm not gonna execute the minimization here the optimization I'm not gonna execute these these I don't care on the circuit back out [Music] locals false I guess it's very high-cost okay that's a really high cost now where am I looking at so here there's a bit of a different post-processing I think the NQ generalization of the Stratus swap test is flattened destructive swab test to precision measurements on that point are supplying a circuit to quantum register in the state mmm well there is also test if I want to know a bit more about de so is this is this the the product it was product but is it I'm not thinking but either the people's product because this is basically calculating the overlap let's Chicago the overlap I'm not so sure because you've got got the people ice product I mean how can work I have something in the plastid grinder and something the plus state so what is doing so the the that what I was what I'm currently using in the code is which might be might might have an implementation mistake but what I'm currently using the code is the notion that you so you measure each pair qubits and then you're doing like so these probabilities your 0.5 0.5 plus 0.5 so 0.5 plus 0 plus 0 5 minus is your own right so it's it's plus plus plus minus and what they're saying here it's not explained anywhere else but you may have the reference that's why probably the reference to the same paper just run to swap test here oh here you go I don't like this the same paper here there seems to be something instructive so tensed mr. swap test not the order of input state is not relevant the software should give the same result and giving a student circuits the resultant sub test is the non-functional at least the result of the so test is the nan function of the outcomes [Music] yeah but that's what I was doing right essentially what we're doing is then and so if pols outcomes are one then we get a failure but then we're talking about expectation values or [Music] okay this is the generalized generalization to n qubits see the question is I don't advance to apply this tool so what am I doing here so so I'm preparing this and preparing the three different circuits because we're measuring three pairs of qubits and then okay I think I might be doing this wrong as in like cackling is for each of the care of the I mean those are exact probabilities known seems to be seems to be okay right so it seems to I mean let's try one thing let's try one really stupid thing which is a certain let's try one stupid thing okay so like a cross-functional solution I want to try to calculate so it's possible so saying door returns so you give it a circuit and it returns the is the overlap so if I create a so expected is a quantum circuit I'm circuit six cubits and this is basically a circuit it's basically a of a harem art to all the qubits and now I do and so this should be I wanna I just want to make the cool enough like we just compare two simple things right and now I do we do Tenth Circuit okay now and we okay now we just should I just wanna cry to get the overlap so I'm just wanna call plus processing on this circuit which it to the job for me to the actual overlap s so a really small overlap why maybe I'm maybe I'm calculating it already so maybe what I'm doing wrong is that I'm doing these on - because I thought so if they are the same it should be it if they are the same like in this case you should have a beak over a lot because those things add up right am i doing something wrong am i doing something essentially or wrong it's difficult for me to think oh wait why am i dividing this by a hundred mm okay the counts I'm getting the counselor of a thousand because it's a thousand shots so you know some getting the percentage so it's exactly it's 500 is like you know a zero five so those things I think that's that's that's the problem those things are not needed I tended to do these I see that was something that I was dragging from the previous implementation why I didn't have these divided by a 1,000 because I rather just divided by 100 so if I do that [Music] wait a second so I'm taking each office so this is already gonna give me for example I'd say if it's 500 right like it's gonna give me like already the 0.5 and so I'm already operating between zero and one right exactly worst case it's there's just a thousand somewhere in so this one and yeah something's off here because if it's something's not working like what if you do I mean if they're completely and Ike if they are completed if anything are something like that okay so you're gonna get like zero so that's sure but I would expect to get a 1 to be honest for these so throughout cooks [Music] I'm sorry I wanted to print out sir good no circuits are here credit okay so measuring zero zero three two one am i measuring the right so I tried to hear certain output don't get pretty because they're inside me I'll make them glow home can I make them global probably I can make them global observe one circle 2 and Glaus 3 and now I should just do one one circuit 3 and so I'm going to print the circuit see if I'm measuring the right things scrolling too much I know this thing's here work and okay so should run on a separate that's kind of over overwriting the other ones insert cell below circuit one so processing where am i doing Gras so why is it like 0.12 let's go ferbos okay so verbose because true okay so the overlaps are 0.5 0.5 0.5 that's weird so maybe I want to decloak the counts other than this earlier or at how let's say each overlap update y 6 0.5 maybe those whoops don't work well so so the overlap so let's see how like you're going to do with something like I want to do a print I want to print a print overlap one one when I print honks I and counts I see what's going on so I I want to see Oh no I gotta win this damn it started Colonel I don't want to click on that song colonel ready launch processing and I should sell turn out what's clear okay good so we're done here and now we're going to execute the this part so that's the first overlap its 258 242 1053 on it so they're both plus okay so it should be like that what am I doing wrong here there's obviously something off there shouldn't be any like it's an equal distribution that's the problem so you see I'm getting like more or less 250 for each and that's not that's definitely not correct what is the circuit doing oh oh ouch this doesn't add the this as an aji overlap test that's what I should do first that's the overlap test right it's doing 0 3 1 4 to 5 and then 0 1 2 yeah yeah should apply over like that sorry so I should do I should do test circuit equals apply overlap circuit yeah yeah okay now works overlap this one makes sense okay so the functional works well so the overlap is one for for such for such a circuit if I would do test circuit H harm arcade and is that game one two and I would get like an overlap but it's almost zero right yeah okay that makes sense okay so keep the actual functionality that works now if we so you might be somewhere else okay so calculate cost function so here we apply the circuit hmm aha me okay that's the problem I cannot add the overlaps I should multiply the overlaps because if my overlap is one which means they are perfect and then my overlap here's one I'm gonna add 200 1 minus 2 this minus 1 which is and what I want is if I wonder if they're both overlaps so maybe that's a problem to multiply this would this be let me remove these prints these extra prints in here because now we know this works and try this with verbose verbose bolts clean the circuit we're gonna or using capitalizing right on I just changed these here yeah okay it's working it's all going it's looking it's a lot of gay guys a little dick oh but it's pretty shitty I brownie I probably like maybe need more iterations like a thousand so maybe just need some more iterations yeah it's going down what where am I okay so maybe I need maybe I need 10,000 me that's the prison it just takes longer to converge oh it converges that's not good it goes just down to 0.78 hmm okay so it's not in this case I'm doing something wrong here what if I just ignore what it is what if I just ignore the overlap to for a second and I just say let's assume overlap two is always one okay yeah that works way better this is something some getting teased so if I ignore doing something wrong with a second overlap I don't know how to be over there then so it's a three point zero nine one point zero seven and that is it's very different than this solution so how would you do that how do you combine the two overlaps that's the question how do I combine the two overlaps cameras maybe that's died yeah maybe it's it's nothing then I shouldn't count it okay I think I think I it's something wrong with how I'm combining the two overlap results because maybe I shouldn't do it this way of course I shouldn't do it this way if I think the whole idea is that if I maybe I'm totally wrong but maybe I'm totally wrong so I'm combining so again was that's--it's a X right and this is so that's that's what I'm basically doing so now a is like a 1 plus a 2 right okay maybe that's maybe that's the reason I can't just do that if one is the identity then so it remains that a1 a2 the some of those should be be so the separate overlaps don't really help me ah maybe that's what the reason it doesn't work I can't just calculate it like this because I'm decomposing and so the overlaps obviously won't make sense so if I calculate how much these overlaps would be and how much these overlaps would be but I'm kind of optimizing each other then separately then it just doesn't work so my whole idea is just not working I guess I mean I guess it's fine it's a good learning right I can't just um because the cost function here is the point here is that the way this is mathematically expressed it already includes the fine you can add the the different so these am Brian is these are these are these my a1 and a2 right and so a definition you've got like those are the sums of those expectation values and so kind of that works to calculate these which is part of your cost function but if I'm using the overlap I'm just using the overlap then there's no way it's gonna work because with the hannam part with the overlap test this okay so what they say is that these can be calculated with me these can be calculated with their overlap test but not the whole thing nine I know but this is the whole thing isn't it how much these overlaps would be again just not getting right so he derived the Khans function by doing these extra normalization stuff which is confusing how much the slides on P then we can subscribe from another number to get desired low number keep you secure server normal in the cost function is due the fact that if this has a small norm the cost function was soon low even if it does not it would be but which is if you just calculate the overlap why do you care what do you care about these and then then to go ahead and it's the some some are handsome and say the Sun should actually work of such comments but it's much it's multiplied this sophisticated this after its own is for some reason universal you come to harm another test for this tutorial we'll just using standard hardware test since so how much the multiplication here yeah I don't know is as possible at all or not is this the multiplication the right thing of each overlap so each component overlaps intuitively it feels like it shouldn't work because I'm I'm calculating the overlap what I'm doing is I'm calculating the overlap of these and and this is just a sort of a one of the components of of the final a right so how would you calculate the overlap what is this expectation values or what is this what is this you transpose here like what is this unit re that's that's what's confusing because pretty much stuck right now so intuitively that doesn't seem that it should work to me because I'm doing the overlap that's obviously not going to overlap because it's just a component of what a is right but if you try to find and then you have a tool and that's like and so you're trying to find an X that meaning that that maximizes the overlap of each of those as we're trying to do okay so multiplication is definitely the way to go but it doesn't seem optimal because addition addition is the wrong is the wrong operation because it just we won our cost to be between 0 & 1 and so we want we want to minimize both overlaps so we want to minimize the product of the overlaps it's almost like saying it's almost like saying that oh wait a second but if I know that maybe makes sense that the overlap is not so low in these things I don't know how to combine the overlaps okay so I'm gonna leave it here I have to think about these I don't know how to combine the overlaps I think that's that's what's what's happening here but I moved a little bit forward so that's good it's always good |
quantum circuits quantum theory quantum circuit circuit quantum circuit then we're going to talk about quantum circuits today Q sharp end that looks like okay let me switch back the line because those things don't look good in the dark theme theme kind of circuits good condom circuits so let's see what I what do I know about quantum circuits circuits circuits is that this is the way you basically this is sort of the graphical way in which you express how do you want to transform your qubits right by you usually start came into the brown and state zero the ground state because it's the easy they are the easiest wants to prepare from physical perspective and then you just you know with time apply gates here and there and then make weird things and entrance on your stain and then you know you do beautiful things which we don't really know what we're doing but let's see let's see how Q sharp guys explain that and let's see what is these it says it's a five minutes read okay let's see consider for a moment the unitary transformation seen on 0 1 0 1 I guess that means from 0 to commit 0 to 1 and then a Harmar this gate sequence is of fundamental significance to quantum computing because it creates a maximally entangled two qubit state okay so we're talking about entanglement already here but we really haven't so what I know about entanglement is that to me intuitively entanglement based know what I know so far is that it basically means that you have a bunch of qubits which by means of applying certain operations you can correlate to each other right so this you know a bunch of stuff out there people talk about spooky actions are a distance and because the idea is doesn't matter once you correlate it doesn't matter how much they are how much is it like how much far away from each other they are in theory the correlation stays there so we just kind of I guess weird from a conceptual perspective it's not something that you see in nature right but the idea is that so once they have entangled them and then you can see in this state in here right I intuitively read these as in you've got 50% chances of getting 0 0 and 1 1 which it means that always if you read one qubit and you know and you really did and you read a zero you know always 100% and the other qubit is gonna read a zero as well because that's the type of that's the type of correlation you feel that's a type of entanglement and you've done so that's all I know but let's see what happens so and this is yeah so this is basically you see just so awkward to read things like that that's why this circuits help basically apply an animal first and then a control not so intuitively what this means to me is always so on this first qubit you bring it up to as position between zero and one was like sixty percent zero fifty percent one and you make that act as a control bead for a control not which literally means that because it can be both once a measure you really don't know at that point without measuring what is what is really like it adds it's like it's adding more it looks like it's adding more uncertainty which in turn you could consider as data right so you're you're adding more information in there or uncertainty is I mean it just maybe probably sounds weird to call it adding more information but you're doing something with other Mart which basically means you have I have the impression intuitively that you have more more stuff to process in there because you basically have F more yeah you have that uncertainty so and that's a quantum circuit diagram the visual language for quantum operations can be more readily digestible than writing down into equivalent matrix once you understand the conventions for express meant for expressing a quantum circuit we will review these conventions below so conventions okay so this is gonna be just so you have the holo arcade okay so top the line is to zero and then they're labeled sequentially so this means this is commute zero this is keeping one the box with the ages Olimar gate acting on a single qubit carnegie's are ordered in chronological order left to leftmost gate is the first applied like this ABC but then you have to remember that if you want to jump into you know switch to the mathematical realm you're gonna you're gonna have to write the multiplication of the matrixes like that CBA much operation obeys the opposite convention as I say rightmost so that's that stuff that I know multi cubed circuit diagrams follow similar conventions we can define it two qubit unitary operation be it will be HS tensor product X and express the second equivalent equivalently ask this right we can also be view B as having an action on a single cubed register rather than 2 1 cubed registers depending on the context in which the circuit is used perhaps the most useful property of such abstract circuit diagrams is that they allow complicated quantum algorithms to be described in a high level without having to compile them down to fundamental gates ok guys what do you mean about that this means that you can get an intuition about data flow for a large quantum algorithm without needing to understand all the details of each other super teens within the algorithm and how they work the other construct that is built into multi qubit quite a circuit diagram is control the action of quantum single quantum singly controlled gates where a single qubits value controls the application of G can be understood by looking at the following example of a product state input ok this is to say the control gate applies G to the register containing this if and only if the control qubit takes a value 1 in general we describe such control operations in circuit diagrams as this yeah say you have the black dot and then the the actual operation G in this case Jake just being like random I think so here's the black circle denotes the quorum bit on a set for the special cases where G is X and G is that we introduced the following notation ok so the control node has its special symbol and the control said ok that's new to me so the special cases introduce the following additional control version so the gates not the control X is a single gate so the control said this way is the same as the control said this way which equals to whatever that means Q sharp methods provides methods to automatically generate the control to version of an operation which saves the programmer from having to hand code these operations well that's good okay that's good that's that's good okay because there is a way right of you if you take the time to dive into the details there's a way you can build any control operation by applying a combination of control arts and whatnot I don't know examine specific but basically I mean that's that's really not needed right now to dive into for us but the control knot is the sort of the first thing that you learn but then you can apply control Hadamard you can apply control to whatever right and you know it's doable and so Q sharp helps you by saying that CTL auto generate the control specification of the operation okay so then basically you can you can generate the control Heather Mart without knowing what gates you have to do and then you have to put that Q sharp does it for you which is pretty good the remaining operations visualizing circuit diagrams because sorry it's just good because that that's so if we want to start building up intuition then you know those are the useful tools right so how you do a control it's literally you know any sort of an equivalent of the eve of the conditionals instead of in classical programming and computing right so if these then do that right so and it's useful to know that you can just use those their remaining approaches to visualize the circuit diagram is measurement measurement takes a cubed register measures it and outputs the results in a classical information that's classical information okay that's just a matter simple Q sharp implements a measure operator for this purpose city section and measurements for more information similarly the subcircuit gives a classically controlled gate where G is applied conditioned on the classical control bit being value of one okay yeah but that's kind of weird you see that here and I mean here here I can't I can I can kind of understand that example from here so after measuring based on that outcome you're still controlling that's something I haven't thought of much you always I tend to think that either the way you build your algorithms is just you know you start off you you put the gates through the gates and then you measure everything but it's definitely interesting of course you can do stuff like that right so you're ready to keep it and then based on that you apply and you apply some the applying oppression to another qubit quantum teleportation is perhaps the best quantum algorithm for illustrating these concepts quantum teleportation is a method for moving data within a quantum computer or even between distant part and computers in a quantum network through the use of entitlement and measurement yeah so here moving in that sense it means you create a correlation so you know that when you know that you know those two kids are gonna be measured in a way that it's predictable once you know the value of one of them interestingly it is actually capable of moving a quantum state say the value in a given qubit from one qubit to another without even knowing when the qubit value is this is necessary for the protocol to work according to the laws of quantum mechanics the content repetition that we say if you know what it is then you've measured it this means that then you have lost the kind of randomness of it so the quantization circuit is given below we also provide an annotated version of the circuit to illustrate how we read the quantum circuit okay so that was that was basic stuff quantum circuits |
of course i'm installing i'm trying to install my plot leap um no no background removal today if anyone knows any alternative to chromacast or anything that works more reliably with obs and i'll be really happy to do that because uh chromacast is just proved to be useless um just fails to start continuously so what are we doing today so what we're doing is we're basically um you know digging a bit deeper into the vq and the view constraint stuff i mean it's just vqe is like the standard vqe right it's just that one of the things that i came across last time um why is it taking so much to the imports do you import this like this i guess so but i came across these i was kind of asking like okay so how do you um how do you do even gradient descent on a quantum circuit how can you do that right like we i was not totally right when i say you didn't know the function you do know the function because you kind of have your circuit right so it's just a multiplication of gates and those have their matrices and so like if you parametrize the matrices then then you kind of have the equation so you have the stuff you want to derivate right like um i wonder can you derivate a matrix matrix it's a derivative differentiate not their fate sorry that's my my english okay differentiate matrix the derivatives of matrices that can be organized into metrics so the same size interesting i didn't know that um the derivative of a matrix by a scalar and the derivative of a scalar biomatrix okay so it's just a derivative of its components okay looks like anyway but it's still like you kind of have um you kind of have a uh you can have a really big matrix right like you have a lot of qubits so it wouldn't be just practical to do that um but then i came across these thing from xanadule where they talk about the um parameter shift rule which basically um or pennylane yeah which basically says that uh oh that's a comment what is this is that a mirror i guess so into the condition constructing of foreign by evaluating the same circuit with different parameters how is this any different than numerical differentiation though i'll mention here what's good example of computers yeah i think i remember reading these here right where we're no that's not that's not no that's not what i want to that's not where i want to go primary shift rules there there you go so this is what i was reading uh this was reading here the other day and i was like okay so basically what they're saying is that you can so the derivative of of your quantum circuit is just like you can do that express this in in in in this way just by like taking two angles like the two different parameters and then just subtracting and then you get the derivative right and what caught my attention is these right since the fact that you so apparently if you make uh it's similar how did the derivative of the function sinus of x is identical to like a half of sinus uh x plus pi divided by 2 minus a half of sinus x minus pi divided by 2. um and i wanted to kind of i kind of had the idea to first of all try to visualize these with same pi and then second of all try to kind of do the same with the quantum circuit basically start with a very simple one and then try to kind of visualize the the expectation value and see see whether that becomes somewhat intuitive or obvious that it that it's got to be this way i mean you can go through the through the full math but basically it seems like that's where you're getting right so okay so there's an example here and but let's see so if i uh well that's still running i don't understand that though maybe should just kernel restart run all how do i plot stuff in my upload leap examples i just want to plot something like like dots or something like plot a matrix plot of matrix map lobby math shell i don't want to no sorry i don't want to plot a matrix i just want to just plot a just plug like i'm liking the language i just want to plot like the because that's going to plot a matrix like it's going to actually plot a grid and i don't want to plot a grid i want to i want to plot a just just dots maybe this is just categorical variables now just a scatter plot okay maybe it's kind of simple it's got a plunger in the legend simple plot stem plot maybe something like these or something like these so what do we do here so you have so you import these and then you kind of have uh oh so you just ah okay so you just have a function give it a value give it a range and then just plot the thing okay cool let's see if we can just do that we'll just do these and then we'll just uh i guess it's just one dot and i just want the uh we'll just get everything why is this still running though and i what i just don't want to have like i don't care about the axis right now and i don't want to save this just want to run that but why why is this taking so long have i installed my bloodlip wrongly or something why is this taking so long successfully uninstalled maple leap 3.1.1 and installed this one 3.3.2 so i already had it installed okay but it doesn't maybe just shoot maybe maybe i should just do this install my plot leap what does this do let's just install it like i guess globally i don't know let's see where is available oh yeah but everything seems to be there it's just why why does this take so long um kernel and trapped so what happens if i just calling this out something's off with my internet connection maybe that's not because i'm just like so what's what's the issue with this maybe with jupiter it took a while though why oh now i have no idea i will have no idea okay cool but now it worked so but if i just do like the uh so what are we trying to do so we're trying to basically take a look at not these but like take a look at where is the the hole where's this guy so this is here basically so we are having these and uh let's plot let's plot these okay so let's say um the derivative of sinus of x just sinus of x so this is just like sinus of t okay so and we'll just do like lean space and we'll just go from zero to like pi in like yeah kind of does this work this way i think so no and like this lead space appealing space end point no start stop and no it's a number of steps okay so i'll just say you know this is like do that in like 100 steps okay so it does like that um or like i don't know three times pi right so we kind of see these so basically what we're we're saying is like if the derivative derivative at this point with this point here right for example it's it's zero right so um what this is what this is telling me is that derivative is like so x plus pi divided by 2 and x minus 5 by 2 okay so so you take pi divided by 2 so that would be like roughly you know one half so um yeah that'll be like i don't know i mean so if i'm at that point that's roughly like five or like four that something and if i'm doing like pi divided by two so that it's like um it's basically what like one and a half so it's like six something right so but like let's say it's kind of kind of here and kind of here so you're kind of getting these two points right and so you're saying the derivative okay so basically these derivatives are opposite always right so if whatever point you're taking at like so in this case this derivative is going to be positive this is going to be negative um now uh sorry um i'm stupid it's not it's it's not like it should be zero so it's this point [Music] okay no okay the idea it's not no i'm i'm phrasing it wrong it's like the points that are like plus pi halves and minus pi halves will actually be or should actually be the same because in this in this particular case right when you're here because you're in the middle of a cycle which is you know a cycle is like uh it's pi right so exactly so if i'm here that's the zero that's the derivative here is zero so if i'm here and it just go like half the cycle back and how forward it's going to be the same um and so that's going to be zero basically if i am like at this point so if i am like say at this point okay so let's let's imagine i'm at this point then half my cycle will be sort of the opposite as the other one so that will actually turn to one so basically what what makes this possible is it's kind of funny really it's it's the it's the fact it's the symmetry right it's the symmetry of the function that makes these possible it feels like it feels like that's it it's the symmetry of the function that makes that possible and so what i would expect to see in in in something like uh when plotting a the expectation value of a simple circuit is that i would sort of expect to see the same uh or similar a similar thing so why don't we just do that why don't we just like um take for example okay what do we do do we pick do we pick key skip we pick skate maybe um because i have everything installed i think so how to import this thing um and so run this again oh no now it's taking a long time again we're playing five buffer messages restoring connection there's something wrong with jupiter i think okay now it seems to have worked um so what i want to do is i want to create a circuit that said let's just say like has a just an ry or an ry rotation like a like a parametrized rotation so it's going to have like a um wait a second actually i think circ has some cert has some nice expectation value thing but i think power string but i think kisked should also have these expectations it's not shell expectation value what is it doing so often operator okay so actually snapshot what is a snapshot all right so probabilities cubicle set the cubits to snapshot the operator um sir can i just go to search documentation expectation value expectation from state vector from density matrix cost function that can be interesting so here this oh that's the actual okay that's actual tutorial i remember doing that where you get that and then creating the answers okay i might as well just copy most of these code print circuit simulation edge function no that's just too much i don't know to be honest i forgot almost how to work with all of them uh at that point i'll just kind of like probably calculate the expectation value uh manually uh so that will be just running you know just running different just running multiple shots and get sort of the average uh but let's just use circus i want to refresh that a little bit let's just do some basic uh some basic circuit like how can i parametrize so no just let's start with gate circuits simulation nice whatever import circ let's create these there's a gate and i want to have a gateway with parameters and i had these i had these in here so let's just start with these a little easier so we start with importing circ or let's do it like import circ and then we are doing basically uh not these but these so we're doing the rational stuff so we're doing uh just one qubit for now so we're just doing one qubit so we're doing one qubit and then we're doing a pending gate so we're just going to append a a y rotation like like a parametrized rotation i feel that has changed a bit actually exponent half turns how do i parametrize stuff so over all repetitions so this is the objective function sorry the expectation value parameterizing the answers that's what i wanted okay ah okay so actually actually with uh with senpai now i remember so what's senpai and then what are we doing so we're doing you create a circuit okay you just do like that so you create a circuit okay but now i don't know how how this is done here so what is this one step what is this defined okay so that's just defined as a rotation x layer so okay i get it so basically so what i will just do is i will i'll just probably get this function kind of uh work it out here i'll just rotation let's say y i just want to try with y uh and so it takes the y pal gate and it just that's these range lines yield rotation um okay that's what what what is okay ah so that's the gate and then that's the gate applied to the qubit and i got to refresh that stuff a lot so if i do these uh circuit a pen i remember i liked a lot the way you build circuits like that and then i say length basically i know it's one but we'll just you know um and then this is basically alpha and if i print the circuit uh simple is not defined import simpli as sp and then we'll just call this sb okay so that's all we got okay cool if i increase these to like three then we're gonna have oh why so many it's a three by three grid okay makes sense so one by one and now what do we do so what do we do um so what do i want to do now i want to vary these and i want to kind of calculate the expectation values and then just plot that right so if i just say that so inputs is like lean space say uh from like you know zero till uh you know two times pi so we'll just we'll just kind of use the full you know range of angles there and then you know do this like in like in 100 steps right so what i want to do is basically i wanna how do they do this now so how to use these how do they how do they run these we're pending the circuit simulation exactly so run partitions hundred what's that measure okay yeah but how do we try different so this is how you calculate one can negative expectation value over all repetitions expectation value no parameters in the enzymes oh man that was that's just what i did okay resolver i want to sweep oh here there you go so so you have a sweep and then print the results of whatever object functions they got you you actually can okay so you can sweep through these say we just want to do actually that is key alpha is what i have there start at like zero stop i like two times pi uh length like a hundred so that is kind of creating this interesting what is this doing though and then get the simulator which i need to still get run a sweep okay so that basically runs a sweep and then we want to print the result let's see what we get so how do i get the simulator set up and running so the first part is i totally forgot it's just like that okay sorry simulator so i'll just you know basically yeah the circuit here we have the simulator see later equals simulator class not simulator simulator so this is not needed so we have this sweep and then we print the results what do we get it's no measure mr sample circuit house how it was this measurement thing nevermind just measure circuits gates gates oh they've updated that a lot okay cool circ measure okay just just like that ap measure uh i guess i want to measure all the qubits so i don't know if they um not the other measurements in here measure oh just like that just like that append all the qubits okay okay so now you have all the measures all the measurements now the expectation value out of these so we'll just kind of have to so basically we'll just kind of have to calculate that so what do they do here so get the result get the he's the energy histogram i don't notice this but like some of all the items okay just calculates an average reshape measurement to an array that matches grid shape convert true true true false to plus one minus one that's the key right because it's the um the z observable here we're calculating the expectation value off or from uh that returns energy because that kind of calculates the energy that's not what i wanted to do um this stuff lists why does it have to be like these i just want to calculate the average of each of this measurements but it's funny because you can see like even just looking at the measurements you can see there's like a pattern in here with like all zeros and there as you move down like you're getting to all ones and you're back to all zeros back to all ones like so there's a bit of these kind of like you know symmetry that we saw with the sinus function a little bit and i think that's kind of what we want to see as well um so what is what is each result what do we get as a result so what's what's this object uh what's result um oh so i can just i can just do like these probably and it's going to be easier how many zeros how many ones that's going to be the easiest way to do that instagram histogram key a key x i guess that's what we want to do valid syntax why this okay so now we have the counters okay cool so so what we want to have is we want to okay so we want to just basically um okay cool so what we're doing is like this is the this is the result okay and so we're now doing is basically the uh expedition value the eval is like r uh zeroed like this is basically um the amount of so what was that like uh so true is plus one and and zero is minus one so times minus one uh right plus r one times one which is redundant and then so we're just summing them up and then divide divided by like a hundred right and if i print eval hey that's the wrong way to so i'll just call it x expect what you see come on just the oh that's for error as well expectation value oh no that's the eh that's the average what am i doing um that's am i that stupid the expectation value is the sum yeah it's the sum uh by divided by repetitions so i mean if i just like print these what do i get okay i'm just just like it's the wrong oh there you go there you go there you go okay so here we have the expectation values cool and now if we just plot them all so what i'm going to say is i want to have so how does the what is this sweet by the way okay so it's printing alpha alpha alpha blah blah blah okay courses so just basically like that and i just if i just want the um i'd like those exact values but i'm not so sure that's the i don't really know how that's implemented structurally so i'll just like say uh basically the inputs i'll end up using island up just using uh np linspace from zero to what do i say two times np pot and then a hundred and so if i just now basically build a um so which is now basically uh you know data is just like a mp array like an empty array and then i just uh do like theta uh append can i do something like that can i just i need an index or some of some sort for result result in results is i need um i need the input and uh what else that's it no so i just i don't know i don't wanna i'm just going to do the dummy way so basically append so i input inputs i and in the actual expectation value and at the end of that can i do that pan push now push and so theta equals np append and i think that's the way you do it data and then an mp array with these this way do it is it don't know yeah exactly so if i turn that into an array sorry then i'll get like a matrix kind of right because if i just say print data 0 then i'll just get or can i just not print this anymore i know i'm not a pen i just wanna i don't wanna append i just want to add np push array element concatenate insert append dependent element but i want to seems a pen is the right way to go but that's uh so if i do reshape shape like in a 2x2 oh it's just a long array you just want to have like a data can it just like it cannot be that complicated uh okay well give me a second how is this how is this working so f lean space and then you're plotting the function but i don't want to plot a function what i want to plot in here is actually uh that plot leap uh a scatter plot i think it's called right x y okay so in in separate arrays okay that that makes that sense then i can just okay i can i can just use these and then it's easy because then i don't need to combine them that's the point so what the only thing that i need that that i need to do is i need to actually um oh basically e values and then i just kind of say well e values append i'll just append these expectation value and then and then print e values and there you go append it's np okay so that should work uh no np array why is it empty well well well well then i probably was just right at the very beginning okay go so we have these values and we have the length of these is and the the length of inputs it's both 100 that makes sense and now i just do it's kind of blood where is this kind of plot it's kind of blood it's kind of blood so just cutter i guess because so if i just uh if i just do scatter so if i just take the subplots let's kind of plot example so plt scatter and then just show okay that's the way you do this so you do just you do plt scatter and then you say inputs and e values and then you say pv show yeah there you go there you go nice so it actually looks like a sign like yeah it really looks like now question is how can i make a um okay let's let's change that let's change these by an x gate for example sorry just call it rotation layer and then we're going to change this like that and what about like an x one it's the same what about like z one well that just like that okay so that makes no sense but if i um yeah because this is a gate interesting interesting i mean it's like if they have the shape then it kind of yeah makes sense right now the question is how what happens if you you know as you increase the angles right um it's interesting so what if we not increasing what if we increase the qubits so let's try to do two qubits so if i do these it's going to create a four qubit kind of thing and that's not what i want uh what i want to do what i want to do is i want to just have two cubiets so i'm just gonna measure that's bad uh so i'm gonna call length two and then i'm just gonna just gonna have one dimension and so this is gonna be always uh zero and it's going to be always i so okay what else am i using lens though not square but like a line you can keep it as natural line um hit line keep it is this is there is there is there this concept just oh actually it can still be i don't know it can still be um grid cubit that's just one i just want these to be about i just want this to be you know in a um precupid and call okay just like that so what if i just so whatever what about what if i just hardcode that like i was just like qubits is like uh oh it's a great cubit off okay so i can just say crit qubit of like one by two okay how do i do that bill so square okay just a rectangular rows and calls i can just do it like that okay right rows and down okay cool so i'll just call it like a rectangular rows one row two columns there we go cool so we have two things that we're measuring and now i want to kind of have like another symbol and i want to have this called better better and what i'm gonna what i want to do now is i am doing the measurements uh i also want to sweep independently better uh so it's gonna be sweet how how do i do this so how do i sweep can i sweep two variables first let's simplify that okay let's just make make sure that we use just one sweep and then sweep so maybe there's just an example somewhere here kind of how can i sweep across multiple keys sweep oh it's a perm resolver okay so there's the maybe maybe maybe there's just you know maybe there's just a way i can do that key just one key linear space sweep for a given key i want to have multiple keys can i at least sweep how do i how do i sweep over multiple parameters so how how does this make sense maybe i'm maybe i should just okay maybe what if i just use the same symbol for both let's do it this way so okay that's cool um alpha blah blah blah but then like the rotation layer it's all alphas actually it's already like that alpha alpha okay that makes sense um alpha okay so that's fine so so just one sweep for one per hour we're going to keep it the same parameters just i want to get it like 3d and then we'll see if we can kind of like there the parameters as well um you know if we kind of get a similar idea what am i doing now so i have these i have these i'm sweeping and i'm running all this kind of stuff and and i don't wanna i don't wanna show this friend for a second for a second i'm reading these and what am i what am i what i'm now like when i'm now actually if i print r the problem is now i get like zero one and yeah i can get like you know any of the four combinations and so what are the expectation values of those right that's the key but it's a similar pattern you see like it goes from like a hundred like a hundred like you can see these patterns right here that it kind of also has this this sort of symmetry um okay we'll get to these i'm i'm i'm i think that's roughly right what i want to do next with next session is maybe explore that more in a 3d uh kind of style i just wanted to keep this video under the hour um i want to see what uh how can i sweep like you know a multiple uh multiple keys at the same time and then kind of like do a 3d scanner plot of stuff like this but i need to figure out how do i calculate the expectation value of those things here of multi-qubits like is it just like the multiplication um of each separate i don't know i guess so like you know one one would be like a plus one plus one right so what uh i think so and then um and what i want to do afterwards is i want to do what uh i got as a suggestion so uh from bruno right was like hey um he thinks qa is very nice uh so i i will probably do so i will probably do that so i'll try to do the 3d thing and i'll try to get like a qa and a key in a exam q a or a example maybe with two cubes as well and see how this looks like what is anyhow different but but we're getting this symmetry so that kind of basically tells me that that re that's that's just the reason why the the parameter shift rule works and while we can actually do these um in vqe so uh yeah but you can you can see here the pattern already with the counter as you can see that it's it's funny because it's not only that you actually see a pattern in terms of the types of answers you're getting like here it's like more spread out in the middle right zero two one and three this is not just in the it's really across the the whole system that you're getting yeah this this symmetry the symmetry symmetric pattern in here that basically is gonna yeah that's that's what tells you that that's interesting i don't mathematically whether there's any curiosity behind this but i can i'll probably do some reading uh around this cool i thought it was not i i hope it was nice um i i like the this kind of exploration uh videos as well from time to time and i'm refreshing some uh some of the you know my circ skills which are almost non-existent goodbye |
uh we are doing today uh wall from physics project i should probably bookmark these yay uh technical introduction so what are we doing today we're taking a look at the path integral staff see if that's the good path to go down to um as next i did the um basic concept of quantum mechanics conformalism quantum measurement and then i took a sneak peek at the operators as well but then i think i think here's where i kind of found a bit of the first friction in terms of um because there's a corresponding circle this is something that will go in probably the very end again at least keep in mind the goal which is learn about quantum mechanics right like i can't that the project is so wide that it just goes i think in too many directions and i'm not saying that i you know would like to go into general relativity and stuff like this i i um i'd love to but let me see is this currently streaming because that tells me that it is streaming but i don't see the s in the app oh there it is just a bit a bit delay um okay good so this was in quantum formalism i think path integral and i just want to go in here because because it's it's a direct reference here right in terms of in terms of understanding how the evolution is modeled here um and maybe just so path integral versus um versus one versus hamiltonian what are you comparing that with i i i'm not so sure what is the i i wanted to understand a bit more what the comparison like what what is this an alternative off the path integral formulation is a description of quantum mechanics that generalizes the action principle of classical mechanics it replaces the classical notion of a single unique signal trajectory for a system with a sum or functional integral over the infinity of quantum mechanically possible trajectories to compute and quantum map to complete quantum amplitude okay so it's a it's it's a description of quantum mechanics um there's a series on quantum mechanics in the wikipedia look at this oh okay that is something that i should definitely bookmark as well um that's no longer rigid in space but a space of functions a functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space but a space of functions functional integrals arise in probability in the study of partial differential equations and the path integral approach to quantum mechanics so that is one approach to quantum mechanics but is it an alternative as an alternative to that's what i wanted to go through today although that is just i it's probably too heavy but we'll see where it goes um and i don't have so much time today but i'll uh i'll try to get to the point sometimes when you don't have time it's one that actually when you learn the most so so let's just go straight but i think this is a an alternative to the hamiltonian model to to schrodinger's equation right so path integral versus schrodinger equation i think that is so shorting your question is the fusion equation with an imaginary diffusion constant was the fusion equation uh and the path integral is an analytic continuation of a method for summing up all possible random walks analytic continuation diffusion there's a parabolic partial differentiate differential equation a parabolic partial differential equation of many micro particles in bone in motion um written as these where is the density of the diffusion diffusing material it's a diffusion okay so it's a diffusion equation i don't know exactly what that means but it i guess what it's saying is that it's an equation that describes some sort of like a diffusion effect whatever that might be um interesting so let's get let's get going here uh let's zoom in a little bit the notion of path integral originates in and it's mainly used in the context of quantum mechanics and quantum field theory also how so so quantum field theory versus quantum mechanics like what is the difference what is more fundamental theory is a theoretical verb it combines classical field theories special relativity and quantum mechanics okay so that is a bit of a particle fields are quantized okay so we're talking about fields which is something that we haven't touched at all um the notion of path integral originates and it's mainly used in the chronic mechanism field theory um but it's used in both right okay where is uh a certain operation supposed to model the notion of quantization but i don't know what quantization is so by quantization is meant some process that reads in a system of classical mechanics or rather of pre-quantum data in the form of a lagrangian action functional and or is the phase space equipment and returns corresponding system quantum mechanics what um what is the quantization it's like all these definitions uh hi kisada yeah i'm i can't um as i said many times i really don't plan the streams i just had a i just had a yeah i just wanted to do it now um quantization i'm mapping input values from a large side to upper valley and a smaller set okay so this is the process of it's like discretization or something yeah um supposed to model the notion of quantization so path integral is uh it's a certain operation that's supposed to model the notion of quantizations it's supposed to model the this transformation or the idea is that the quantum propagator in fqft the value of the functor you know i don't even know what a factor is on a certain corporatism is given by an integral kernel where what the is all this stuff something like the integral of the exponentiated action functional and cos is the path integral well that actually was under construction um that is too technical the idea goes back to richard fireman who motivated the idea in quantum mechanics in that context the notion can typically be made precise and shown to be equivalent to various other quantization prescriptions kind of clear encoded by an action functional should be disregarding the fact that um [Music] feminine pack formula elementary elementary description in quantum mechanics a simple form of the path integral is realized in in quantum mechanics where it was originally dreamt up by feynman and then made precise using the filemancak formula most calculations in practice are still done using perturbation theory see the section blah blah blah the schrodinger equation says that the rate at which the phase of an energy eigenvector rotates proportional to its energy the rate at which the phase of an energy eigenvector rotates is proportional to its energy therefore the probability that the system evolves to a final state after evolving for time t from the initial state is these so okay so here we're talking about evolution probabilities chop this up into time steps and use the fact that the integral you get all this stuff which i have no idea how to read and assume we have the free hamiltonian looking at individual term we can insert a factor one okay that is way too deep for me at the moment maybe maybe we should start by trying trying to understand better the schrodinger equation i don't know let me try to see now if i can find some some easier introduction to to path integral so quantum mechanics into growth introduction uh i'm i'm currently just trying to find you know ways to i'm trying to find an anchor where i can then kind of start going up so i'm i'm really just digging a bit deeper into all these so i was i was yeah i was looking at these no i was not did i no but maybe that's actually good or not although i would never say that wikipedia is a good point to start because usually it's quite heavy but let's give it another try so the pathological formulation is a description in quantum mechanics that generalizes the action principle it replaces the classical notion of a single then this relation has proven crucial of theoretical physics because manifests lorenz covariance time and space you know unlike previous methods when we allow us to easily change coordinates between very different canonical descriptions another advantage is that it is in practice easy to guess the correct form of the lagrangian of a theory which naturally enters the path into girls for instructions in time uh then the hamiltonian way possible downsides of the approach include that you need 30 this is relative conservation probability the probabilities of all physical possible outcomes must add up to one subsequent formulation divided approaches we proved to be equivalent to other formalisms of quantum mechanics and quantum field theory is by deriving either approach from the other problems associated with one okay so so it's it's something that can you know it's it's equivalent to the hamiltonian stuff and um you know the basic idea the python integral formulation can be traced back to norwich vina who introduced the vena integral for solving problems in diffusion and brand new motion this idea was extended to the use of the lagrangian in quantum mechanics by dirac in 1933 the complete method was developed in 948 by richard feynman some preliminaries were worked out earlier in his doctoral work under the supervision of john archibald wheeler the name sound familiar the original motivations tempt from the desire to obtain a quantum kind of formulation of the wheel of fame and of observer theory using lagrangian rather than hamiltonian at the starting point okay so let's get into these um quantum action principle let's say what is the structure here particularly rather famous interpretation so maybe we can do these two and the path integral and quantum mechanics time slicing inter path integral free particle and and this is so this is a formalism that tries to capture the dynamic part of quantum mechanics right so how how the the how does the system evolve so to say um sorry give me a second i'll be back in a second back there um the basic um where was i back here in quantum mechanics as in classical mechanics the hamiltonian is the generator of time translations this means that the state at a slightly later time differs from the state at the current time by the result of acting with a hamiltonian operator that makes yeah that kind of makes sense multiply by the negative imaginary unit so there's the hamilton operator which basically tells you how the system evolves right so you apply that operator multiply by the imaginary unit minus i and and then you get kind of the next uh state of the system for states with a definite energy this is a statement of the uh the broccoli relation between frequency and energy and the general relation is consistent with that applies as preposition principle i have no idea what this is the hamiltonian and classical mechanics is derived from a lagrangian which is a more fundamental quantity relative to special relativity hamiltonian indicates how to march forward in time but the time is different in different reference frames and so the lagrangian is a lawrence scalar while hamilton is a time as the time component of a four vector so the hamiltonian is different in different frames yeah because i guess it's purely kind of relative right like it seems to make sense it's it's relative to the observer right how the system will evolve it's if you take a look at um at the system from another reference frame then the hamiltonian must be different because you'll perceive the evolution differently um this type of symmetry is not apparent in the original formulation of quantum mechanics the hamiltonian is a function of the position and momentum at one time position and momentum at one time and it determines the position went about a little later the lagrangian is a function of the position now and the position a little later okay so the lagrangian but how how is that so equally for infinite decimal time separations it is a function of the position and velocity so the relation between the two is by a legendary transformation like in their transformation involved transmission and the real value complex and the condition that determines the classical equations of motion the lagrangian questions is that the action has an extreme moon in quantum mechanics the legitimate form is hard to interpret because the motion is not over a definite trajectory in classical mechanics with discretization in time the legitimation becomes this where partial derivative with respect to q calls okay i don't know i'm not and the partial derivative now is with respect to p at fixed q in quantum mechanics the status is a proposition of different states with different values of q q being momentum i guess or different values of p position and the quantities p and q can be interpreted as non-commuting operators the operator p is only definite on states that are indefinite with respect to q so consider two states separated in time and act with the operator corresponding to the lagrangian if the multiplication is implicit or as much as multiplication is the first factor [Music] and so it takes the fourier transform in qt to change bases and so it takes the fourier transform in qt to change basis to pt that is that is the action on the hilbert space which is uh i don't know the h factor contains all the amount of information it pushes the state forward in time the first part of the last part just for it transforms to change to appear cubases and and domain p basis another way of saying this is that since amazon is naturally a function of p and q exponentiating this quantity and changing bases from p to q at each step allows the matrix element of h to be expressed as simple function along each path this function is the quantum analog of the classical action this observation is due to paul dirk the one co square the time evolution operator and then that is that is already dense as well quantum action principle okay but i get it the the the difference being the difference being that the hamiltonian is based on a as a function of of like two static values right like the momentum and the position while the lagrangian stuff which is then what the the currency is a bit more fundamental yeah but i'm not so sure i understand what the concept of the chapter is of the section famous interpretation derek's worked in a provided to calculate the sum over paths and did not show that you could record the schrodinger equation dx quantum action was for most cases of interest simply equal to the classical action appropriately discretized this means that the classical action is the phase acquired by quantum evolution between two fixed endpoints you propose to recover all quantum mechanics from the following postulates the probability of an event is given by the square modulus of a complex number called the probability amplitude and the controls of all paths in configuration space and the contrition of a path is proportional to these where s is the action given by the time integral of the lagrangian along the path okay so this concept of action seems to be so in physics action is a numerical description of a part of how a physical system has changed over time it is significant because the equations of motion of a system can be derived through the principle of stationary action and a simple case of single particle moving with a specific velocity the action is the momentum of the particle so when we talk about action we're talking about the stuff that is changing the system and in this case the momentum of the particle times the distance it moves added up along its path or equivalently twice its kinetic energy times the length of time for which it has so and physics action is a numerical description of part of how a physical system has changed over time so kind of what's the you'd say what's the not to confuse with action at a distance yeah yeah why you can't use these words i mean they say because equations of motion of a system can be derived through the principle of stationary action in the simple case of a single particle moving with a specific velocity the action is the momentum of the particle times the distance it moves okay so what's quantum action so the quantum action principle then is trying to describe numerically how the system is uh changed or changes so for states with a definite energy this is a statement of the prison revolution between the frequency and energy and the reactions the hamiltonian in classical mechanics is derived from a lagrangian which is more fundamental quantity that was a function of the position momentum at one time the quantum mechanics origin transform is hard to interpret so [Music] so the and the relation between the two is okay so that is the action on the hilbert space it's funny i have to actually search for the word action to try to understand how is this being defined here so in quantum mechanics the state is a superposition of different states with different values of q and p and the quantities p and q can be interpreted as non-committing operators the operator ps only definite on states that are indefinite with respect to q the sum over all state states integrates over all q and so it takes and so it takes the fourier transform to change bases that is the action on the hilbert space so okay so this function is the quantum analog of the classical action so what is this function another way of saying is that since hamilton is naturally a function of p and q exponentiating the quantity and changing bases from p to q at each step allows the matrix element of h to be expressed as a simple function what did i do what happened i pressed something in the mouse and it just went nuts a simple function along each path this function is the quantum okay so exponentiating this quantity and changing basis from p to q at each step allows the matrix element of h to be expressed as a simple function along each path um this function is a quantum analog of the classical action so what this is basically saying is that the so h being the hamiltonian i guess right so the h factor on this stuff here right i don't know what the legitimate transform he has to do but the idea is that the h contains all the dynamical information it pushes the state forward in time and the first part of the last part are just free transforms to change to appear cubases or an intermediate p basis i don't really know but essentially what this is saying is that this h contains all the dynamical information but somehow what you can derive out of all these is that if you manipulate if you exponentiate this quantity and do the change basis if you do these then you can basically express you can express the hamiltonian as a function along each path so this means you can somewhat probably decompose these and so so that's the action i guess the function that is defined over all the paths the function that is defined over all the paths is what's then the quantum analog of the classical action so it's it's it's what's yeah so that's just telling that it's just a confirmation or just you know saying that is that that hamiltonian it does encode all the information that allows us to describe how the system's changed over time i guess this is just this it's been formally proved i i really don't know and and maybe this is where one starts to move from the hamiltonian way towards the the path integral way by by actually realizing these this gives time so they're further noted that one could square the time evolution operator in the s representation i don't know what the s representation is state space representation i don't know um and this gives the time evolution operator between time t and t plus two whatever while in in the um hamilton representation the quantity that is being summed over the intermediate states is an obscure matrix element in this representation it is reinterpreted as a quantity associated to the path okay so you can then somehow reformulate that and then you have a specific operator for each path and the limit that one takes in the limit that one takes a larger power of this operator one reconstructs the full point of evolution between the two states the early one with a fixed value and wanted to fix all of qt so the result is the sum over paths with a phase which is a quantum action crucially direct identified in this article the deep quantum mechanical reason for the principle of list action controlling the classic limit okay the deep quantum mechanical reason for the principle of list action controlling the classical limit this one the only important part i don't know so but okay so there's something to go back to i guess so so for some reason that is something that allows you to derive the principle of least action classical uh okay so feyman did a bit of extra work on top of these now to find the overall probability amplitude for a given process then one adds up or integrates the amplitudes of the third postulate over the space of all possible paths of a system in between the initial and final states let's base some corner to another it is correct to include paths in which the particle describes a little bit curly cues curves in which the particles shoots off in outer space and flies back again so the path integral assigns to all these amplitudes equal weight but varying face or argument of the complex number so fimen actually then build the bridge towards the schroedinger equation and then the patterning information kind of filter represents the transition amplitude of the classical correlation function as to the classical correlation function as a weighted sum of all possible histories of the system from the initial to the final statement i'll be right back yeah now microphone should be back um so where are we um okay so basically fireman just kind of build a bridge between these right so this is a bit more historical i don't really understand all the details about these but um path into economy okay it's times lasting derivation so one common approach to deriving the path integral formula is to divide the time interval into small pieces once it's done the charter product formula tells us that the non-cumulativity of the kinetic and potential energy operators can be ignored for a particle in the smooth potential the path nucleus approximated by its exact paths um which in one dimension is a product of ordinary integrals for the motion of the particle from position x a to time t a to x be a time t b the time sequence is like this can be divided out into a smaller segments of one fixed duration the process called time slicing and approximation of the path integral can be computed as proportional to but what are we integrating uh we're integrating the lagrangian as the classical lagrangian of the one dimensional system considered in terms of the wave function in the position representation the path into the formula reads as follows what is that okay okay so let's let's um what is that what is the lagrangian then so what are we what are we integrating here what are we integrating because that's that's something that bothers me i don't i don't really know what the lagrangian is at least not at the level i know what the um what the hamiltonian is right like rancho mechanics stationary action principle so lagrangian mechanics defines a mechanical system to be a pair ml of a configuration space m okay and a function l lqvd called lagrangian so lagrangian is a function by convention convention l equals t minus v where t and v are the kinetic and the potential energy of the system respectively of our kvt momentum velocity and time q belongs to m and v is the velocity vector at q v is tangential to m uh given the time instance t1 t182 lagrangian mechanics postulates that the system's trajectory describing evolution of a system over time is represented by a curve um must be stationary a stationary point of the action functional okay here we have this s the thing with the action stuff so if m is an open subset of r and then of the real of n dimensional u reals and then t 1 t 2 are finite and the path is a stationary point of s okay so it might be actually really worth coming back to to be you know that deep as as these right um um lagrangian mechanics so it just because i want to understand that better so it seems like the path integral is then based on the lagrangian stuff not the hamiltonian stuff so that's based upon you're integrating a function it's a functional integral right that's what it's a functional integral so you're integrating a function functional integration functionality probability so that might be a point worth taking a look at let me just bookmark these and then uh lagrangian mechanics and again all i'm doing now these first sessions is really trying to navigate i'm just really trying to navigate the whole landscape of you know all these complex things and they try to understand complicated things trying to understand a bit where's the where's good point a good anchor point to start from because we've gone from the project here regard to to the way they're trying to model things and now there's the dynamic part which they try to seem to map to the path integral formulation of quantum mechanics um and that's what led me to the path integral staff here that basically um talks about talks about action and the quantum action the quantum version of the classical action and you know being then some some some sort of the action that's just that that's described by each path and the action being the amount of change that a system would be undergoing and within this path and so if you add all these that's the analog for the quantum system would be that you add all the actions or you integrate all the actions uh um all the amount of change that goes and i guess that some of these things can cancel out right like if a path will go one way another path would go another way somewhat they could kind of cancel um so this integration this addition there but this integration then is so this is the stuff here okay so so should i bookmark this as well so we are integrating so we're slicing the paths and they were integrating um over this l thing and l is the lagrangian of the one dimensional system this is the lagrangian that's then associated to that specific to specific path right yeah that's kind of let's try to close some of these pieces together in a later session that was a quick one but um at least it helped me move a little bit forward i think that that point in time we're just still navigating the whole landscape it's a bit complicated to make progress but we'll get there have a good day |
I completely agree with your focus on QM because it is an area that has so much to offer and the theory has so much more to give. |
Firstly this is good progress and discipline.... In my opinion the Wolfram Physics Model is the most promising advance since GR, SR and QM back when giants walked the earth .... It is important as you do this open study to just keep diving in and going backwards over the material more then once. Can I suggest a method?<br><br>Watch once with written notes on a piece of paper. Then do the stream rewatching looking to connect the dots. I rewatched the fundamentals more then once I have to admit because my maths wasn't up to it and I had to do some study since my college level was shacky at best. <br><br>What has impressed me so far is you doing the maths yourself .... In my opinion there is still much fertile ground to discover here especially in the models understanding of particles and QM..... It has already made some exciting ground to do with gravity waves that is ground breaking .... <br><br>I think the models space atoms so to speak have serious implications for quantum states and radioactive decay rates. <br><br>QM and Quantum information is a vast ocean and so far we have just hugged the shores...<br><br>Kind Regards from Australia, Thanks for the support and ideas. I will def incorporate the watch and rewatch idea ;) I've watched some vids innthe past that I could lready rewatch prob. |
Awesome |
we're looking i'm gonna be looking at the position and momentum stuff a bit more in detail and let's just get going so it's probably gonna be a short um it's probably going to be a short session today but we'll see i just basically not to lose track of what i was doing last time it was about position and momentum function and so it was about the derivation i wanted to at least take a start taking a look at the derivation so a three-dimensional wave function in momentum space can be expressed as a weighted sum of orthogonal basis find this is what i want to take a look at it so essentially what it is saying is it's it's fairly simple it's just saying you know if i have let's try to let's try to i think this this is like a paint thing right pain pain pain paint online chase paint there you go so you know it's like let's imagine you know this is my um oh i can use my fingers this is my bases and i have you know i have this vector in here maybe i'll just change color to something red so i have this vector here and now let's say you know this is this is this is position and now i wanna i want to go the to the blue one which is momentum and let's say that i'm saying well you know this is a position vector so i can you know i can express it as let's say let's say there's another base and i'm gonna i'm not gonna even draw like um i'm having a dry perpendicular to orthogonal to the original base which would be this by the way so it's my fingers too thick so let's say this is this is the basis yeah so below so now basically i have sort of these you know there's this component here this component here so you know let's say this is uh p x and this is p y right so um you know well essentially this is the same as just saying p y times sort of the the the the y basis whatever that whatever that is right it says this is y and this x from a p perspective y p x b right plus my ph this is awful anyway p x x b like that that's the point right so this this would be the basis vectors so this is what we're saying it's like in and that is that is these in momentum space that's all this is saying it's like you can always just express it as in some other um in some other bases that's all this is saying so okay orthogonal basis functions this is this is kind of like um conversational machine learning space can express the way that some orthogonal basis functions yeah i'm just i'm assuming i'm intuitively you can exchange those values by functions and so that's what you're getting here or as an integral um [Music] now why would you do an integral you do an integral when you're why would you do an integral though would you do an integral because would you do an integral because it's um a very big space i'm not so sure i understand this step and then the position operator okay so essentially the position operator is what takes observable of a particle so what we're saying with position operator is your product response to the position observable of the particle when the position of particle is considered with a white one not domain then the space of the tempered okay let's give it a second so its eigenvalues are the possible position vectors of a particle let's see this so in quantum mechanics the position operator is the operator that corresponds to the position observable of a particle when the position operator is considered with a wide and not domain the space of temperature distributions its eigenvalues are the possible position vectors of the particle in one dimension by simulator we denote the unitary eigenvector in one dimension if by the symbol we denote the unitary eigenvector of the position operator corresponding to the eigenvalue x and x represents the state of the particle in which we know with certainty to find the particle itself in position x therefore denoting the position operator with the symbol x in the literature we find also other symbols for the position operator for instance q from lagrangian mechanics x and so on we can write uh for every real position x one possible realization of the inventory state with position x is the direct delta function distributed in distribution centered in the position x denoted by delt this dx so quantum mechanics the order continuous assembly of all the functions it's called interposition bases in one dimension just because it is unit basis no idea what that means whatever that means basic properties i can states three dimensions so the generalization to three dimensions is straightforward oh let's start from the eigenstates uh basic properties basic properties no eigenstates the eigen functions of the position operator on the space of terms representative since informal proof okay so this is just a proof that they should be delta distributions the eigenfunctions it means that there's just like it's like a big spike you might obey this ah what does the sort of what does the position operator do when you multiply to a system it basically tells you the probabilities that it will be and you know in in not not that but like the expectation value or yeah square will tell you the probabilities of the different positions and if not um how to understand sort of the the position the position operator the position operator this is this is something that is just hard for me to have like a mental model for what is in the position operator is the operator corresponds to the position observable of a particle when the position produces conceived okay that's what i just read how do you use it positions are operators uh how the position operating the position bases are correctly defined in quantum mechanics if one deals with wave functions the hillbid space in question is for a protocol in the dimension the position product corresponds to the ith coordinate uh it is exactly defined by whatever that is now if we generalize things slightly we have a state space which is a hilly space with state vectors this current its possible state of system in this case the position operator is usually defined by okay um but here's the problem this definition depends on on a particular basis but how those kids are defined while they are defined exactly as eigen vectors of the position operator so to define the position operator we need we need it in a certain basis but define the vectors of that basis would be defined then as the eigenvectors of the position operator the circular hmm so what is the except is there any accepted answer no but this one has a lot of our boats uploads in quantum mechanics position appeared in the hilly space are defined contextually the hill with space and the operator position along x k is defined in that space with the obvious domain you can adopt a more abstract viewpoint if you find the momentum in the position of prayer that's satisfying ccr whatever that is [Music] in that case one assumes that there is a continuous basis that's why you do an integral there's a continuous basis i i know like that continuous bases existed but this is even what's even the formal definition continuous basis mathematics so so it's like if if you think about one i mean operators i understand operators as it's a function over a space of physical states into another space of physical states an operator is basically a function that helps you go from one it transforms the state into another state so how how is position operator okay so position operator is basically an operator that tells you like how is the position evolving right okay yeah that makes sense so it encodes that it's how how is the yeah how is that evolving yeah yeah um but then what is the meaning of the eigen functions of of that operator because these are then functions where the eigen functions of the position operator how do what is what is the what is the interpretation of that of the eigenfunctions do they correspond to the bases like what is the so these are supposed to be the eigenfunctions and then what we're doing is i get it y is an integral because it's a continuous basis okay this makes sense i'm not sure what the d3 means in here [Music] but okay but essentially it's like as a version of these it's like the specific okay i think in a way what this is doing is saying it's actually decomposing the operator could that be it's like saying it's like saying take that eigenfunction and multiply multiplied by so this is your starting point as your this is starting point it's basically the the wave function in position space and so uh and so we're doing is we're multiplying each of the um so so you're taking your eigenfunctions of the position operator not the position but this should be the momentum operator right i don't get it [Music] or not so you're saying so you're defining the word we're defining here the the momentum operator and we're defining it as the integral so we're doing is we're saying well if we take what the position operator is doing is just evolving as involving the the okay it's evolving that so you're saying take the eigenfunctions but these are the eigen functions of the position operator keep multiplying and adding them or integrating them and then and that's it and that gives you the momentum that is what's uh a bit odd first of all why are we multiplying the eigenfunctions by the actual wave function in position space and why is this giving us the momentum space i don't see the intuition i would assume yeah the position operator so and that's the momentum space a three-dimensional wave function momentum space can be expressed as a weighted sum of orthogonal basis functions i just don't i just don't get why this gives us the momentum the momentum interpret the momentum wave function maybe i just need to see some concrete examples of these but okay there's two things so one single example anything understanding understanding what what the eigenfunctions are then so the eigenstates of the position operator are delta functions to each observable in quantum mechanics there's no predictor responding to it i don't understand what's the meaning of the eigenvalues of the exoplanet synthesis hermitian arguments correspond to real numbers what's their physical meaning if they describe the particle localized to the particular point blah blah blah blah blah blah if these random operators do not commute the eigenvectors of the hamiltonian are usually different from the elements that both presumably is what i see in books applying the percent you know just the other states of the election operator how do you find solutions okay eigenvalues are the values that are measured in the experiment uh otherwise the position obtained when measuring it every measurement will produce a different result different eigenvalues unless the system was prepared in mag instead of measured quantities okay yeah yeah yeah because the statistics the state of the system does not have to be an eigen state of all the operators which is pretty much the point of the uncertain principle there's an eigen zero i still don't get it what the meaning is um because we choose to work in the basis of eigenvectors of x wave functions we have a hilbert space which is a vector space in which we can choose any bases we wish we mustn't choose to work is that this diagonalizes the position operator why once this is chosen in state um um momentum operators in chrono basis okay so you might ask how can one derive the expression of the momentum operator p in the coordinate basis this is done as follows the by definition what [Music] i don't get it i don't i don't get this what i don't get is intuitively why why so i don't i don't get why this is where i'm stuck at the moment why does these why does this expo why doing these leads to the momentum it just doesn't make sense these basis functions these orthogonal basis functions where we're actually just what we're doing is we're taking the eigenfunctions of the position operator so essentially what you're doing is you're and then you're integrating over that i think that's the point the point is you're integrating hmm um the the the the trick is somehow in the integration um [Music] but and again i just have no idea what oldest d3 here is convert to momentum space momentum space wave functions okay so maybe this is explained here a bit better in free space we saw that the time independent schrodinger equation is the following which has solution blah blah blah so this is in free space cool we know that for a wave traveling from left to right and our last array both with energy so we could in fact simplify this a bit with um okay adding this k plus minus 1 and then just doing this as a particular solution for any direction we get more physical insight into this by writing the momentum operator okay so we get more physical insight into these by running the momentum operator as an eigenvalue equation ah so you're doing basically okay so you're actually just doing an equation and then and then deriving it from that that's so you're saying okay so you're saying the derivative what is f of x where's that even i hate that like jesus i guess it's just whatever ah god huh so plane waves which are eigen functions of the energy operator for free space are also the momentum operator so so these can surface because i have a while i'll define tremendous momentum so there is some sense in which this is not a physically realizable state to normalize it when you determine that momentum for it transforms whatever that doesn't help that didn't help so let's say i have a state vector in a position space with an orthonormal position basis if i now use an operator p on these bases okay so that actually that's the first that's a question that i have as well like what happens if i use the operator the momentum the momentum operator if i now use an operator p on these basis i will get the basis which corresponds to momentum space and projections of the on these basis vectors will now be no i don't think so that's the point i don't think that's that's true you can't so what what will he what will what will you do if you what will happen if you apply a momentum operator uh to a wave function that is in position space or maybe that's the whole point of the equation is to say that if you evolve somehow if you were evolving no the way so the wave function vector is supposed to be a function of time only when you're right you're not considering the projection of the function nor the position neither in a rhythm space but just to say the system at time t which is nothing but a positive quantum mechanics or how do i function you will have the function coordinate uh or momentum or any other observable once you project your state vector on the basis of the observable you have chosen for instance in coordinate space then you do these oh my headset is running out of battery so i might actually my sound might just go um oh sorry i actually had missed the chat because i only have one screen integrate out all the r's and you get the wave function in k space yeah that's kind of what i what i think ah so d3 is all space so everywhere the relevant wave function reaches needs to be included there's a is a noun there's a space derivative yeah of course it is of course this is a derivative i mean it this this seems intuitive that it's a derivative um and applying p hat will take the derivative of the r space or space expression i'm not so sure i understand all that so but thanks for the thanks for the hints sorry because i i'm just working on one screen these for the next for the coming week so i i for some reason i just can't see the chat unless i maybe i should be able to pop out the chat in the screen somehow can i pop out the chat show viewers now i have to think what i i have to see if i can pop out the chat and at least have it floating on my screen because then i can at least talk to you guys so thanks a lot um yeah yeah that's just you know exploring here stuff and just trying to learn stuff uh on the go but i mean yeah yeah actually that's probably that's probably uh the best thing to do uh i mean i'll uh you know i'm i'm then putting all the videos in youtube so i'll just i i will come back to that and then take a look at the chat um yeah i think the struggle is always what matters and anyway i'm just probably running out of time anyway i'm going for half an hour now and i don't have so much more time today sadly but let's let's just go through these so the wave function uh yeah okay so get that um kind of okay this is still i keep i i keep struggling i keep struggling with these and again as i said my uh headset keeps informing me the sound is going to go off in a bit so apologies if that goes uh on mute maybe it will it will it might actually switch to to the computer speakers we'll see but then it's going to be really loud because they're really shitty um whatever let's assume that's okay um i'm just i'm not really agile no not a job i'm not really um well educated in these i feel like yeah i get it the wave function the oh okay but it's a vector yeah okay sorry i keep getting confused the wave function does typically oh let's say just we have this okay let's ignore the kets and stuff for now um the probability of this is if you want to switch from coordinate space to momentum space you want to have the following probability amplitude okay i get it i mean i i get it in the sense that what this is just telling me is this is just a projection that's that what you're doing is a projection but i'm i'm trying to understand this without the the vector notation so [Music] i will see i think i have to wrap up now um for each t having inserted the completeness relation the specimen certainly now knowing you'll find that the projection of a function of space is different from the granite space um that doesn't doesn't satisfy my my it doesn't answer my question i feel that it's just a way to say well just just do the projection assume you know these just do the projection and then that's not what i was looking for i was looking for i was looking for something a bit more intuitive at this level like i have yeah i know i mean there is a derivative relationship between position and momentum that's for sure um it's an obvious one classically speaking okay so you're just saying that's just the integral because it's the because the derivative of momentum yeah maybe that's that's just the most intuitive way to understand that and probably that's what you're saying right um in our space them in our space the momentum operator is this is a space derivative so you're integrating in order to recover that when i do electron modeling calculations around the wave functions and space functions like those chemistry orbitals that is that is uh that is that goes definitely i take the application of those that goes definitely way above my pay grade at the moment but yeah thanks for the uh thanks for the thanks for the insight in there let me see yeah i gotta actually have to i have to probably wrap wrap it up here um i don't think i have cleared anything specifically in this session but i will i will eventually i have to sleep on it it just that's what's bugged me but it's it it might just be that it's just maybe i'm trying to understand something that does not really need that uh level of intuition i don't know yeah yeah yeah it's it's definitely struggling to find a bit longer periods of time to actually focus a bit more because like 30 minutes 40 minutes is not enough sometimes uh or most of the times but we'll see better than nothing cool um yeah i'll wrap it up and just uh basically do another session as soon as i can in the next couple days thank you |
today we're going with the streamer mode i'm trying with that let tell me tell me in the comments if you um or just don't tell me just just don't tell me um what are we doing today so we are gonna be working on portfolio optimization today um but we are gonna be doing it a bit differently so i've recently come across uh a paper wait a second how do i find this oh god this had to pop the hell is this um how can i find that so that was a paper from uh so that was a message from sam uh from sam the cto of multiverse qc multiverse computing um basically so he uh because i i tweeted like about like hey what's the you know what's a good overview on uh on on on portfolio optimization stuff oh there you go and so and he basically uh okay so that's the pre-print about this particular paper and i i think that's worth taking a look at this another paper that they shared i also want to do these by the way the the google thing um there's another paper they uh this thing is getting the audio right there's another paper um yeah the overview here so i will also want to do these i should probably bookmark those because i'm not going to be able to find them soon so i should probably just uh can i bookmark it tweet i think i got to do this right just add it to bookmarks and then i'll find the thread so it's basically so this dynamic portfolio optimization with real data sets is in quantum processes and quantum inspired tensor networks um and it's uh surprisingly quite similar to um where it it seems let me check the pdf if this is the one that i saw and so here in in sorry and the other one was the quantum computer for finance obvious prospects and uh i've been i i have the impression that this is a really good one to take a look at can i download this without paying i hope so i hope so it seems like because i haven't loaded it i think but i don't know where it was can i just download it so i don't have to yeah so i don't know which one should i do um i kind of can do the overview or i could also just do this one i think they're more or less the same length and i'm interested in these maybe because it's similar to the one to the paper that i'm uh doing with uh that i'm tinkering with uh from chicago quantum where i'm struggling a little i'm struggling a little bit um because it i i think they just have some some gaps in the way they describe how they went through the math and i just maybe maybe it's worth taking a look at this it's gonna maybe you know give me a fresh perspective on that um something that i can use to kind of reproduce these maybe right thinker with that so maybe i'll do this now and then next time i'll do the the overview just to kind of background some core problems in finance because that can give me some i think additional perspective some basics and quantum computing so there's a bit of the basics in here as well the case for quantum computing in finance maybe it might be worth reading these part as well actually so the case for quantum computing finance various quantum algorithms offer substantial speed apps relative to classical algorithms this means that when the number of classical beats needed to specify the input data is increased the number of operations needed to run the quantum algorithm increases slower than the best known classical alternative in this section we outlined some quantum algorithms which are potentially applicable to financial problems in what follows and specifies the number of possible inputs aka the size of the problem which can be codified using um log and classical beats um one can not talk about the great breakthroughs that started the the field of quantum computing without mentioning grover's algorithm which finds a particular register in an ordered database um in uh square event steps by contrast the base classic algorithm requires uh and divided by two steps okay so this they mentioned grover designer can be adopted solid transition problems um and also implementing monte carlo methods really grower can be adapted to to implement monte carlo methods grover the carbon methods alternatively the quantum approximate optimization algorithm finds a good solution to an optimization problem in a polynomial time this requires exponential time on a classical computer um yes so that's basically the variational approach to i guess solving the cue ball right optimization techniques that are also applicable to machine learning algorithms indeed training can be considered as special case of optimization for neural networks machine learning also makes also makes frequent use of fourier transforms here again quantum computers can result in sustainable speed ups while classical fast fourier transforms runs in n times log n steps the quantified transform is the s it's it's log n square um let's go to the log n squared k of t can be useful for some artificial intelligence methods such as quantum principle uh component analysis and quantum support vector machines the hsl algorithm folding is linear systems of equations can qft be used for for pca as well how would you find how i don't know because matrix operation is essential to any machine learning blah blah blah blah okay but it's interesting that there's no uh mention of annealing in here i'll listen in in this 2.4 like this and healers oh oh there is something in there uh no what's the first mention in paper simulate knewing quantum mechanization transition problems are the core financial of any financial problems this is the case for instance of portfolio optimization which we'll discuss in the following um because it is an np hard problem it is extremely difficult if not if not impossible for classical computers to efficiently determine the best choice of portfolio there are a number of different ways to implement quantum optimization algorithms on a quantum computer the most prominent of which is quantum annealing okay so they do you mention it's it's the most prominent one at the heart of the quantum transition algorithms method known as adiabatic quantum computation which we'll describe in the following first we must map the optimization problem to the physical problem of finding the ground state of a hamiltonian which encodes the cost function to be minimized we prepare the system in the ground state of an initial hamiltonian h0 chosen because its grandstand is known and easy to prepare and that's what d-wave does with the um superposition state we then add hepatically deform h0 to hp over a long time over a long time t the adiabatic theorem states that a system initiated in its ground state will always remain close to its instantaneous ground state provided its lowest energy levels are non-degenerate and that the evolution is slow like it's awesome because they actually explain the whole core of the whole like of the thing here just in a sentence and that's not even explained so easily and clearly in the d-wave web page when this is true measuring the standard system at the end of the evolution has a high probability of turning the ground state of hp which is this is a universal method this is a universal model of quantum computation which means that it can in principle perform any quantum algorithm that's actually quite nuts to be honest like how it's extremely general model as it can be modified to white intermediate hamiltonians and can be made to fulfill local adiabatic conditions this process multiclassical multi-classical assembly annealing where thermal fluctuations allow the system to jump between different local minima and the energy landscape as the temperature is lowered in quantum dealing this jumps the dream quantum tunneling events this process exposed the landscape of flow coming minimum more efficiently blah blah blah but like how how is quantum annealing universal it's a how is it a universal model for quantum computation this is reference here this is not like this this is totally like counterintuitive this is totally counterintuitive to me but anyway maybe let's not get too distracted actually i should is that a long paper oh 30 pages okay is it is it basically the concept that you can you know like any any circuit any circuit just can be compiled into an actual hamiltonian or like into a like a unitary matrix and so you can anneal your way there and then perform measurements i guess that's why right you could in theory just like this could be a nice exercise to try like can i can i turn can i turn like uh for example a grover and like into a matrix and then uh and then run it on d-wave that would be awesome actually it could be something to try um but we are gonna work on these today dynamic portfolio optimization with real data sets using quantum processors and quantum inspired tensor networks um i'm gonna aim at just you know for the session now just as i usually do fly through the paper um and so kind of you know even if i don't understand something i'll just kind of you know go ahead and then see see if there's anything interesting that we can do with it maybe some some sort of comparison with the chicago one will be interesting maybe see if that kind of leads into a cleaner implement a cleaner implementation of what they're doing maybe they even do it differently because that's like that's really recent that's like june 2020. um and i see there's people from bbb uh baby in spanish pba madrid sebastian cool in this paper we tackle the problem of dynamic before optimization determining the optimal trading trajectory for an investment portfolio of assets over a period of time taking into account transaction costs and other possible constraints that's interesting because the other paper doesn't seem to take that into account uh this problem well known to be np hard essential to continue quantitative finance or maybe maybe they do but they don't mention that explicitly after after detailed introduction to the problem we implement a number of quantum and quantum inspired algorithms on different hardware platforms to solve its um discrete formulation using real data from daily prices over eight years of 52 assets and do a detailed comparison of the obtained sharpie ranges oh so they use the sharpie ranges as well profits and computing times and particularly we implement classical solvers uh gecko exhaustive d-wave hybrid quantum annealing two different approaches based on variation of quantum eigen solvers and ibm q one of them brand new and tailored to the problem okay interesting different approaches based on so two two different approaches based on vqe i'm i i already know i'm gonna learn something so i'm liking this um and for the first time in this context also quantum inspired optimizer based on tensor networks in order to fit the data into each specific hardware platform we also consider doing a pre-processing based on clustering of assets from our comparison we conclude that d-wave hybrid intensity networks are able to handle the largest systems where we do calculations up to 1272 fully connected qubits for the monster diff uh the most sensitive purposes finally we also discuss how to mathematically implement other possible real life constraints as well as several ideas to further improve the performance of study methods the abstract like it really makes me feel like that's a dope paper actually way more than the chicago quantum one like sorry but it's let's see if it leaves lives up to the expectations i really want to read through the entire thing like i i feel like i feel like yeah that'll be cool so in quantitative finance portfolio optimization is the problem of selecting the best distribution of assets that optimizes some objective function typically this objective function tries to maximize expected returns and minimize the financial risk the problem gets more complicated if we do it dynamically optimize the investment portfolio over a series of consecutive trading days in this dynamic before optimization the goal is to determine the optimal trading trajectory over the considered period of time the optimal decisions that should be taken or should have been taken by a broker in order to maximize the overall return at the end of the time period the dynamic problem is more complex because transaction costs and transactions market impact must be taken into account as well as other possible constraints in practice it is well known that this is an intractable problem that falls into the complexity class and be hard parallel to the both has been understood recently that quantum and quantum inspired computing can help in solving hard financial problems for instance the quantum computer should be able to solve more efficiently and with more accuracy problems related to pricing of financial derivatives okay prediction of financial crashes detection of arbitrage cycles credit scoring and an identification of several types of fraud among many applications portfolio optimization is no exception blah blah blah pre-existing studies okay yeah none of this work then to solve the problem on real data sets and and no comparison has ever been done openly and democratically between different methods and hardware and hardware platforms what was the chicago quantum paper from chicago quantum portfolio optimization was this one before this one oh it's also 20 20. oh it's july 2020. wait a second that was first actually it's especially similar july submitted in july and that's june 2020 right sorry not this one this one it's june 2020 okay okay let's go in this paper we implement several quantum inspired algorithms for dynamic portfolio optimization and run them for the first time with real data corresponding to daily prices over eight years of 52 assets in particular we implement d wave hybrid quantum and annealing of a variational quantum migrant solver on a quantum processor of ibm q a new vq inspired algorithm which we call vqe constraint also on ibm q and a quantum inspired i lost tensor network yeah optimization algorithm cool benchmark algorithms using two classical methods a gecko solver what is a gecko solver a python based optimization suite okay just let check just check ah where's the chair getting stuck oh now shown the transition of mixed integer and different okay so this is an actual package this is it's an exhaustive solver we expose how pre-processing is performed and reduce the problems i mentioned earlier using a clustering algorithm we then do a detailed comparison of all the results focusing on the obtained sharpie ranges and computing times from our comparison we conclude that as of today the wave hybrid intense and errors are able to handle the largest systems where the calculations have to blah blah um interestingly we see that there's no clear answer as to which is the best algorithm or how we're plotting to deal with large systems as this depends strongly on different figures have merit it's organized as follows the paper section to keep an overview of the dynamic portfolio optimization problem good section oh we show it how can we express it oh let's do this let's jump let's jump to section two okay um let's see if it's easy to understand the dynamic portfolio optimization problem can be expressed in a form amendable to a quantum computer in what follows we present the problem in some technical detail where possible we use the notation of 11. i don't know technicalities i guess generalities in the dynamic version of the so-called modern portfolio theory or mean fair i mean variance analysis would deal with the issue of allocating weights to a number of assets over a period of time in order to maximize the overall maximize the overall return at the end of the period more specifically for n assets we consider an n dimensional vector of weights like that's interesting like the because here they really keep putting emphasis on on on the return like on a given period of time like towards the future whereas in the chicago paper i i maybe i just didn't read that well but it didn't seem to be something that was kind of factored in in into the equation more specifically for n assets we consider an n-dimensional vector of weights wt each of its component wnt is the weight of acetate and at time t yes that's actually so that's actually more yeah you see that's that's in in the chicago quantum paper they they talk about w as the w as being the weights the acids like the weights vector but like they never mentioned like that as like having a time dimension to it as well um uh let's see where each of its components is the weight at time t where ti and df are respectively the initial and the final trading rebalancing times being the number of trading steps we also define mu t as its forecast returns at time t and the sum or whatever this is d d as its covariance at time t it is the end length vector while this is an n by n matrix um so okay so this is the the covariance i guess the covariance of each acid against all like all the others right for a given training trajectory a given set of vectors total return is given by okay so the overall return is the sum of the so this is the assets forecast return at time t so yeah so basically they multiply the forecasted uh okay the transpose like it's a it's a uh so the assets forecast returns times the actual vector with the allocations and the risk of the trajectory is defined by the sum of so this was the covariance so notice that this is the the the variance of the portfolio return at time t in practical situations one typically measures risk at time t in terms of volatility so if this is the variance right if this is variance of the portfolio and then they just go and say like half the variance in practical situations one typical measures risk at time t in terms of volatility scrolling uh which is which is the square root of the variance of these variants nevertheless equation two is a convenient way to quantify the risk in the dynamic setting since it makes the optimization problem better behaved being the pre-factor half of a convention the goal of modern portfolio theory is to find the trajectory which maximizes return for a fixed risk it is common to request a total investment at any given time is it is common to request a total investment at any given time is fixed for example with k the total investment let us define at the step the normalized weights so that's a total investment how is that total investment deal with k the total investment let us define at the step to normalized weights so that their sum at every step is equal to one ah okay i get it i get it so the the thing with a stripe like that's actually not the the it's it's not it's not the percentage that you're putting into every asset but it's the actual investment money ah okay so then you define that without the hat like without without the without the line and and that's normalized okay makes sense so that they summit every time is equal to one so that's basically yeah in terms of this normalized weights the problem can be solved by finding the trajectory which minimizes okay so here they really give like the they really go straight ahead like to a hamiltonian that it's actually something that's like a cuba friendly thing right um by analogy to quantum mechanics we shall refer to the above cost function as a hamiltonian is the risk version which tunes the eagerness of the investor to explore risky trajectories and role is the lagrange multiplier that imposes the constraint in equation three as a penalty term we have introduced the n-dimensional vector u with u sub n equals one for all n which makes the constraint more compact the lagrange multiplier was fine to denote to satisfy equation three okay let's try to unpack these um what is saying here the equation five can also be written as the sum of of these h sub ts where each h sub t is yeah basically the same it's in here yeah plus the hamiltonian is diagonal in time this implies that the optimal trading trajectory is simply the concatenation of the optimal portfolios at each time note that the w sub n t are interpreted as percentages of the total investment for instance having 0.1 means that we invest 10 of our total amount k in the n asset n at time t additionally one may also introduce a cap k prima k prime on the maximum amount that can invest which asset okay we measure the quality of the portfolio by the so-called ratio sharpie ratio this quantifies the amount of return per unit of risk of the trading trajectory um so let's let's let's take a look at this for a second because i so mu sub t is the n length vector this is an n and then length vector uh sorry mu sub t is the assets forecasts returns at time t so you're taking the and you're always multiplying it obviously by uh by wt which is the vector that encodes the percentages uh of your investment and uh so you take the expected return then you add and it's a negative term right because we want this to have like why is it negative right like is it because we want this to be the component that's like because we want this to minimize when i haven't like want this to be as small as possible that's what i can come up with but it's definitely not like and then yeah yeah and then applause and then we're adding like so then you're you're adding the risk component that they define here where they say the risk is defined like dad i think that's what's substantially different than what the chicago people do is it so notice that so they take the variance and they divide it by half so notice that is the rise of the portfolio return at time t in practical situations one typically measures a risk at time t in terms of the volatility the square uh this scrolling is just killing me the volatility which is the um uh up again up again the square root of its of of these variants nevertheless equation two is a convenient way to quantify the risk in the dynamic setting since it makes the optimization problem better behaved being the pre-factor one divided by two a convention okay the goal of modern portfolio theory is to find the trajectory okay so they just say that's a convention it just makes the what did the chicago people say what was that this one here so developing the cube oh they go as page six or whatever so they go through all this stuff and then they so what they do is they yeah that's kind of similar similar to what they're saying by the square root of the orions because it's two in here so what they say is that uh single answer before this month consider the linear terms in a cubo in particular we have a lower triangular matrix so those in terms of zero model and the inverse sharpie ratio and cubits and there's a penalty on the complex but where's the where is the actual hamiltonian yes this one here finally the optimal portfolio so v supplies the variance and and co and cov is the covariance term between ssl y and j one recognizes the initial formula um and it's not suitable for d wave and it's current iteration we find that you got quantum score solves this problem and can be presented as a quadratic form that's what i don't get like that's that's really what i don't get the chicago quantum score thing like what is the first place i mentioned these so here they they so what they do is they just r sub w is the weighted portfolio so this is like friends or what and then alpha is a real number and we chose an equal weighting and most experience which is an equal weighting where n is the number of assets included and we choose often year one these are not requirements but they do make the competitions into a slightly easier it is a wide open question as to finding optimal weighting and optimal alpha but they don't explain how they f how to formulate the actual keyword today it's it's it says cqns is basically the variance minus the expectation value of of your average investment this makes just no sense like in chapter four they just blah blah about this stuff and then i guess that is consider the universe u of n assets when dealing with a single asset before they let's say linear terms in q one particular one we have it below the triangular metrics as you know the products of the formula doesn't pick up not linear terms if you have two or more assets then we have more work to do you look at uh there's no okay we create a unique cube for each size portfolio by applying the weights directly to the matrix so qi and qj can remain binary we divide the linear terms by n so they sum up to one yeah that's kind of it it's a bit confusing because it seems like the they don't make a difference between when when is the one of the weights normalized or not we divide the variance terms the diagonal entries by n squared times m minus 1 to avoid duplication of what and divide the covariance terms yeah so they basically use the variance and then the covariance as the off-diagonal entries so these are the the coupler strands i guess to avoid duplication so i guess they essentially do the same but they don't compute like i i don't i don't see why the risk like i it doesn't seem like they it doesn't seem like like these guys at the risk component to these somehow or at least it's not really obvious what was the embedding part um we embed the c q and s and d wave by writing the expected returns into the linear terms both ryan's and covariance are there of the angle into the quadratic terms yeah from that's it that's it okay as we increase our asset size we see that for fully connected cubic ua requires multiple qubits and chains to leverage the available connections they do some fine tuning fine tuning where they don't have links between portfolios that they don't have like a high covariance because it means that they are just not such a difference okay let's say this makes more sense to me but how are they going to embed do this into a keyboard then it's going to be interesting but i understand that this is the return on investment right expect a return on investment as a minus because you want you want to maximize this because if you maximize these then it means that you're minimizing the the expectation value of the hamiltonian then you add the risk with a plus because this means the higher the risk is the lower the return like yeah you want to minimize you want to minimize return maximize you want to maximize return minimize risk so it's going to be a plus term and what was the u thing here um we have introduced a new dimensional vector u with u sub n equally equals 1 for all n which makes the constraint in equation 3 more compact i think the lagrange multiplier raw is fine tuned satisfy equation three that is interesting so this is this seems to be something this seems to be the component that it's added in there to penalize for portfolios that exceed the mind for choices that might might exceed like the total investment we want to have so instead of limiting the amount of assets you peak you're limiting for the amount of for the total amount of investment you want to make which kind of makes more sense because you kind of say i've got like half a million to invest right and i don't care whether you pick like three or four i just don't want you to spend more than that like okay let's go ahead we'll probably come back to these uh i just i'm i'm curious about the cupola actually what they detail that so okay they use the sharpie ratio and not like whatever made up thing uh we measure the quality of the portfolio by the so-called sharpie rage i mean that's maybe it's interesting so here there's no like there's not much discussion about like whether this is something suitable or not this quantifies the amount of return per unit of risk um notice that the numerator numerator is numerator as the return and not the profit which would imply removing all the possible additional costs such a transaction suggests transaction costs to be discussed later okay notice also that for example on rebalancing step one rebalancing step the denominator is a normalized volatility understood as the square root of the variance of the normalized returns square root of the variance um of the normalized returns that's the volatility like i'm gonna i'm gonna have to believe these i i i guess i'm not familiar with the with the field right so i really don't know how would you how would you represent volatility otherwise a large sharpie ratio means a large return for the risk that is assumed whereas the rate close to zero means the opposite and negative ratio means losses instead of profits okay transaction costs so how are you gonna add these okay so can i jump to the cue ball somehow and then i'll come back to this because uh so this is the simplistic problem and then they go ahead and they want to add transaction costs i don't know how they do this but like uh i'll go back to these okay i'll go back to these and this continuous versus discrete formulations discrete variables which is more relevant to big industry players as investment funds typically trade in large discrete amounts in this section we will briefly discuss the case where the asset location can be approximated by continuous variables in this section we will briefly discuss the case where the asset allocation can be approximated by continuous variables okay so that gets probably too complicated um the girls blah blah blah i i'll go i'll go back to these i promise um okay this is where i wanna i wanna get to the qr matrix so see i i start reading ql here so discrete acid locations in industry the rebalancing is done at discrete time steps and it is common for funds to trade assets in large discrete packages in this setting the problem is naturally recast as a quadratic unconstrained binary optimization cubo for these we chose a binary encoding of each variable okay so a binary in terms of like whether you invest in there or not right uh for these we choose a binary encoding of each variable in wnt sub nt in terms of n sub q so beats in terms of n sub q what is n sub q so beats x and t q of n q and sub q beats x and eq there's several options for this encoding as discussed example in reference 11 for simplicity in this paper we choose to work with a binary encoding where the variables can be 0 1 by construction we have that the maximum investment asset maximum investment per asset is okay so it's like invest in everything yeah that would be like investing in everything which is naturally included in the formalism investments go also in discrete packages of amount one substituting equation 25 into for example equation 12 results in a cube problem for the n total bit variables finding the optimal for your weights of any given time is therefore equivalent to finding the ground state over the variables x and dq of the classical hamiltonian so replacing what is substituting equation 25 so this thing into the equation 12 results in this equation 12. oh yeah that part here okay so yeah yeah replacing it here yeah and then it turns into the cubo thing right so you have each variable exactly because you have basically but what is 2 to the square to the power of q excuse the number of assets [Music] i know q is what you're trading here okay so that unpacks that unpacks into uh into what how is that a vector okay meaning like each of these terms would be an element of the vector of the variables okay finally that will follow weights at any given time is therefore going to find the ground state of the hamiltonian the beat vector corresponding q matrix which can easily be derived from equations 12 and 25. yeah it as i said it feels like so it feels like that's kind of what is it right like you you're using the returns you're using the returns as the um the the linear terms but in these paper what do they use of the linear terms the variance this rides here i don't know it's not clear to solve this problem one on a quantum computer we quantize equation 26. by promoting the beat variables to qubit operators with eigenstates zero and one q is the cube matrix right yeah and that's kind of the expectation value i guess that's what it's saying or i don't know okay so here they go ahead and translate it into the ising spin kind of model thing well you have the couplings okay that's it basically and then there's methods overview so and there's always various methods and hardware implementations to solve them your preferred decision problem is discrete formulation these are the following classical quantum annealing quantum universal and quantum inspired a d-wave hybrid let's let's go through this so as it's well known quantum annealing is a type of quantum algorithm based on the ideas of a diabolic quantum computation and it's particularly well suited to solve optimization problems this processing with the classical simulated annealing or thermal fluctuations so we know this we know this we know this johnson manager and by line this one is quantum manually provided by d-wave in particular the so-called d-wave 2000 q processor which gives access to 2048 non-coherent keywords coupled through a so-called chemera graph this architecture allows us to solve problems for up to 65 fully connected qubits due to embedding overheads um the twaf hybrid algorithm uses the hyperclassical quantum strategy to overcome this limitation allowing us to deal with much larger problems in a nutshell d-wave hybrid breaks down problems which are larger than the capability of quantum processing into parts then we combine to produce a solution [Music] tensor tensile okay then the vqe part like i want i want to see if there's some some concrete there's these tensor networks data preparation so i want to i want to see i want to i want to dive into the d-wave once i i know i said i would go through the whole paper but i'm curious about the d-wave details as the first step we benchmark our different objectives which are probably using random data so this is data preparation the bare return for each asset is given by the price at time t of s at the n is p sub nt minus the price and then divided by that price okay so actually they do so actually i was doing it wrong i was just i was checking the prices and i should like in in the in in by the way isn't he using github i upload it i upload it in github um uh what confirm confirm yeah here you go so i uploaded it here and and i did this wrong so i can go ahead and play with it if you want but like it's it's really um so what i'm just loading is the prices in clo and and at close date i think and i should and i'm i'm using this to work out the rest and i should use i should then probably create a returns which is basically the price you minus the price from the day before right from the time before yeah importantly mathematical expressions often call for logarithmic returns instead of the bear returns these are defined as these there are several reasons why logarithmic returns are preferred over the bear returns in particular they follow a normal distribution but most most importantly for for us they are time additive and hence justify the sum in equation one in trading situations though the bare returns are usually small implying that bear and logarithmic returns are interchangeable at the expense of a very small error okay that's interesting there's some dimensional reduction technique so deways can actually find close to optimal projectors even even with all the problems variables are used i mean cluster variance number of clusters okay so what they okay so that so i guess what the pre-processing is doing is it's taking the 52 assets and clustering them in six buckets so you have less uh investment options you have less variables to play with right is it what they're doing and i guess they cluster them based on the price evolution yeah okay kind of makes sense the idea is that this should make it easier for vqe i guess blah blah blah blah blah classic we'll also make the noise and data by applying a hundred prescott small thing which extracts the data train we then compute the occluding distance and the data okay so there's a bunch of i mean there's some actually decent pre-processing in here i see there's some kind of like eliminate like smooth smoothing the data curve that the data like there's some actual decent stuff done during the course of mana basis running average this has the advantage of eliminating daily fluctuations which is delicate variance is an estimate of daily fluctuations of a group of assets including the matrix poop up so once we get an optimal portfolio trajectory for clustered effective assets we unfold the investment by assuming an equal investment on each asset okay so then what they do is yeah they take so once the algorithm picks an actual um cluster then they assume that you do an equal investment in all the assets in this cluster make sense results okay but results like do you wave a sharpie right just computed by the different methods it's funny because in small problems vqe [Music] is giving europe seems to be even like a better solution does it one specifies specifics of different data sets okay so that's what um the xs and the xxl okay this is the yeah these are okay so this is the actual amount of combination so an excess an excess problem is easily solvable by uh i mean classically uh vqe can't even like it's crazy that vqa and even the exhaustive which is supposed to be the newest this new vqa idea they just can't like go past s like oh no that's v key constraint sorry exhausted is classical v key constrained okay so v key constraint gets to the size m but it's still like it really you can really see the d like already with size s like the d wave and the tensor networks just go over the over the roof in terms of the sharpie ratio like that's way better 8.9.6.04 it seems like for xxl like really the tensor networks are winning actually tensor networks win already from step m from size m that's interesting okay but uh so these are the results being discussed which we can go through later next steps okay so here that how to mod constraints [Music] market impact intuitively when a large order is passed it can affect the market for instance a large buy order may increase the price of an asset because it signals high demand whereas a large cell may decrease this is of course would alter the optimal trading strat trajectory as discussed in revenue 11 this can be implemented with market impact coefficients so they have they add another piece of data in there for escape this is like this is really detailed okay so i i i do wanna you know i do wanna end up like running these like on the d-wave stuff like and and you know just using the data set that i have okay so i gotta probably correct some things i definitely gotta correct the um the returns and uh what it what what these really seems to indicate is that uh like the cubo is basically the returns the the expected return is the uh the expected return is basically the linear term but they also add another linear term in here which is the the risk right as a sort of a penalty and then there's penalties where was the what was the question was it was the question 25 no it was equation 12. 12. here it is so you have so you have the um the return the expected returns and then you have yeah there you go there there you then you have the so is this does this mean the transpose let's figure this out like uh i think that's a transpose what is the risks this is the risk the risk of each investment as they multiply the investment by this which is which was the the covariance so so it is actually covariance but it's half of the covariance so so here they use the covariance as well but then what is the other what is the other stuff what is this part this is the coherence matrix right i guess that just means these in column form because if that's a matrix that would your yeah dive right into the portfolio return so where is this defined here so this is the is the acids covariance although covariance with what with the market maybe that's the assets convergence with the market and so what they do here is was this this was the return minus the return free my the risk-free return those things might be quite equivalent but they use the variance in the cover covariance term between acids i and j so i'm not so sure i think it's essentially i think it's a bit different because i think that is a risk like it's an isolated risk i think that's the isolated risk i think that's the isolated risk like comparing sort of it's it's the covariance for the market so that's the the volatility of the acid in isolation and then these other components [Music] here what is this other component here i saw the lagrange multiplier okay let's see it's it's fine to satisfy equation three okay but what is it what is it what is it there that's just the constraint that's the constraint on the maximum investment so okay let's see let's let's think through these i i feel i'm just reading and not like thinking you know so forget about the square part that's just to make that that just like a bigger constraint so rho is a lot of multiplier that imposes the constraint in equation three the branch multiplier first of all why they call it this way so this is supposed to impose these so basically what it does is um so and ut is [Music] u sub n equals one so what this does is it takes yeah it it basically i don't know actually how how like it's not obvious what are the quadratic terms in here to be honest so what are the quadratic terms are the ones for the couplers in the cubo the off diagonal are the so what is these first of all what is the crunch multiplier so in mathematical optimization the method of lagrange multiplies a strategy for finding the local maximum and minimum function subject to equality constraints it is named another recently lagrange the basic idea is to convert a constraint problem into a form such such that the derivative test of an unconstrained problem can still be applied the method can be summarized as follows you need to find the maximum or minimum function subject to the equality constraint gx equals 0 form the lagrangian function x lambda equals f of f of x minus lambda times g x so this is a subtle point of our crunching function what as a result so the great advantage of this method is that it allows the optimization to be solved without explicitly explicit parameterization in terms of the constraints what the heck is this what is a lagrange multiplier so let f be the objective function g with the constraints function both belonging to c let me know solution for optimization problem maximize f subject to g equals x at g x equals zero so this is the constraints that you have then there exists uniquely crunch multipliers such that whatever d times the lagrange multiplier theorem states that at any local minimum maxima or minimum of the function evaluated under the equality constraints if constrained qualifications apply then the gradient of the function at that point can be expressed as a linear combination of the gradients of the constraints at that point so okay but this is so then i guess this is something that is uh precomputed for this constrain that and that ensures that like intuitively speaking what this is doing is somewhat like so it's trying to enforce that the total sum of investment is k it's capital k and and that's the point is like that the approach here in comparison to the chicago quantum one is that you're constraining on the total amount of investment yeah but that's equivalent right because if you're just normalizing it so the size of the portfolio you're picking is really dictating the total amount of investment you make but it feels like it's better explained it here um but still i it's not clear what the quadratic terms are for the cubo hero substituting equation 25 into equation 12 results in a cubic problem for the n total n times nt times nq so you're replacing the saying that w n t is basically the sum of of these okay so i guess i guess what makes that the quadratic term is the fact that when you square these then you're going to have the actual um variables being multiplied to each other so once you replace these in here you're going to have you know it's the same like these times the same and so this is when you're going to have like the variables multiplied by each other oh sorry not here is it yeah actions here what is this though why did i miss that eh oh that actually for instance having additionally one may also introduce the cap k prime on the maximum amount that can be invested for each asset the two quadratic terms ah that's the transactional costs this guy's evolved okay um yeah i think this paper helped me sort of i think walk through the problem better than the chicago quantum one the i i have the feeling the uh q what seems to be the same or similar but this one so there's something that constrains uh based on you know this lagrange thing which i don't really understand it adds this so it's way more detailed at the transactional costs it has a um seeing a bit of a different conception of re constable risk and then it's using the um expected return as the linear component whereas in here it seems to be using the variance i guess if if this is really what goes into the cubo right because they don't mention they don't talk about this at all they say the linear terms by n and apply the linear fine transformation the write the variance terms to avoid duplication or that whatever that means and there is a sign on the linear terms finally we apply a scale factor to the cube and write it into a matrix for per participate your wave on exporting portfolios of different sizes we present a different metrics to deway for each desired size portfolio we add a penalty for exploring portfolios of different sizes while maintaining accurate values for the desired portfolio size the intuition for this follows closely from converting cuboid into an ising model in the curve queue into another model we consider the transformation on the binary vector x um since we're only looking for the location of the lowest energy is equivalent to convert back to x and find the actual lowest energy that's our intuition leads to consider a fine f in transformations in x our goal at this point is to find a shift which we can apply to the quadratic form this corresponds to translation and it's this closely related to the term above in our formulation we do not use balance although our mathematics takes it into consideration in this case balance will correspond to a boost in our coordinate system give it given the university of assets from which to choose will define a shift factor s and for exactly n or bigger than one assets which is the best score derived from a classical simulation in this case a genetic algorithm is a multiplier empirically i multiply m is generally around five intuitively errors can be multiplicative and certain values close to two new teams some sort of shift added to quadratic terms there's another shift at it covariance features the market and data based on log returns covariance seems be assets we've done all these drive a queue for each portfolio size like i don't get why they need to add these shifts and why they have a specific cube for each portfolio size that's kind of in the results like do we have some kind of maybe references maybe they have some code somewhere available outlook hmm it's overall like a these paper feels really decent in comparison to the to the other one and so the difference the difference differences are here i'm trying to understand why so why did the chicago quantum people need to apply this shift somehow why do they need to have a cube that's different for each portfolio size um whereas maybe that maybe there's something hidden in this lagrange part of things here that's kind of where i don't really understand and then here they started marketing impact and stuff like that like wrench multiplier it seems a much more compact way of doing that um [Music] it's it's nevertheless so here they talk about again they they they talk about the they talk about the um constantly talk about k as the investment limit versus near the paper they talk about and i think like the number of assets you want to invest in which seem to be i mean there must be related right um especially no no no no they don't have to be necessarily related right because you can just invest everything in one asset so i think these uh if i'm getting this right like that is better because it's it's not limiting you in terms of the number of assets you want to do and invest in it it's limiting you in terms of the amount you want to invest in but like the optimal like the result that it's picked like can be you know really stretched or it can be just really compressed into like just one or two assets that's that's that's the feeling that i have and i think it kind of makes me more sense that's what we're trying to limit so that's this like crunch thing here so the lagrange multiplier which i so single constrain examples so suppose we wish to maximize x plus y subject to the constraint that x squared plus y squared is one um in our case like you want to minimize the function that's your cube function with the constraint that it like all adds sub no okay with a constraint that like yeah yeah with with that constrain the k the k thing right like that the sum is like it's equals k the feasible set is the unit's circle and the level set of f are diagonal lines of slope minus one have no idea what they're talking about okay so they're basically the minimum occurs in this point for the method of lagrange multipliers the constraint is g x y g x y would basically be okay so you do you you have to come up with a g x whether the g function that equals zero so you kind of you know pass like move the one there and hence the lagrange what the current is saying is that there is then these uh f of x y plus like this lambda times the g function now we can calculate the gradient okay so you do the gradient of these the derivative right so you do derivative of these and then you end up with the systems of equations and now notice that the last equation is the original constraint the first two equations yield that x equals y equals minus 1 divided by 2 times lambda lambda cannot be zero and by replacing these into the last equation then we have that lambda is plus minus one over square root of two which implies that the stationary points of l are these evaluating the objective function f at this point yields square root of two and minus square root of two does the constraint maximum is square of two and the constraint minimum is minus square of 2. so you're taking the constraints what are you doing you're trying to you okay so what what's the essence of these here the f plus plus that that's kind of get i get the process in here but like what what was it saying the method can be summarized as follows in order to find a maximum minimum of function f subject to the equality constraint so f x minus g x so this is a constraint and find the stationary points of l considered as a function of x and directional multiplier the solution corresponds to the original constraint the transition is always a subtle point of the lagrangian function which can be identified among the stationary bonuses select engine function constrain which is called equation multiply it's very less important because it measures the change that occurs in the variable being optimized given a one unit change in the constrain so okay so this like so we'll dive into these in the next sessions i don't know how long is good how long we're going to be working on this but so what is the crunch multiplier is it's measuring the so it's somewhat related to the that like to that constrain but that constraint applied to the function of what the function of um kind of calculating the returns or so you have a function that is basically in this case what's the function like what's f of x like what's the function here for calculating the lagrangian like that's what i'm missing but i guess you just have to read that maybe when it's not that late and maybe you know not one hour and a half into the whole thing um okay but it's it's something that's then related to that constraint i get that and we use this as a penalty um why why does this work though like why does these really enforce that like why does this really this is really i still okay so the question that i have to answer here is how can these really uh how can this really enforce that constrain that we don't go over k or that like especially when w t is normalized right so this is basically so w sub nt is this divided by k so it's a relative investment already is the percentage investment and this u t u sub n equals 1 for all n which makes the constraint in equation three more compact it's the combination of this ut and this row will like enforce this constraint but this is not like i'm not entirely sure how can this really guarantee i guess this means you're so i guess this means if um if if you're gonna have a solution if you're gonna have a solution that goes over that right then then you're basically um you know that should give you a bigger number so so those are being penalized but like how does this ensure that you really invest k like that it equals k that's that's what i don't really understand it's like how does this work that it actually ensures the investment equals up to one it doesn't peak like small like a smaller portfolio with like a less investment that's what i don't really understand here that's the only thing like how does the the the constrain is really enforced uh okay yeah maybe you know i really wanna actually trying to run a cue ball um yeah cool that was fun that was definitely fun yeah if you're liking these videos i mean uh probably nobody's watching by now already but if you're liking that and you want to subscribe to the channel you're going to make me uh you know really happy um i i enjoy a lot seeing that people like the content scene that people want to subscribe to uh you know be notified of more videos coming up but i'm just i'm going to be uploading more often in the next in in the near future so stay tuned for this i'm really i'm really enjoying that um you know digging a bit into the quantum annealing part it's also fun um and and sort of like taking a look at the financial stuff it's just i mean you can just get so easily lost in here um but i want to take a look at vqe the vq constraint part but i really want to i really want to get the the d-wave stuff done i just want to want to run one example somewhat but i'm feeling really lost i'm feeling really like i don't know kind of what the next step is so what i have is um i gotta okay i gotta fix that to calculate the actual returns and now i would really have to build the cubo right like that's kind of the next thing that i want to do and uh uh to build the cuba i guess i just have to take the uh the returns and those are the diagonal terms minus i gotta add the risk though to these right minus a return plus the risk but that's the only the diagonal the diagonal terms and i want to calculate this stuff like what are these off diagonal terms like how how do we calculate these like what's the actual problem for blah blah find the optimal portfolio what's b times therefore going to find the ground state over the variables the vector and q q being the cube matrix which can be easily derived from equations 12 and 25. so the keys are in equations 12 and 25 12 being this part in here and 25 so this is what i want to really focus on 25 is this which is kind of basically yeah yeah we'll try to figure this out okay so next time we're really getting into the actual coding and then you know what we can do is actually i'll try to compare both cubos like this one and the one from the chicago guys see if there's any any any difference like i don't really remember this cuba has this shift stuff man like they'll i'll i'll definitely i i i won't really get this done so this has the shift things that i need to figure out and the other one it's got this like wrench and stuff so those are the next big big big challenges cool |
Also check out the 1 page quick notes on comparing the Sharpe and Hamiltonians from the two papers (posted on Twitter), Actually ended up spending the entire session on breaking down the hamiltonian :) will read through your notes when ill go through the comparison |
The real way to solve this is to determine what is the Objective function (or cost function) also the Hamiltonian or Energy function. <br><br>In our case, we start with the Sharpe Ratio which is a fraction. We clearly state the numerator and the denominator in equations (1) and (2) and also in some more detail in section 4.1. However, converting a fraction into a QUBO is not directly possible. So we come up with an approximation like the CQNS. This goes from a ratio to a difference. CQNS is defined in equation (6). Broadly the Hamiltonian equivalent is - Expected Return to a power + variance (or the risk component) + a shift (or penalty). Here the penalty minimizes the energy of the QUBO at the number of assets we want to find in the QUBO of Nn x N x N. Thus we have n NxN QUBOs each minimizing the energy at n assets. <br><br>In the Roman paper: They have a time evolution of the assets such that at various times the assets can be rebalanced. The also say that the rebalancing could have a fee. They show results with and without the rebalancing. They show the Hamiltonian first and claim it can be used to maximize return for a fixed risk using the risk aversion multiplier on the risk component. Thus they convert from a ratio to an energy difference. Their Hamiltonian cost function equation (5) is thus the - Expected Return + risk aversion multiplier x the Risk + penalty. I think the penalty ensures the return for the assets is minimized in the QUBO. Their QUBO is N x Nt x Nq. So they must have t NxN QUBOs each minimizing the energy at time t., @Uncertain Systems Equations 21 to 24 explain how the Hamiltonian is derived for the time evolution in the Roman case. It is interesting, that their formulation for a time evolution portfolio optimization using Lagrangians, Euler-Lagrangian and PDE ends up being practically the same as the CQNS with shift if the change in asset allocation over time or omega-dot=0. In other words, if you don't change your asset allocation as in our paper., Thanks for the comments! Next vid on the qubos on Monday!!! :D stay tuned. I'll go through ur comment during the vid |
A number of things after watching your video. 1) we are keeping the weights of the assets constant and using a CAPM model. It is not a model where you are correcting the portfolio. 2) the wTXw is the official format of the QUBO actually (it is one half of the matrix) wT is the transpose. 3) The Hamiltonian in Roman's paper is slightly different but you can see that it has quadratic terms, and linear terms which would have to be put into the QUBO format. This is tricky and requires some math, to get into a QUBO form. You will have to look at which are the quadratic terms qiqj and linear terms qi from both sums, to put it into a QUBO. I will have to share my derivation for it to fully make sense, but I wonder if this video helps. <a href="https://youtu.be/PdNc4dqNFbY">https://youtu.be/PdNc4dqNFbY</a> |
so we're back again on the topic of the quantum generator for Cyril Network when I began so we basically in the last video did a an overview we know for you on the topic in I'm sort of the general set up now I think I kind of realized that those two things might end up being two separate circuits um this one being one and this one being another it's just because the way they communicate is if here the density matrix which you can check out the density matrix videos as well and you will understand probably why maybe a little bit better because I think it's a it's a neat way a neat way to have a model that you can then optimize especially the switching is always in machine learning right so so this is this is point number one point number two is what I wanted to talk about today what I wanted to investigate today is the concept of this granion thing so and you know the overall theme of this channel is really not to get into the math steeply but I can and this paper is just like covered with that but it's just funny right because at the end of the day some let's try to break ties right so but incorrect what this is at the end of that we check that in this in the in the previous video what a gradient is nothing else than sort of a metric or on a specific when your specific quantum state that the hell's you in which direction maybe it's going oh it's going to its minimum I don't know whatever that means but I guess the idea is you know what you're trying to do with this is you're trying to minimize that so you wanna know what a gradient is so you know in what direction to be fall off the parameters so that's kind of your learning iteration um and I'm being super fast here because I don't I'm not deep into the details myself either but it's like don't be this is so funny you really I guess it's cuz it's a paper but basically what they're trying to say here is so they're trying to introduce a bit of annotation and how do you calculate the kradic Radian karate and how that's pronounced of a I would say maybe quantum gate box I don't know whatever the discriminator is so how do you calculate the how to calculate how clean to grade the gradient of these for example right um and so what they say is first they start they introduce a way to define okay so let's how do we break down that D right of the box D because they will have to break this down in order to calculate this creating this gradient and I you know at the end of the day what I'll try to do is I'll try to get a practical example as a last video in this series for the cornichons because this is super abstract it's what happens when these papers is it trying to try to generalize that and then that makes it super tough to get into the details here exactly quantum gradients so that's what they are doing so it's saying here is basically all this blah blah here is nothing else in saying so you've got a gate you and then let's think compose it and that gain you has a parameter set up or however that's called because that's the one you're optimizing that parameter and and so all this is saying is let's basically assume that your gate is broken down into like sap operations each of them has may be its own parameter which doesn't have to be the same that it can be a version of it or it can be the same in some cases this is really a general notation so just really all honestly we'll all you have to take a look at is this one where it's like if you've got a gate which is equal to this gate and this gate and this gate in this gate and each of them has parameter um so and that's all what the column is saying so just forget about this I mean they have it you know for for scientific paper you go to do that but it's just that that brings it far away from from practical usage um this notation is used to signify that the composition of a unitary transformation and it's elementary parameterize gates see and then it says that's kind of a general form of each of those gates okay that's so what I don't know is for the Hamiltonian he's really in that context a gentleman assuming assuming each element is generated by a Hamiltonian so mr. Google what is quantum gates by inverse engineering of Hamiltonian I think an open if revelation Hamiltonian the gate overflow competing I think I've read that before ready car mechanics in quantum mechanics Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system this addition is the total energy of the system in most of the cases under analysis that is way beyond what I'm trying to achieve in this hamiltonians for quantum computing but in this section we consider not gain based on a single qubit as an accessory started in the literature in the literature and in response to which why we label in one zero zero one two basis States and not get response to those instructions which over the time interval accomplish the following here often better are arbitrary in the unitary matrix you the corresponds evolution is so maybe this is just one like a generic way come on for time independent Hamiltonian in the diagonal presentation it yields the energy levels that's getting too much into the physics and I think but maybe this seems to me intuitively it's just a another way of formulating gate in terms of maybe some kind of energy levels or something like that so assuming a gentleman is generate about Hamiltonian because I think my next so my next video on this topic will be try to get an actual concrete example of what's going on inside those boxes and I think that's the problem here is they are making a lot of assumptions in terms of what shape those gates have and I think that's just they're trying to make a general expression for what those gates do and then so they can then go ahead and do their math here which is basically so what they are basically saying is there isn't that expression and then they're making the the derivative of the gate of course they replace here and then at the gate for that stuff and mm-hmm and so they make the derivative and then they say if an initial state and Q you don't think that I understand that stuff it's like most of the things I just read and pretend that I understand but I don't really understand so you're not all in there it's just that at some point I should take a look at these things for like five or ten times you can across maybe something an intuition level so we define an initial state and Q cube is the gradient with respect to the parameter theta is then given by this so they mathematically can be there at this point so I mathematically come up with okay what's the expression of a dirt if because that's what the gradient is really just enter into derivative um I seen yeah so I mean this is a bit you know this is sadly this is one of those things where the math really kids into the like digging it gets into the way right so um so derivative of function so this is really you know and that's that's but I consider those things basic maths but you might not and it's fine it's just so you understand the derivative if it's an operation that you can do in a function that tells you basically it kind of tells you the the inclination in a really roughly twenty point of your function so that's what they're trying to do here is they are they're trying to find that let's say inclination or how to find that inclination so they can then minimize it that's gonna be I'm curious how I'm gonna make a no math summary out of all these cuz it's a tough one but but that's basically what's happening here so they do this in the maths and then they try to walk they way back backwards and say so if this is what would again what are the set of gates what is litigate so we need to achieve that and that's when you get those two things here that's what you get done like this is exactly what's happening right so you've got the in a we show the general structure of quantum Craig radians and the structure of kind of has since this is something else I don't know at this point it is convenient to introduce some canonical quantum gates in the harem arcade the not gate XK the said Kaede and it's also useful to define a single qubit W gate so that's made up K by them the dusties as shown in figure 5 the gradient of a parameterised quantum circuit can be sampled from the Z expectation the expectation value of an ancillary can be such that such that probability of being 0 - probably being won using quantum grants to train documents and that's all they say about that because then it's already like ok so how do you train down some this is not it's not of our interest how do you how did they use the corner guardians to train that's just I think out of the scope for these I the I'm trying to I'm trying to understand because I thought that would be there would be an interesting concept to generalize right so how do you make gradients and they decompose this so it says we now have all the elements required to evaluate the gradient the gradient of 13 directly in the quantum computer the operator P from section 2 B corresponds to the set operator of 4 okay what is section 2b the operator P no um but basically it says the your printer P corresponds to the Z operator of four when computing crazy ingredients for said create decide operator here sad okay it's a measurement that's what he's saying so it's a control fisherman like is this what it is here it's a controlled measurement may be the permit rise discriminator D and generator G can be decomposed into endian ng gates such dad in North America radians who introduced a single qubit register right okay that makes sense so this is what you're gonna measure this is this it follows that all elements of the gradient of the discriminator can be evaluated for each label in resources for each label n sources are ng by the quantum circular figure seeks with an appropriate XK to account for the sign of the cost function in the later case the next grade is applied on the on the outer edges after discriminator to get the correct sign of the cranium okay it's a technicality the circuit that needlestick radian stays off the generator know of this generator for each label is shown in Figure 6b 6b okay so this is the tree okay so that's the that's how they break it down okay so they break it down no it's just that it's not a major might exist that it's a control set but they've totally so apply the gain in the plane what is what is these right since the compost in three K in three like steps first one then this whatever Hamiltonian this is and then this one and that's the controlled gate so we also seen that the expectation value for each where it was I um that's it we know that the sign is so basically they just they just told it derivative mathematically and then they just write this down in circuit intuition and they come up with a circuit awesome but I'm not happy with that I'm pretty not happy with that it follows that all elements of the gradient discriminator that's that's definitely that's definitely a tough one that's definitely that's definitely a tough one but really what are they at the end of the measure I mean they're gonna measure like zero or one and that's it and that tells them I think I'm gonna I think I'm gonna what I'm gonna probably do is we should probably do is I should probably I should probably research it would be the search on that topic assigned this means independent from this paper but maybe I'm not gonna find because it seems really a dog here defined so the gradient with respect to the properties given by this so the derivative of gate J with respect to parameter as since the problem is the same concept of the derivative is always a bit abstract right so you you you kind of understand it with simple functions when it comes to quantum gates it's like what does this what does this really really tell you right such that this is OK J with respect to parameter J so these this here when you see here that's what we're trying to calculate is intuitive how they come up with that I don't know maybe based on this right on the definition of the gate so I'm isn't chained that's the the diving into this confuses me even more because it just goes into full abstract mode and then just talking maths so okay so you expand and expand blah blah if we define an initial state on Q cube it's the expectation value of an observable okay so these I just happen to know that this is probably the notation for the density matrix because I'm working on the videos as well at the same time I'm doing this expectation value of an observer all SD and I think TR means trace which is basically the sum of the diagonal other sorry it's just you know that's I'm reading this out loud and I just don't want to that's not that's not the outcome that I want to make their own it generate out of these sessions so the because basically they just go ahead it's like boom that's how you defined it the derivative of all done and then it's punished committing to introduce some canonical gates well awesome because then these these gates so okay so what I will what I will probably what I will what I have to do is probably this is this has been a painful video and there's nothing much out of it but the next step is I'm gonna actually try to reason it from the circuit maybe from this maybe from this circuit here I think that's gonna that's gonna be more valuable so there's you know there's a hot important and this is kind of split into three steps and there's a control to thing here and then there's an X for accounting for whatever and then there's another condition controls that are more let's see if let's see if that works let's see if that works |
today or we'll do the quantum measurement part but um i'm aware that i'm flying over these just like at the speed of line pun intended no but uh like just flying flying over the staff without really you know sticking to a specific point in digging deep but that's like on purpose right that's kind of at least from my perspective the way that i um that i kind of like to to approach topics like these let me just because i can talk and do things at the same time i just wanted to tweet out that people can join if they want so twitch dot tv slash uncertainties there you go hopefully that's the right link yeah that's like so link perfect so yeah basically basically that's what i will do so uh let me open the chat awesome um so last time with the um and you know and i've and i've i haven't taken a look at all this stuff in here right so i will come back to these it's just the fourth session and and i'm just really you know the goal is to learn the quantum staff and and kind of learn quantum mechanics right and and just to kind of re refresh the goal of this whole project right is not to understand the project in its entirety it's it's you know my primary goal is to kind of learn more about quantum mechanics and i thought a great way to do this is by by kind of exploring this project because i know that their stuff they're doing is quite cutting edge and actually it it's what i like about this is that it's an exercise of um you know a big group of scientists are saying we've got quantum mechanics and we've got this model and kind of try to map it right so this mapping exercise is a great exercise to kind of i believe it's a great exercise for me it was a great chance for me to try to understand sort of what are the parts of quantum mechanics that are relevant and kind of like you know get a bit of a non-traditional view on on all this kind of stuff so and then you know just build up my knowledge base from there and i yeah i i thought i'm not taking notes or anything i'm just really this is just really the way that i that i do stuff so i know that it doesn't work for many of the most of the people but at least i record the sessions and i have the videos so if you ever want to come back to one of these learning sessions and then they are there in the youtube channel as well um uncertain systems and i'll organize that a little bit better because there's a lot of mixed content from from from the past as well but anyway today i wanted to spend some time on the quantum measurement stuff so we did the basics and we did quantum formalism which here in quantum formalism my last time i already learned a lot of stuff that i want to come back to which is basically um and then afterwards kind of after thinking about this be more depth i did realize that there is more to these that i have to definitely dig into right because i come from just pure strictly speaking quantum computing side of things where you have a state and then you have a quantum circuit which basically dictates how stuff evolves and i think the way that i was reading through some of these sections here would be misleading because in in essence you have this this is just a toy example right of a system that evolves and then each node is a is is a configuration of this is basically your state your system state and so this is the multi-way graph this means what this graph is telling you is that you know from this node this one rule you can apply which is the rule that turns it turns this state into the a b state from this point in time there's two rules you can apply so you apply one rule you'll turn the b into an a if you apply another rule you'll turn the a into an a b and then so you kind of have two possible alternatives and so i was having a hard time going through this because i think i was too much hardwired thinking about the actual quantums not like a quantum circuit which is not later on i kind of realized that's not what that's not the point the point here is if you think about this is a quantum mechanical system there's the concept of you know the the the the evolution the time evolution of the system and so that is described described by the hamiltonian and i think that what this is trying to say is that you know and also there's a there's a point in time a point here somewhere where they talk about like um the evolution amplitude or some some sort of like the probabilities of evolving from one state to another and i think that's what the hamiltonian is really encoding in here right it's instead of just being a circuit is it's just telling you how the system could evolve and so that's what the multi-way graph here is telling you so that's why there's a lot of stuff that i wanted to you know um that i definitely have to come back to and and deep dive into afterwards um there's also the whole thing with lagrangian stuff um and that i definitely want to take a look at so we'll we'll do again we'll we'll basically do a that that's the way that i do this right so just go end-to-end and then we'll just kind of go back and try to put pieces together and then deep dive into some of these aspects um but so what i took away from this section last time was basically this duality of you know you've got states and then you've got like the the way they map the formalism into the multi-weight graph and and we haven't talked about all the other hypergraph stuff because that is just everything that comes before these like and i think that it's more related to uh to you know general relativity and whatnot and and the quantum stuff is really happening at the multi-way graph level i guess where you can take a look at the possible evolution paths of your system and um and then there's another so that was the first half in the second half is like exactly talking about like how this evolution how can you reason about this evolution somehow and how how does this then somewhat mapped maps to to um to the standard quantum formalism but i literally did not take out anything else from that whereas like oh yeah that makes sense like it was totally you know at least i grasped i think i grasped the overall structure of the chapter and that's what matters um so let's go through quantum mechanical let's see how they map that and what's because that's that's that's where things get interesting as well right because it's um it's also where you have um some of the biggest questions around quantum like or open challenges around quantum mechanics in general right like the whole the whole measurement problem so let's get to it above we gave a brief summary of how quantum measurement can work in the context of our models uh did they here we give some more detail okay i i don't know i haven't i don't think i have read anything before about quantum measurement but could be in the dance in a sense the key to quantum measurement is reconciling our notion that the definite things happen in the universe with the formalism of quantum mechanics or the branching structure of a multivariate system but if definite things are going to happen what might they be here will be again considered the example of a string supposition system through the core of what we say also applies to the full hypergraph case consider the rule a aab ba we could imagine a simple classical procedure for evolving according to this rule in which we just do all updates we can say based on a left-to-right scan at each step yeah that was that was what was marked in the previous chapters just like the red notes right um the gender was it like something like the generational evolution or something like that um so you apply all the possible rules you can from right or left from left to right and so you get this path but if how we know that there are many other possibilities that can be represented by the multi-way system right so that that tells you that's just a path it's not even a path but that's just a subset of the nodes they don't even have to be consecutive or like happen at the same like consecutive levels um most of the states that appear in the multi-way system are however unfinished in the sense that there are additional that there are additional independent updates that can consistently be done on them for example with the rule a aba there are four separate updates that can be applied to aaa right but none of this depends on depending on the others so they can in effect all be done together giving the result put together would put another way all of these updates involve space like separated parts space like separated parts of this string so that they are all casually independent and cannot consistently be carried out at the same time discussed in 5.1 doing all this across the state together can be thought of as evolving in the generational steps that's what they said as one so you have generational states in in multi-way cases there might be a single sequence in some multi-way cases there may be a single con sequence of generational states so these are the ones marked in red and in other cases there can be several branches of generational states okay really why because if we consider these two rules a a b and b a so a maybe that's just another example but like i'd say yeah okay you can apply all the rules but there are different ways in which you can apply all the rules at the same time the presence of multiple branches is a consequence of having a mixture of space like in branch like separate events that can be applied to single state for example with the rule a b and a a to a b and a to b b um the first and second updates here are space like separated but the first and third are branch like separated okay let's try understand that the first and the second updates here are space like separated but the first and the third are branch-like separated i don't know what they mean by this so a b a b so these are oh the first these are the four possible updates you can have so uh yeah okay i get it so this means these two these two updates here the first two can be applied simultaneously but then these other two cannot a view of because they basically they they affect the same element in the string right um the view of quantum measurement is that that it is an attempt to describe multi-way systems in generational states sometimes there may be a new classical path sometimes there will be several outcomes for measurements of states okay so so the measurement has got to do with the generational states that is not what i expected maybe because i still have an incomplete picture of of the whole evolution concept in here i'm just probably too i'm just probably overthinking it too much but now let's let us consider the actual process of doing an experiment on a multi-way system okay but now let us consider a process right now or a quantum system our basic goal is as much as possible to describe the motorway system in terms of a limited number of generational states without having to track all different branches in the motorway system at some point in the evolution of a string substitution system we might see a large number of different strings but we can view them all as part of a single generational state if they in effect yield only space like separate events in other words this string should be assembled without branch like ambiguity they can be thought of as forming because this is international stages in a little traditional state right that's what they're saying here i guess it's like at any point in time you can see the state aa or abb but it doesn't matter because they all converge to this one if we think about this generational concept right um in the timeframes and quantum mechanics we can think of the state in the multi-wave system as being quantum states that's what they said in the previous chapter the construct we formed by assembling these states can be thought of as a superposition of states okay so [Music] casual invariants causal invariants like such a message basic word causal no causal invariance that implies that through the evolution of the multi-wave system any such superposition will then actually become a single quantum state the construct will form by assembling these states can be thought of as a superposition of the states there's something a bit off with these like it's still maybe i kind of coming back to the whole circuit thing and it's just i think that might be i might be just fooling myself but because when i'm building a quantum circuit i i am in a way designing the way the evolution is going to look like right so if i say apply a hard mark gate like i i know that i want to be getting into a superposition of a zero and a one it's not that i there's nothing missing in the air i don't understand i i can't i i just don't find a way to map the generational evolution concept with the quantum circuit example for example just take a simple one qubit circuit right like you have one cubit and then you apply a harmonic gate and so that takes you into a zero plus one state but i'm not 100 sure if that means because you know because i could think of you know i could think of like like the the hallmark gate as a rule would be put the zero into the zero plus one state put the one into the one into the zero minus one state um but there's just one update it can do so so that is already a generational state so but it's a it's a state that it's super position so that's why i'm not i'm i'm maybe i'm just making the wrong analogy that's or i'm just looking at the wrong example with a quantum circuit but that's still a bit hard so the constructive form of assembling the states can be thought of as a proposition of the state causal invariance that implies that through the evolution of the multi-wave system any superposition will then actually become a single quantum state in some sense the observer did nothing they just notionally identified the collection of states it was the actual evolution of the system that produced a specific combined state in describing a quantum system or a multi-wave system one must in effect define coordinates and in particular one must specify what foliation one is going to use to represent the progress of time and this freedom of freedom to pick a quantum observation frame is critical in being able to maintain a view in which one imagines defining things to happen in the system with a foliation like the following at any given time there is a mixture of different states okay so now these are mixtures are they like is the word mixture here used as in like like an actual um mixture of states are not like a superposition right so with the foliation like the following at any given time there is a mixture of different states and no attempt has been made to find a way to summarize what the system is doing because there's a proposition and a mixture are essentially different right consider i have a foliation like the following so each in this case each and this picture generally generational states have been highlighted and affiliation has been selected that essentially freezes time around a particular generational state in effect the observer is choosing a quantum observation frame in which there is a definite classical outcome for the behavior of the system freezing time around a particular state is something an observer can choose to do in their description of the system but the crucial point is that the actual dynamics of the evolution of the multivoid system cause the choice to have implications in particular in the case shown the original multiple system in which time is frozen progressively expands the choice the observer has made to freeze a particular state is causing more and more states to have to be considered as similarly frozen in the physics of quantum measurement one is just the idea of that for quantum measurement to be considered to have a definite result it must involve more and more quantum degrees of freedom this i didn't know quantum measurement to be considered to have a definite result it must involve one more quantum degrees of freedom what we see here is effectively manifestation it's found at this i i don't know what this means in facing time and sunlight inflation the picture of all we are effectively doing is creating a coordinate singularity and and defining our quantum resolution frame and there is an analogy to this journal to to do a freeze time of my friend once again first time in a relatively sticky reference frame for example as an object approaches the event horizon of a black hole its time is described by a typical coordinate system set up by an observer far from the black hole will become frozen and just like in our quantum case we'll consider this database stay fixed whatever i don't know but there's there is a complicated issue here to what extent is the singularity and the freezing of time a feature of our description and dual extent is something that really happens this depends in a sense in the relationship one has to to the system in traditional thinking about quantum measurement one is most interested in the impressions of observers who are in effect embedded in the system and first and for them the coordinate system they chosen in effect defines a reality but one can also imagine being somehow outside the system for example one might try to set up a quantum experiment or a quantum computer in which the construction of the system somehow makes it natural to maintain a frozen time foliation [Music] the picture below shows a toy example in which the motorway system by its very construction has a terminal state for which time does not advance but now the question arises of what can be achieved in the multi-wave system corresponding to the actual physical universe and where can we expect that and here we can expect that one will not be able to set up truly isolated states and that instead there will be continual inevitable entanglement one might have imagined could be maintained as a separate state will always become entangled with other states the picture below shows a slightly more realistic motorway system with an attempt to construct a foliation that freezes time god i need to understand that those foliation stuff would be better and we see there is in a sense the structured multiple graph limits the extent to which we can freeze time in effect multi-way system forces the coherence or entanglement just by its very structure we should note that it's not the case that there is just a single possible sequence directional stage because point this is a possible classical path here an example where there are four generational states that occur at a particular generational stance and and and that might be then the superposition really and now we can for example construct affiliation that at least for a while for this time for all of these generational states it is worth pointing out that if we try to freeze time for something that is not a proper generational state there will be an immediate issue a proper generational state contains a result of all space like separate events at a particular point in the evolution of a system i feel i feel i have to maybe step a little bit back and and and and really try to map like really try to understand that concept to be better of the generational states with like their like a proper example um if we try to freeze time for a state that did not include all space like separate events there would quickly be a mismatch with the progress of time for the excluded events or in effect the singularity of coin observation frame would spill over into singularity in the casual and the causal graph leading to a singularity in space-time in other words the fact that the states that appear in quantum measurement are generational states is not just a convenience but a necessity or put another way in doing quantum measurement we are effectively setting up a singularity in parental space and only if the states we measure are in effect complete in space time will the singularity be kept only in branches space otherwise it will also become a singularity in physical space-time or big words this is too much in general we'll talk about coin measurement we're talking about how an observer manages to construct a description of a system that in effect allows the observer to make a conclusion conclusion about what has happened in the system and what we have seen is the appropriate time freezing foliations allow us to do this and while there may be some restrictions uh principle possible to construct socializing motor system but in practice as the pictures above begin to suggest after a while the foliations have to we have to construct and get increasingly complicated effect what we're having to do in constrain in constructing deflation is to reverse engineer the actual evolution of the motorway system so that our elaborate description we're still managing to maintain time as frozen for a particular state to out-compute the system itself and so we will be asking the observer to do a more elaborate competition to maintain the description they are using and as soon as the computation required exceeds the capability of the observer the observer will no longer be able to maintain description it is worthwhile to compare the situation with what happens in in thermodynamic processes and in particular with the parent entropy increase in a reversible system it is always in principle possible to recognize say that the initial conditions for the system were simple and low entropy but it practiced the actual configurations of the system usually become complicated enough that this is increasingly difficult to do in traditional statistical mechanics one talks of coarse grain measurements as a way to characterize what an observer can actually analyze about the system in computational terms we talk about the computational capabilities of the observer and how computational irreducibility in the evolution of the system will eventually overwhelm the computational capabilities of the observer okay i i'm this is getting heavier so what do we what do we see here so what what is let's try to let's try to unwrap some of these it's again the generational states i'm missing a good example of these with a quantum circuit kind of like i have maybe no i was gonna say maybe it's got to do with noise and in general you know the mixtures that you know the evolution of the system is somewhat dictated by like a noisy thing but it's also not true because these nodes are basic states not just states they are basic states so so what this is trying to say is that oh god um what is it even trying to say that that the measurement in a measurement you can only see generational states i think that is at least something that i can say out of these and then and then making a measurement is picking a specific way of foliating the multi weight graph that's just your reference frame right like so maybe because i'm i'm inclined i'm a bit inclined to take a look at just the operators part because i because maybe that is going to help me understand or build an analogy with the circuit stuff so there are states and operators now models updating events are yeah okay updating events are corresponds to operators in the standard evolution of a multi-wave system all applicable operators are in effect automatically applied to every state oh okay so that is actually i think that is gonna oh that's short enough i think that's gonna help me build that bridge because that's that's what i was looking for right now so so there are states and they're operators right let's see what is the reference here oh come on um in our models the dating events are will correspond to operators so the operators are the rules right and and that's kind of yeah that's that's what i was saying right you can you can ride you can write rules that like specify the harmonar gate saying zero to the plus state one to the minus state in the standard evolution of the multi-way system all applicable operators are in effect automatically applied to every state to generate the actual evolution of the system that's what i don't understand but to understand the correspondence with standard quantum formalism we can imagine just applying particular operators by doing only particular updating events consider the string substitution system a b a b a so a b to a b a and b a to b a b and the system we are affected to operators so one and o two correspond to these two possible updating rules we can think about building up an operator algebra by considering the relations between different sequences of applications of these operators in particular we can study the commutator in terms of the underlying rules it's going to response to i don't know what a commutator is um uh the first header is also playing golden initial state are different we can then say that it stays related from the branch pair but then at the second step the branch panel resolves and the branches merge the same state and in fact we can represent this by saying that i wanted to commute um okay so that that whether they compute or not um that all branches can result in a single classical state just like in standard quantum formalism the computing operator is associated with seemingly classical behavior but there's a key point here even if even if causal invariance applies to branch pairs and will eventually resolve they might take time to do so and this is and this is delayed resolution that is the core of what leads to what we normally think of as quantum effects um once a branch pair is resolved there are no longer multiple branches and a single state has merged but before the branch pair is resolved there are multiple states and therefore what one might think of as quantum in determinacy in this case where branch pair is not a resolve the corresponding commutator will be non-zero in a sense the value of the commutator measures the branch like distance between the states so branch pair is not a resolve the corresponding commutator will be non-zero i don't think that's going to help anything you know so this is a brush like separations and there are other pictures in tongue one two stays on that if they are part of the same unresolved branch pair and does have a common ancestor the multi-way graph gives the full map of all entanglements but in any particular time corresponding to particular slice of or of foliation defined by a quantum observation frame and the branchial graph gives a snapshot that captures distant instantaneous comparison of the negative ones okay um it doesn't help me but i mean at least it confirms that the updating rules are the operators right or you can think of these as operators um but then this would mean that a cert that uh that a circuit is is picking a specific updating order you know so see what i don't understand is what what will be the analogy of that multi-way system like what is the multi-way system trying to because if i don't understand that then i i also don't know how to make sense of this whole generational stuff so for this i think i have to go back to the quantum formulas and stuff um because for a circuit right i have not only a specific set of rules right like the harmon the harmonic operator the control knob operator but i have a specific i have a specific order or i have a specific path it really is a specific path like a quantum circuit would be of a specific path in the multi-wave system because i know at every point in time what is the specific set of operators that i'm picking and where are they applied within the state right like you know in this case you have or somewhere below here the other stay you had a system with an example with like rules that both apply to so that'll be that that will be where is that there was an example here with the word exactly so you have these these two right so you could think of i could think of these as like let's say the harmon gate you actually have two rules no i mean it doesn't matter like whatever right like actually each operator is really each operator is really a set of rules right or would be i mean i would build it like that right so i'd say and and they're based on your computational basis although that feels somewhat wrong because i i i have the feeling that the computational basis got to do with the at the end of the defoliation ah that's maybe not true either i think the computational basis is at the end of the day that choice is the choice of language you're picking the choice of of notation you're picking if you're to pick zeros and ones then then so be it right um and so the rules are going to be defined like turn a zero into um the plus state or turn a one into the minus state but then there can be another say the x game right it's also going to have it's also going to share things like these so you're going to have a bunch of all different possible yeah evolutions but then what's the point so i think just a quantum circuit so i don't i don't really get maybe and as i said maybe it's because i'm missing that dynamic component of the quantum mechanics right let's say how is it that we describe a system evolving because for me i just you know coming from the circuit world just have that in my head i have a circuit which this model is just a specific path in here because i know exactly what rules i want to apply and where i want to apply them you know even if i could apply a rule in multiple spots in here rerun rule i might i'll choose to apply this one right um but i feel i think i really i think before before i go on i need to deep dive into this a bit more yeah before i even go to wave particle duality and stuff like that what is that even i mean that's also fairly short and quantum mechanics it's funny because that that is already kind of i don't know what when i see such short chapters i'm scared because it's probably just event horizons and singularities and space time and quantum mechanics cosmology expansion and singularities reversibility reversibility motion and special activity space and time structure space so i think i have to come back to this speed and the first part i kind of understand so we that that i'm i'm like i keep getting lost in the same space so each i was wrong so when i said that it's not a specific path but it's a specific sub graph because i might choose to apply a hard mark gate which might split into two is to me to two notes because i have a zero right let's try to do that again with the paint if i can manage to open the application without breaking the computer and the streaming that'll be awesome so so i start off let's say i start off with the state 0 0 right and and my set of operations are the hana mart and the x gate right but i have a specific circuit in mind which is you know apply a harmonic gate to this qubit and then apply an x gate to this qubit like if i would consider all the possible ways this can be done right like which is what these graphs seem to be doing it's like well okay one possibility is apply the hadamard rule to the first qubit but that actually leads into you know into these because this is these are really two rules but it could also just be no way yeah so but these two would merge right but i could also just apply the x-gate which in this case i have two possibilities no i also have the possibility of applying it everywhere and i also have the possibility of applying the hadamard gate to both oh god and the automar gate to so that would just grow right so i mean some of these merch and would not but then if i have a specific circuit in mind then i know i'm picking a specific subgraph in terms of the evolution i'm saying i want to go down this path and then next gate here so kind of like these are the two these are the two should change color but these are the the the two notes that i want to have everything else i don't care i've picked that specific i've picked that specific path that has led me to these having two separate notes i think this is i think i think this is probably what what's happening here probably but then why build that feature right like why why do you need to build a multi-way system for these um that's that's what i'm [Music] and that's kind of maybe the step that i'm missing towards like the quantum mechanical part of it right like to to kind of like um um let's go back to these so let's let's i think we have to unpack some of these things so here we talk about the states being quantum base the nodes being quantum basis states that makes currently sense based on what i on this feature i don't get so much the entanglement stuff because i wouldn't say these things are entangled even though they probably share some ancestry i mean they are correlated but it's not an entanglement correlation so they are like the thing is they are coral like the two things are correlated in a way right so each pair of states generated by a branching and its graph are considered to be entangled i don't care about the geometry geometry right now here we have the count of the paths let's say that we want to track what happens to some part of this branch like hypersurface each state undergoes updating events that are represented by edges in the multi-way craft and in general the path followed in the multi-weight graph can be thought of as geodesics in multi-way space so here they get into better worker down respond to energy the space time caught causal graph however it's just a projection of the full multiplied causal graph now note that every node in the multi-wave causal graph represents some event in the multiple graph but events are well produced branching and turns let's just zoom out turning power parallel energy in energy is identified with a hamiltonian h so what this says is that in our models we can expect transition amplitudes to have the basic form in agreement with the result quantum mechanics okay so let's start with transition amplitudes let's just dig into this so so so is your quantum mechanics transition amplitudes what are transition amplitudes path integrals let's do this extended sassy quantum mechanics transition amplitudes okay so this seems the right no uh so in quantum mechanics the physical system corresponds to a hilbert space states correspond to not in a one-to-one way to points in here with space and the physical postulate is that the transition amplitude a complex number from a state corresponding to v into state corresponding to u is given by where the physical meaning of the transition amplitude is that if you take the squared absolute value of its complex number you get the actual probability of the system going from the state corresponding to v okay so there's these dynamics are defined but i don't understand them so it's it's a postulate right but this is a probability amplitude that's that's that's just the measurement stuff so in quantum mechanics the probability amplitude is a complex number used in describing the behavior of systems the modulus squared of this is a probability density but probability amplitudes provide a relationship between the wave function of a system and the results of observations of the system but i don't want to that that's that's the that's what i don't want to have i don't want to have the the observation stuff i want to have the evolution stuff transition amplitude it keeps it keeps keeps taking me to the path integrals which i don't know what is it the traditional amplitude in the most commonly used operator approach the transition amplitude is expressed as the vacuum expectation value of the product of particle creation and annihilation operators these operators obey certain commutation relations so what is this oh quantum field theory so this is just a portion of these i'm not gonna no how can i even so what is what is the path integrals quantum mechanics what is the path integral the python formulation is a description of quantum mechanics that generalizes the action principle of classical mechanics it replaces the classical notion of a single unique classical trajectory for a system with a sum or functional integral over an infinity of quantum mechanically possible trajectories to compute a quantum amplitude this formulation has proven crucial to the subsequent development of theoretical physics because manifest lawrence covariance is easy to achieve than in operative formalism okay so this is a separate formalism than the operator formalism and i guess the operator formalism is just the staff with operators should probably dive into one of these god i think i've opened uh an unconventional text from our mechanics kind of feel theory starting mechanics i mean just symmetry principle without reference to classical mechanics and mathematical foundation the bridge conservation paper so i i i kind of get it that it's like so i guess what this might be trying to say is that in in in in the formalism itself there's just an obstruction that you to what you usually think classically which is like you have an object and it just has a trajectory and in this case have a quantum system which doesn't have a trajectory but it just has a collection of all the possible trajectories they can go through um and what is this but is this what the hamiltonian is defining or not that's that's my that's the point hamiltonian so what about connecting hamiltonian hamiltonian hamiltonian pathway does it disappear as a suggestion oh that's just a graph okay now hamiltonian hemapungen mechanics no hamilton mechanics hamiltonian mechanics there is a video and there's some lectures i was thinking about introducing or watching some videos as well uh also the iphone again with microsoft information of classical mechanics historically contributed to the formulation of statistical mechanics and quantum mechanics and melatonin mechanics was first formulated by william rowe and hamilton in 1833 starting from lagrangian mechanics the previous revolution of classical mechanics introduced by joseph luis de grange like lagrangian mechanics hamiltonian mechanics is equivalent to newton's laws of motion in the framework of classical mechanics so in hamiltonian mechanics the classical physical system is described by a set of canonical coordinates where each component of the coordinate is indexed to the frame of reference of the system the qi are called generalized coordinates and are chosen so as to eliminate the constraints or to take advantage of the symmetries of the problem and the time evolution of the system is uniquely defined by hamilton's equations where this is a hamiltonian which often corresponds to total energy of the system for a closed system is the sum of the kinetic and potential energy of the system in the training mechanics the time evolution is obtained by computing the total force being exerted on each particle of the system and from newton's second law the time evolution of both position and velocity are computed in contrast hamiltonian mechanics time evolution is obtained by computing the hamiltonian of the system into generalized coordinates and inserting it into hamiltonian equations this approach is equivalent to the one used in lagrangian mechanics the hamiltonian is the legendary transform of the lagrangian when holding q and t seeks blah blah blah no idea what i'm talking about the more degrees of freedom the system has the more complicated its time evolution is and in most cases it becomes chaotic cockle it in the hamilton from a lagrangian okay so this does actually take me down the classical path a little bit and then there's the crunch of mechanics stationary action principle that might be introduction it's funny because that's just from newtonian telegrams and mechanics that is super dense so what about solar cryo mechanics is there some sort of tutorial oh god lesson one basically branching mechanics uh that is quite wikipedia as well maybe i should just do this off of the wikipedia stuff which is pretty hardcore though but okay so so looking at the hamiltonian mechanics and then then there's this um but then isn't isn't it then schrodinger's equation right what defines the equation isn't isn't it what is this what actually defines the movement this is the hamiltonian operator the wave function is a linear partial differential equation that governs the wave function of a quantum mechanical system the series equation gives the evolution over time of a wave function the quantum mechanical characterization of an isolated physical system so that is that is what um preliminaries okay particle in a box example so here there's some examples with the harmonic oscillator quantum harmonic oscillator oscillator okay so here there's some examples of so this is this is maybe what then leads to these to the transition amplitudes right or like what if i google these two things like do they anyhow schrodinger equation [Music] transition amplitudes oh is the transition after this the path integral formulation that's then a different formulation so this relation between schrodinger's equation and the path integral formulation of quantum mechanics um this article relates to shrinkage equation with the pythagorean symbol non-releasing one-dimensional single particle hamiltonian composed of kinetic and potential energy so background schrodinger's equation the bracket notation is like these camber and stat that's the hamiltonian operator whatever i have no idea what i'm reading um the path integral formulation the bottom information states that the transition amplitude is simply the integral of the quantity over all possible paths from the initial state to the final state where s is the cl is the classical action so the reformulation of this transition amplitude originally due to dirac and conceptualized by phaman forms the basis of the path into the formulation part of the product formula says that for non-self-regeneration we have the transition amplitude can then be written as these although the kinetic energy and potential energy will produce do not commute the charter product formula cited both states it's a classical lagrangian god so they are related and then the the transition amplitudes is this is really it's got to do with that and so this is something that you can calculate as well from schroedinger's equation based on what i currently seem to have understand so these would be yeah so these would basically be a system that just evolves kind of i think i have to re-read these and then i'll probably have to do the lagrangian and the hamiltonian mechanics stuff i i feel like this this is going to be inevitable to actually understand why i mean we could we can do this just not starting from the lagrangian mechanic stuff it's just i don't want to dive too deep into these but that's quite complicated my grandchild mechanics i don't want to watch any of the quick videos here like the lagrangian method brilliant.org yeah that's also not intent mechanics doesn't it doesn't look better either anyway i think we'll just go from so an introduction to location mechanics maybe something like that it could be worth exploring oh that's a boat man i'm gonna buy a book yeah i think we'll start from lagrangian mechanics i want to try to get a sense of how to get how do you go from lagrange mechanics to hamiltonian mechanics to to then the schrodinger's equation sort of and then then this whole thing with a path integral formalism um and um but again it's so it's it's in in a way the hamiltonian tells you sort of how how the system is evolving and the system can evolve in multiple different ways right so you've got it can evolve you know in these and that right and that's kind of your yeah well in essence that's probably what the the wave function is telling you right it's like i'm either in this state or this like no not either in this state or in this state i'm saying i'm in a linear combination of states um are those equivalent then to these paths i i don't know too many open questions too much open questions that's complicated i i have to sort of tunnel a bit more because otherwise i'm going to get lost but basically i think at least i understood a little bit better the the function of these and then how the mapping between these and the quantum system and then what the what is then the analogy with a quantum circuit in there which is just pick a type graph of these that you specifically pick because you're telling the system how to evolve essentially you're programming the system you're telling the system what actions to to take what to do right how to evolve the system essentially but i'm not it's not entirely clear how the evolution maps to the rules right i think that's what they're trying to explain here is how these rules are mapping so probably i should read that again because they talk about the angles of this turnings and i think that is basically the rule applications right it's the action and oh look at this so this is the path in the performation of quantum mechanics i think i have to re-read that second part a little bit more next time and then maybe i should maybe i should maybe i should just to consider a path in the multi-way system going through some multi-way space to know how much turning to expect in the path we need an effect to integrate the lagrangian density along the path this will give us some form of but it's exactly what the started patting the formulation of quantum mechanics maybe i should just maybe i should just just go there just path integral formulation and see how that relates to schrodinger's equation yeah yeah yeah yeah probably also there's some some oh so there's some nice link that i can just follow here with references and stuff so we can start here awesome oh god that is actually that is actually a big kind of worms already for the next couple of years so far here this is heavy this is heavy okay but i better start to get a sense of these so there's just a formalism right in terms of how to study the the this evolution and i guess there are multiple formalisms the same that you've got multiform multiple formalisms in classical mechanics anyway have a good night |
If you can catch up to where the Wolfram Physics Project is right now who knows... many discovers await us all )))) |
Thanks for show. I watched some YouTube videos by Stephen Wolfram and it helped me with your video today. There was a foot note in this video you may want to study, [110] called Cellular Automata which he explains. Just a beginner but learning. Thanks. |
i really really don't know whether this is gonna work or not uh we'll go to twitter oh here i am here's my face uh so it seems to be working um what are we gonna do today so i kind of i think i've i need to be kind of the uh the more exploratory kind of stuff you know kind of asking questions and i recently i kind of realized that i've been doing too much stuff that it's been more like you know community oriented and i'm a bit selfish so i really wanna i really wanted to try to understand a couple of things that have been bothering me for a long time so like when i started the channel i started with a couple of videos with d-wave systems and then um i keep keep reading and hearing people saying it's it's it's bs it's you know but like it's not really clear to me and i really wanted to kind of um go go deep with it right and i think the the culminating point for me was uh when i um commented on on a chicago quantum post recently uh let's see where was that i don't know if reddit is a good chat room for quantum computing to be honest was it these oh yeah this is quantum check request and so the the so so the quantum detector said like that's bs um the reason is like i found the tweet funny that we're working to reach thousands of subscribers in 2020. this is kind of our channel um uh what did i say yeah yeah yeah so they basically say um you know you can keep up with the research and the progress in their financial portfolio optimization including we including with quantum annealing computers um so there's kind of two things that i'm not fully um understanding here right so let me just open some notes maybe so i can keep track of these so one is basically what is really oh what what is really quantum any link right i hope you guys can read these so this is something that like i know more or less i remember but i'm not like sure either is this it is this is this way the bs is so this is kind of a question another question would be is portfolio optimization oh what am i doing sorry is portfolio optimization with uh with the with with quantum computer with the quantum computer at all like right um or is it like or is it just d-wave i mean there's a lot of questions that i want to answer and i should probably keep those uh because so there's the medium article which i'll uh okay i'm recording just a tab so i'll just open it up in a second and then there's there um there's an actual paper called portfolio optimization of 40 stocks using a d-wave quantum annealer maybe i should start here with the paper um because i when i ask for these it's like hey um you know where where are they comments where are their comments like did they comment did they where are their comments man i i asked oh yeah here exactly so i i just hate i just hate twitter threads uh it's not really um sometimes they're not linear and that kind of bothers me a lot because i i tend to mis uh i tend to miss basically um some of the replies so um they say you know uh our first portfolio results favor our three custom algorithms simulated an alert genetic algorithm and fat-tailed monte carlo when lucky in our tweet yesterday our quantum answers underperform the 60 asset benchmark i don't know what this means though the underperform doesn't mean there doesn't mean that it's actually working or so maybe i should just you know go ahead and and try to read these first the actual paper i'll put it now in the screen in the tab that i'm uh that i'm like basically okay so yeah that's probably the best thing that i could do with the time that i have today um trying to get an understanding of this but that's my goal and i'm i really want to try to spend some time on these um as i wanna as i said wanna uh really kind of get to understand uh oh sorry you guys didn't see what is there a way to have like notes on the tab or something i should switch to actually recording the full screen maybe i just wanted didn't want to pause process the the the video afterwards and so um maybe anyways i i wrote down the questions what is really quantum annealing um is there where the bs is and and is portfolio optimization uh with quant with a quantum computer like bs or not like um and what it what does it mean for these cpps at all i mean it's like yeah you can do things right like it doesn't mean like if you're not aiming at getting something more just for some reason more efficiently than a classical algorithm you just want to do it because maybe you know it's not about efficiency but it's about like the the actual end result um and i fear that's kind of where the portfolio transition stuff goes it's less about efficiency um it's less about efficiency i mean based on my experience in this whole thing right now it's really really hard to justify that a quantum algorithm or to find sort of quantum advantages really in terms of speed and so when it comes to when it comes to portfolio optimization i guess what you're looking for is you're looking for better models so you're looking for something that works better than the classical models maybe because of the entanglement maybe because of i don't know um so like if this is where the bs is like i'm fine with it right like i i'm i'm all in for uh trying to understand uh or trying to you know play with things that might be just different in nature um but if the ps is is you know that that just doesn't work then let's figure this out okay so let me open the the paper so i hope you can see this portfolio um okay so this is july 6 2020 provide the position of 40 stocks is in d-waves quantum annealer we've got introduction validity of the formulation classical methods and using an annealing kind of computer and hopefully we can reproduce these i really want to go and reproduce these like um introduction because i think they claim that they claim that you know it's everything reproducible so i i definitely want to do these um although i might need access to the wave uh which uh to be honest i don't but like let's see how far how far can we get introduction blah blah blah validity of the formulation classical methods uh brute force genetic algorithm random sampling heuristic approach simulator as a monte carlo using an annealing quantum computer the optimal portfolio developing the cubo to number of assets in the portfolio um a fine transformations of the cube i guess this gets into the mathematics visualization matrix okay that can let's see okay so a discussion explosion uh so what can we see here can we see completion time by method so they are claiming this is faster or at least the the seems like this the completion time by method is is smaller i don't know what c q a c q n s by method r versus quantum annealer computations quantum and classical so the quantum [Music] lower is better okay so the classical is better okay let's start from the beginning maybe let's get let's get to the bases okay uh yeah okay so um the abstract we investigate the use of quantum computers for building a portfolio out of a universe of u.s listed uh liquid equities that contains an optimal set of stocks uh okay so the goal is to build a portfolio so you're picking a portfolio you're picking basically out of a full of assets you're picking a portfolio that it's optimal i guess in terms of maybe some kind of return schema or something i'm not i'm not an expert in in in in um in the stock market so i'm gonna i might have to learn something extra here and there starting from historical market data we look at various problem formulations on the d-wave system so starting from historical market data we look at various problem formulations on the d-wave system here after called d-wave to find the optimal risk versus return portfolio an optimized portfolio based on the markovic's formulation and the sharp sharpie sharp ratio a simplified chicago quantum ratio a cqr is a chicago cornridge and then a new chicago quantum net score ckns approach this first classically and then by our new method on v-wave our results show that practitioners can use d-wave to select attractive portfolios out of 40 us liquid entities okay so the goal i guess if you've got like 40 entities and then you're going to pick a portfolio of some i don't know maybe you know six or five or whatever out of these that it gives you an optimal uh return versus risk kind of um what i would kind of so the it seems like they're simplifying that this is a bit you know this i might be cautious when i see this right a simplified chicago quantum ratio so they're they're simplifying something based on a i don't know what the what this is mark of its formulation uh i'm gonna open another topic because i'm not gonna see this but i'll if i find something useful sorry portfolio theory okay so there's a hall there's a whole it's a whole like portfolio theory and at least name risk and expected return okay so there's basically some mathematical equations here let me show you these guys so this is basically uh okay so we should probably read through these as well later try to understand whether this has any influence in the results but like okay so you're picking they're picking model they're simplifying it it's the and then and then they run it both classically and um good um and quantum uh so the challenge we approach in financial portfolio is to maximize expected returns while minimizing variability of expected returns or uh risk so the defined risk as the variability of the expected returns this is a buy and halt strategy and not a meet or high frequency trading strategy it relies on previous period risk and in our case one year of daily adjusted close data and the underlying variability and relationship of equities we believe investors can improve their chances by selecting the right combination of stocks among the major challenges in financial portfolio is how does an investor balance long-term investment inverse between expected return and volatility and this is why we tackle this question for a variety of methods this problem is particularly well suited for an annealing solution either classically classical simulated thermal annealing or quantum annealing since we wish to consider n equities in which each equity may be included in a portfolio or not so this yields exactly two to the power of n possibilities okay so uh for a potential list of equities as small as 40 this becomes nearly invisible in a work say on a workstation when we approach the entity of the s p 500 uh we very quickly run into solution space which is computationally infinite that is we do not have enough member in the observable universe to run through a bridge for a solution this work is structured as follows okay so there's some exploration with a sharpie sharp ratio where that is the range of covariance of a portfolio with the market over blah blah blah okay so this is then details on the actual model um we'll come back to these in a second from here with developer chicago quantum reacher okay and it's the covariance of the i've asset against the entire market this is a slight improvement over the sharpie ratio in terms of computation as we need not consider nominal assets okay so they so these index simplifies the computation that's what they're claiming uh we can also formulate security matrix form we explore this form okay so they go ahead and do a matrix form of cqr um which i guess may be applied but that's why you took 8 model-ish like applied to a state or or i don't know what they want to do with the matrix we explore this formulations by a variety of classical methods this hybrid causes some different set of mathematical problems in formulating a consistent quadratic form finally we settle on the chicago quantum net score which is given by okay so it seems like they've got some trouble adapting the mathematical formulation to feed into the d-wave stuff and so they adapt it um and then okay we'll dig into this i'm just really going to fly through uh in this first video of validia formulation and its current capacity to use source problems which are from the in terms of analyzing model this our practical challenge is not is to provide from d-wave and an acceptable model on which you may begin its computations consider following image image one voice image one here comparison of cqr and cq ns scores against the sharpie ratio what are you comparing i don't understand that uh our formulation has a propensity toward contoured conservative side in investment terms however it also demonstrates that the pres the present formulations are near the efficient frontier of investment portfolios this from an empirical perspective the solution passes master we develop the method in more detail okay so this is basically what they're saying is the method makes sense but uh we can dig into this later as well then classical methods so it does seem like it it does seem like uh they're trying to use indeed the wave system here is for um uh tractability like so they say for 40 portfolios we can you can you cannot just brute force these because you've got so i guess i'm guessing the idea here would be you go through each possibility you calculate whatever ratio you're using in your model and then you just you know sort those and pick the top five right that is of course intractable with 40 because you have 2 to the power of 40 possibilities so you can't you cannot just force it um and the claim here is that the wave system can can't do these because yeah uh i gotta pause for a second i'll be back in a second you will notice difference okay so i'm back uh you probably didn't know this but i just um yeah uh took some time off uh it's a couple of things to take care of but anyway um better this way maybe so what i was going to say it seems like it seems like the issue here is um or they're trying to use the airport to uh sorry the airport i'm at the airport but they're trying to use the t wave system uh to uh basically be able to actually do something right because the 40 portfolio the proof force is not doable so here's a performance discussion the smaller asset universe are able to do to loop through the all binary solutions and then perforce roughly for the assets um that's not possible this generic algorithm uh okay it gets to a local minimum deeper than our monte carlo method with 950 million samples and does so very quickly our difficulties in tuning the parameters for number of evolutionary steps probability of leds in in size of initial population even with essential random guesses and these parameters are generating which is a low enough energy level so that uh we can determine whether the quantum annealing social media so we can determine whether the quantum annealing solutions are legitimate okay so here they're what they're saying is that with the generic algorithm they can at least get something comparable like they can they can compare with the two web system uh output to at least uh check whether the out the the the the system is not like bsing or something random sampling as mentioned earlier slightly more than a workstation camera without any place algorithm just randomly sample as much as we can um we're able to sample roughly to the power of 29 portfolios of potential to the power 40 this means most of our effort is spent around portfolios of size 40 plus minus square root of 40 percentage wise this doesn't cover much of the entire spectrum uh but we approach point four percent of the meat science portfolio okay i don't know okay heuristic approach but what's the point of using the d-wave system then original test of our problem comes in the form of simulated thermal learning solution using statistics of random mattresses we're able to tune in the parameters of our simulated kneeling solution to deliver very deep local minima additionally this style solution only covers minimizing grease skin portfolio based on the cooling rate of the thermal annealing okay so using an annealing quantum computer the optimal portfolio in our case is one which maximizes the sharp euratia however as presented by the sharpie richards of portfolio is not computable as a qbo so then they have to basically um the main thrust of this research is in fact okay so the aha okay so the challenge is to formulate it in a way that the d-wave system can eat it um blah blah blah developing the qbo we'll get into the details later on i'm just flying over things right now um because the point is like if if they have to do some simplifications we'll see the results and i'm definitely i'm really we're really going to replicate that embedding scaling and hardware considerations [Music] inspector shows how the system encodes and embeds assets into physical key weights an attempt at changing the formulation by manually embedding terms to respect the reordering of assets does not yield substantial improvements um ethane transformations of the qbo qubo and export profilers of different sizes will present a different matrix to doa for each desired size of portfolio add a penalty for exploring portfolios different sciences one obtaining accurate values on example the addition for these follows closely from converting a cubio into a nicer model this transformation blah blah blah okay so this these are more technicalities um ah i think we're gonna sneeze no visualization initial landscape is critical and learning how the wave finds a solution it also is our understanding of how matrix transformations adjust the landscape to improve the probability of something correct as the values matrix c q n is calculated if you place a penalty on the smaller portfolios by adding the sheath subtracting the shift turn places the penalty for larger portfolios um okay results are explaining the waters as follows download one year of daily market data for a specific date a set of n assets in the universes current as of that moment halt data for all experiments calculate covariance of each asset with the market say how much this varies with the market in respect to the market based on log returns calculate covariance terms between assets calculating they're lying in summary values including sharpie ratio and chicago point in score for an all asset portfolio okay so kind of hold all the 40 assets okay so what they do is they okay collect okay so that would be the yeah because that's probably the word i mean that's definitely the worst you can get like just pick out pick everything right and you want to pick a smaller portfolio um that maximizes or minimizes probably the the ratio the right the cubia for each portfolio size portfolio size this i don't get um this part i don't get uh but it's because i didn't get into the keyboard representation part but run a classical progress scale with me in our case a generic algorithm to see one best portfolio and its values okay so run basically one of the classic alternatives so then we can compare it with the d-wave results execute like this an appropriate range of portfolio sizes usually result to see the generic encryption compare values classical so they use the d-wave to seat the generic algorithm okay these ii so they are not using the d-wave system to choose the portfolio to choose the answers but to inform the generic algorithm and genetic algorithm i'll have to go into details i guess the following figures 34 give some idea of how well the quantum computer performs using the ckns against the sharp eurasia we see that in the sample that have purchased the efficient frontier in a few cases highest return for that level of risk most points chief one will also miss the baby shouting uh we see classical random sampling on average for all portfolios which is where most of our portfolios where run this shows that d-wave is not picking randomly or average solutions but good ones life classical expected to understand the deviation i don't know if you can hear this as a baby in the background discussion and collagen positive semi-definite considerations practitioners of numpy will know well that numpy is prone to running errors in particular we find that numpy computes covariance matrices with slightly negative eigenvalues what [Music] okay where's the codes so let's find the code next steps references thank you where's the code maybe in the medium article so [Music] please see our archive article uh here on youtube you can play this here there is our maybe if there's no code i just should try to do this myself so kind of based on the paper go back to the paper um what about it what if i just search for these code say anything github maybe researchgate [Music] maybe in the webpage oh that hurt so portfolio contact team blockoff and security use cases more car logging block offerings during use cases it's okay maybe maybe there's something no uh do you wave for your portfolio and it's just one red hole quantum markov is a dui sample code could this be there could this be that who's this user so data averages covariance prices say anything in the code that kobe it's embedding composite okay could that be the code who's this person i don't know if this is the course i don't know whether this is these perforated wave sorry i've been checking i was checking other tabs i i apologize for this i started the video like that and i can't i can't change the um i guess i guess we maybe can just start working on these to be honest so is there enough detail i guess this basically so here there's the there's the formulation right so they they say we can reformulate cqrs in matrix form explore these relationships around classical methods do you wave requires a linear quadratic form we attempt to rectify this by exploring the following which i don't understand and i mean i really don't understand but i don't understand one thing so it so so what what are they calculating with the what are they calculating with uh with a d-wave computing computer so rw is a weighted portfolio and alpha is a real number and most experiments we choose an equal weighting uh brute force genetic algorithm our general consumption gets a lot coming in deeper than carla method random sampling heuristic approach one second yeah i running out of time but i tried this that days don't understand where they are calculating with the which means the sharky rage however shutter is not computable so trying to throw sensors in fact how to formulate a qbo which when presented to d wave pretty similar to the classical sharpie so they're calculating the actual ratio or the numerator can be expressed as a simple dot product where we expect the return and the relative weight of the ice acid respectively the denominator okay so the d wave computer is supposed to give a weighted like an answer of like the weight of each of the portfolios that minimizes that ratio and still they use these as a starting point for the generic algorithm later maybe keyword two number of assets in the portfolio this is what i don't really get we're dealing with a single asset portfolio we only consider linear terms in the keyboard and transfer matrix i think i have to so do you wave i think i have to go through this first maybe try to get something there up and running that's what i'll that's what i'll do in the next uh in the next video stanford quadratic and constraint binary optimization is from the science restaurant falls there the key problem is defined isn't an upper diagonal matrix it's going to express more consensus okay all right terms could write a fishing variable but what's the what's the way that do you wave define an objective function objective value q1 i guess q0 q1 being the qubits and then so i'm guessing what they're doing is kind of define that rate as the function and then as a function of like different portfolios and then try to kind of get the um okay so each qubit will be an asset probably i guess that's the point and then you want to have the 40 assets represented and you want to get an answer of uh what's the probability associated with each asset i guess that's what i guess that's what you want to do but i don't get like one thing that i'm missing here is how does these then fit into the um genetic algorithm genetic algorithm it gets to a local minimum deeper than our monte carlo and that's so very quickly our difficulties including the parameters three number of evolutionary stats for ability of altitude and science and potential population it was essentially random guesses at the parameters which is learning to learn developing the keyboard i'm betting scanning hard considerations um because i don't get that step like i i don't get that like uh around a classical so execute the d-wave using a proper range of portfolio sizes and use the d-wave to see the generic algorithm this is what's not clear like i get maybe the d-wave is used in here to work work around the brute force or the the the the fact that you can't just brute force the thing but that doesn't still give you an optimal yeah i don't know i i it feels like there's some pieces missing here so i need to find i need to find more about these in terms of what are the other like is there any reference code anywhere that i can to take a look at otherwise like i i can what i will do is i'll try to implement the dwa stuff they talk about in here uh and then but i really don't know what the genetic algorithm is that they're using and how does these uh so we further give results of the cqns by method once one will notice that the waveforms are in fact depending better is awesome with the current methods but unperformation i got with i'm using dwf as a seat guarantees that you know from better okay so that's the reason as we disallow anything worse than the c to propagate through generations of solutions uh okay yeah so it's basically okay so you create an initial population based on what d-wave tells you and then you you tune the actual uh you you tune the actual uh genetic algorithm not to go worse than these um and so you know you can only get better okay yeah i i don't know i don't even know where whether you can call this bs or not ps or like where is the problem in here i might be like to be fair it's like i don't know where um i really don't know what is there to evaluate at all it's like it feels like they're using the d-wave system just to get a better first gas and and it it's it's maybe less about efficiency but it's about the quality of the gas right if it's about the quality i don't i don't think that's bs but like if it's about efficiency yeah that's another that's another level it's a discussion at another level i guess but i don't know let's try to implement this okay and then we'll make we'll get conclusions at the end of everything so we'll see yep gotta go bye-bye |
please take a look on quantum money and semi quantum money i can share you some papers, Happy to do so if you share some papers ;) |
Hi, I am Jeff from Chicago Quantum. |
for exercise three I truly don't remember what makes an operator linear was the mathematical definition of this so I I guess just look it up and then I mean this is between known functions so I guess it should be easy to figure out for exercise 4 is kind of just a basically demonstration exercise I guess you find Properties or relate with the commutative definition is which I also don't remember um but I guess that's what the things are that there should be fairly easy one same with five um and then six is just following I guess a recipe to solve the and I got stuck last time that's a differential equation I guess that'd be easy as well with number seven against the trick here is just apply the operator to that function and see what happens so I think that's already the first part and then finding diagram values is just finding how that change |
and here we here we are again at the Yellow Submarine project so it's taken a bit of a turn but I'm happy basically just to catch up with the previous videos I kind of opened up the box of of CV quantum computation which I didn't expect and sometimes I learn a little bit more about that before I dive into the code here and I will try to run that myself to sort of break down a bit the the project so here in the github rapport you have like the actual solver class and then exactly so we went through that in the first video and I was basically so Josh they basically reply to one of my tweets saying that there is a part of documentation where you can see some neat gives let's see let's see how that looks like so okay cv quantum gate visualizations color endings so basically okay so you see kind of what the gates are doing that's crazy okay hmm so in a conventional quantum circuit qubits represented by wires are operated on by a quantum gate but quantum gates which collectively perform computations similarly continuous variable quantum computing quantum computing uses cue modes that represent bundles of interacting photons okay bundles and interesting photons to perform competitions and cue modes will leverage Gaussian and no Gaussian gates casa members and gates can be described with seen the face space this space is shown by the position and the momentum ax axis ok so this is what I was reading in the appendix of the paper I think it's this paper here that could be it cuz yeah exactly so stacking a look at this and I was basically that's a pretty good summary um okay so so now okay so it's about the phase space whatever that means but maybe I mean I didn't a is quantum mechanics right so I think talking about the phase this is something that it's pretty sort of at the technical level at the physical level but it still makes sense and then this space is shown by the position and momentum axis so I'm guessing that's we're gonna see in those visualizations position and somehow [Music] Gaussian get such as the squeezing and rotation gates act linearly involved okay squeezing and rotation these gates can only reach positive quasi probability distributions and can be classically simulated okay on the other hand non Gaussian gate such as the Kerr gate and cubic phase gate acting on linearly this property allows them to be in a negative quasi probability distributions in negative quasi-probabilities versions whatever that means mmm and not be classically simulated in this notebook we'll learn about various single mode Gaussian and no Gaussian gates and apply them to a state using Strawberry Fields mm-hmm vacuum state the vacuum state is the lowest energy Gaussian state it has no displacement or squeezing in the face space here we learn how to create a vacuum state in a quantum circuit with one cue mode okay so this is more like initializing stuff I mean I'm gonna skip that for the moment this interesting sort of like an engine like a Gaussian engine okay should we know the default standing story field is the vacuum state above cake is listed to show the operations surface there are three main backends that can be useful okay no but maybe it's interesting so three main backends the back end selector will depend on the task you want to perform we've used the Gaussian back end to run our vacuum state circuit we can ultimate will use the back in to run the circuit the third bucket is a tensor flow packing DF it leverages tensor flow to create quantum machine learning models using matplotlib we then plot the fingering function to the vacuum state which is the gas which is a Gaussian distribution the Wigner function quasi-probability illustrates negative and positive regions in phase space will notice below the gaussian states such as the vacuum and squeezed state are in the positive regions of the face base while mungus and sis can reach negative regions of the face space the y-axis on this plot can describe the peak quadrature or momentum operator and the x-axis describes the X quadrature of position or position operator so Y axis is momentum and x-axis is quadrature our stat axis we can make a couple of observations from the gas in the solution above so this is the Wigner whatever that is which basically takes your state into a Gaussian distribution some what I'm I'm having that feeling that this is pretty much going to be about creating kind of kind of probability a la probability distributions sort of like similar to I mean it's not it's not it's not what a d-wave people do because this is more that's more like the energy function saying and but I wonder how I can do a quick search I wonder was this anything here regarding entanglement and superpositions and stuff like that or is it really just all about exploiting this stuff um this the y-axis on this plot can describe the peak of Sherman in prayer and the x axis the X quadrature okay so so here we've got the Wigner Wigner vikner Wigner whatever and then so this is basically the like a gaussian so it can make a couple notice that it's positive quasi-probability everywhere later I want is quasi-probability across the probabilities which is a mathematical object similar to a privatization but which relaxes some of blah blah blah axioms of probability theorem okay into this yeah whatever it's not like sitting it's not in theory like a purely pure like it's probably sufficient okay later on we'll have a look at the non-gaussian case have regions of negative quasi-probability the vacuum state is also centered at zero and both in the position axis and the momentum axis on both okay okay squeezing gate this quiz and gave can also be thought of as a transformation on the position and momentum axes the squeezing gate scales x these and scale speed to this and the ring a function we can see the proper distribution being lengthened on the momentum axis and being pinched in the position access through that's why it's called squeezing squeezing mmm-hmm rotation gate the rotation gate shifts the X quadrature to X cosine whenever you're - blah blah blah blah blah in other words the gate simply rotates the face face just rotates that distribution the displacement displacement game okay mm-hmm cubic face okay so this is pre so those those animations are definitely definitely helpful definitely helpful displacement Gaede the displacement gate has a specified complex value alpha it shifts X by a value proportional to the real component of alpha and P by the in the complex component of alpha interesting I don't see what's the difference as alpha increases in value the status is placed more to the right [Music] cubic face gave the cubic face game shifts momentum by this finable parameter but does not perform that does not transform X in the so the position in the face space below okay here we can see that the vacuum state has been shifted to a state where it reaches negative quasi-probability where's the negative because this goes up we use a font backing and find the argument kind of theme the carer gate before we understand the car Gail operates we should first take another look at the rotation getting the basis the rotation gate acts with these in comparison in comparison the current gate transforms the care gates transforms with these the square values what makes the state non Gaussian in this term a transition transforms the shape of the state significantly we can see it's coherent structure morphing after the car gate is applied at different strengths for more information about safety gates the city gate section is representation that's pretty cool honestly really it's just um so that's what that's what those things are doing so basically the engine Gaussian engine see okay so I kind of I kind of see that so you've got those two things and you've got like fancy transformations we've got momentum and position and then you've got those fancy transformations in a way that like I guess what it means here is the difference is that it's really continuous whereas in kind of the traditional quantum computing you it's not like continuous exactly and what I wanted to do which I think I'm running out of time for the videos to go and take a look at some of the example circuits but that was me that's been a helpful this very helpful that's been helpful one so let me check sorry no such results kay teleportation minimize mnemonic relations kind of gets into the state limitation Oh car moccasin so we're gonna take a little we can start taking a look at this actually why not five minutes ten minutes maybe so there is indeed some sort of entanglement being Houston here so Connor dollar petition sometimes referred to as state the repetition to avoid confusion with the gate dealer petition is a reliable transfer of an unknown quantum state across spatially separated cubits or key modes through the use of a classical transmission channel and quantum entanglement consider fundamental quantum information protocol as it has applications ranging from blah blah blah we know all this in general all connotation circuits work on the same basic principle to distant observers Alice and Bob share a maximum economists aid in discrete variables any one of the four Bell States or in CB on computing a maximally entangled state for a fixed energy and have access to a classical condition channel okay L is in possession of an unknown state which he wishes to transport to Bob makes a joint measurement of the unknown state and her half of the entangled stain by projecting onto the bail basis by transmitting the results of her measurement to Bob Bob is then able to informed transform his health I was like I said to an accurate replica the original unknown stay by performing a conditional face Philippe or this or displacement for chemos see the implementation while originally designed for discrete variable chronic predation with qubits especially separated quantum to be described above can be easily translated to cpq modes the result is shown in the following circuit okay so you've got whatever bsp q zero one two measurements but uh but here it seems like so there's some differences i I mean I don't know I'm guessing but maybe this is a measurement of the position and a measurement of the momentum may be could it be and then based on these an X and a septate I performed yeah but you're cheating Lucas's are not those are not are those CB gates or one this process can be explained as follows here Q modes q1 and q2 are initially prepared us the unphysical infinitely squeezed vacuum state in the momentum and position space respectively so this is this is the vacuum stayed [Music] before being maximally entangled by a 50/50 okay so the beam splitter is what does the entanglement seems like so when I check the beam splitter this is how you do entanglement in CB quantum computing okay these two chemos are now especially especially especially separated squeezed vacuum state with q1 held by alleys and q2 held by Bob with the two connected via the classical communication channel C 1 C 0 and C 1 to teleport her known stayed to Bob Allison upper friends a projective measurement of her entire system onto the maximally entangled basis States so she entangles this with q1 with another beam splitter before performing two homodyne measurements we'll have to check what that means again in the X and the P quadratures respectively okay so yeah as I said measuring position and momentum the results of these measurements are then transmitted to Bob Hope reference for the position both a position displacement conditional on the X measurement okay so this is a this is a displacement and this is a momentum displacement yes it seems like I'm gonna dive into this more detail the next video I don't wanna I don't wanna okay I want to compare that calmly I don't want to make the video too long it's been definitely helpful I this seems to me like it's base a similar concept in the sense that you entangle staff and you just then apply gates based on them on the world on how you on what you measure right and then so in this case is both a position displacement and a momentum displacement because this is also the way you define the commodes as having the those two those two components or variables this momentum and which in which in inclined instead of discrete quantum computing you would do a like a flip or you would so you you know you would do a flip or you'd basically the flipping like you change the amplitude totally from the zero to the one side or you would do a face a face change so to say I think so I think that's really what's happening here okay but it starts to make a bit of sense let me do let me do another I'll do another the next step I'll really do it was a blackbird so some important notes it's definitely again it's definitely badly chosen name bsk um let's the planes beam splitter okay so we'll take a look at beam splitting and then we'll take a look at and we'll make a comparison between these and the other the other ones and then maybe we'll take a look at other simulation okay cool perfect |
and this is what we're going to be doing uh let me see if i can open the browser i did yesterday have a bit of a quick um session just off stream where i merged a pull request that um was from you know jordan who uh basically really you know really liked the project and kind of wanted to contribute a little bit here and there so and what he did especially we separated um he separated all the visualization stuff into it into a separate module um there's a quirk export but needs to be this needs to be reworked because this was still under the assumption of we have gates as mattresses and and so you have a gate that's applied to certain qubits and you can you know the idea is you can record the application of gates and then because quirk actually you know will require you to keep the the the to know what the circuit is not just the state right remember that all everything else that we do in terms of printing stuff is with printing a concrete state we're not printing a set of gates so um so what this means basically is that now we have this file so we changed a bit the way let's see a new terminal well i finally ended up i finally ended up so i finally ended up um putting the actual gates in here and you know there's two things i want to do next one is work on the rules and actually think about you know um so we have rules how do we you know how do we do everything else that you usually can do with a with a quantum circuit building library like circle or kiskeep like at least we're missing parametrized rules right and then i also want to build this type of rules that are a bit more um you know sort of symbolic rules right like that that they go and match and assign values to variables and then you can do things like um entangling operations between registers like a plus a or a plus b you know like add add the value of the top resistor to the bottom register and stuff like this um so we'll have to what i thought i'd do is i just literally actually have half classes like i just i just have i think the best thing to do here is to have basically [Music] a um sort of the the operation class let me share in twitter so i wish i wish there should be this would be something more original building param trieste gates or operations so i wish i i wish i kind of had the um i don't know what i wish i don't know what i what i just said um okay this is definitely not the browser that i won i won the other browser there you go we'll actually open jupyter and go to notebooks and go to testing so yeah so basically we now have like we don't have the rules here anymore and we we just like have the x and instead of just call it x sub um as x rule or h rule just have x h and that stuff right so we party from this file from this module and then the visualization is just um on this mod module um that basic visualization here that we had and uh yeah i know i i know i started working on the factory but i feel like that's something i might just well just do off stream because it's a bit more boring um and then you know kind of testing stuff is a bit more boring and i thought you know i kind of had a will today to do something a bit bit more challenging or more you know kind of design oriented right thinking about how you would do that um because there are two things that you know one could think of which is in circ for example you have the parameterized case you use simpy well and circ actually distinguishes data and operations by by by you know calling gate the actual definition and the operation is you know as the the is the actual gate applied to a specific qubit um named cupid symbol no nothing about symbols common gates um okay just literally as an argument actually okay rotations because i might as well just use the same the same uh okay so there's exponents right and that this rate this this the the angles the two things that we should have in there so probably probably what this means is that we should have a um maybe i shouldn't call these specifically gates or just have it defined here because here we have the hypograph yeah probably have the defined in here as well so kind of have so here we have the node class and then we can have also basically the class um do i call it gate i i'm i'm a bit hesitant to use the word gabe because it's got this very specific you know um qubit level kind of thing but if i call it operation it can be it can be i can call it operator but that's also a bit overloading you know stuff like that there's this the operator concept in in physics and quantum mechanics um yeah i'll go with gate but i i i might really i might really change these um and so the initialization of the gate so what is the gate gonna have right um so the gate is gonna have basically the gates going to have a um like a unique identifier right which is i'd say these here which is going to be internal as well make sure that you know one can refer i guess so just for serialization maybe exporting and stuff like this i think that's just good practice so so the the unique identifier will be will be the over here and then what what you also want to have is yeah you're going to have basically um the the rules will be defined and given by the gate right so it'll be like a function that just says uh okay rules and that basically um or just initialized i'm not so sure what's the best practice there it's like what's the point in what's the point in making it a uh sort of a getter instead of just you know [Music] okay because then it's not really um this is basic inheritance and stuff like that right so i i i what i want to do is then i want to define you know i want to define a um a class you know x yeah do i want to do this no i don't want to do this i definitely don't want to do these um or how is it done what's the design pattern that they use in in in circ like is i think circuit hop [Music] so for example from cirque ops okay so these things are defined like that circobs so they are in operations defined okay it's identity okay so it control gates tests the weight gate arithmetic operation common gates so what do they do they no they actually classes okay so they're actually classes and then they have apply unitary act on so they they it kind of makes sense because they've controlled face exponent so they actually you know it's like yeah so they say see they define it as a they did they define these things as classes and then there's probably a gate features single qubit gates of this like stuff they inherit from from these generic gates right if i uh the stuff they inherit from these generic gates so actually what i should be doing is i should be building it you know sort of bottoms up so not not having it here but actually really you know really defining the gates here right and say um so really kind of have really kind of have the you know hmm the x i'm thinking how do you even name this the x-gate x-power gate you know kind of then the rules the the the the rules would be this one so they can't be changed that's probably the best you know so this would be like you know um rules this would be return these right so that returns the uh that returns the uh the rules for the x gate um actually just just like that just the rules so return the rules um there's no no uid that needs to be passed in here that that makes no sense i think but it does make sense is how do we implement the mapping right so how do we implement so here's a the rewrite thing right so this rewrite stuff is basically you know um it's got an input map so what we can say is the input map actually you know the input map actually is um it's given to the gate i like the design pattern i like um i like the fact that you can uh where is that uh where's the world it's just pilot gates this is ivan gate by the way how do i implement tests identity pauli pauli common gates controlled operation dense pali it's just these palligates there you go so so you've got for example um class paulie ride blah blah blah and then they have the on right which is which is like this is the mapping right so returns an application of this gate to the given keywords what what does this do it returns single qubit police string gate operation with with cubits zero um yeah i just it just yeah the mapping is stored right so that's that's what it does the mapping is stored and um um and then but then how is the circuit field so how is the how's the application then run like i don't know um i don't know what the what are the advantages because then you have the circuit and then in the circuit you you basically add operations that i don't that that operate on cubiets so a circuit has a set of operations that know where to go um and here i'm you know the the the concept of the hypergraph thing here is to be to just be able to work at the operation level right not to this is not about like building a circuit and letting it run and see what happens it's about like you know i i apply transformation i want to see what the result is and i do this kind of like rather on the fly it's not about running shots and whatnot so i'm not so sure these way of working with you would would feel right like i'm i'm you know i'm working with a hyper graph and i used to say hey i want to apply the escape to these kiwis and so it's not that i want to create i think i think that stays probably like these so you know um that's still why i'm a bit hesitant to implement it like to implement this whole thing really as a class i'd love i'd love to just have i'd love to just have it flat like plain you know it's uh not so sure i'm not so sure about the class thing like i'm not really not so sure about the class thing you know like if i have the x-gate in here so i i i so so this is so this is the this is the it's sort of the the rule object or the object that defines the x-gate it's it's got a gate it's got a gate name call it name okay so i'll call it name name and i think where was that used um yeah here only so um call it name but then i think it just should have like like a params but yeah just simply like a perms thing here where you kind of you know define you define a you just defined a list of parameters and it should be sorry it should be no no it should probably be just um just an exponent right an exponent which is just one so this will be a value from zero to one and um and then it can also have a an angle for face rotation or just rod which like i don't know if it's something that we just want to have a normalized way of specifying that like also a valid from zero to one how much you want to rotate um yeah because we're not like we're not restricted to using linear algebra stuff so we don't have to now you know do this whole thing it's just but that rotation is also not because so it's a match and replace kind of concept that's how it started right so we have you match a pattern and you replace it with another pattern now these is indeed an actual state so if i wanted to have if i wanted to specify that what i want to do is i want to um [Music] do an exponent or something i think that's definitely not i think it's this is not even needed like these i think this is just something that we have to for example let's do the square let's try to do the square root of x right so so squared of x um gate right so the square root of x what it does is it turns so yeah it turns a zero into into this like half plus half i zero and half minus i and the one i do yeah so it basically it turns a zero into a um half plas have is that j i think j or i don't know how do you define that and um so you're actually you're actually defining these already as part of the rule right so yeah so i think actually there's just no there's just no need to specify these things um yeah so you're saying you're you're doing it like this right that's this workout so oh sorry guys square root of x x so so that like it's it's hardcoded if you want to have a gate that is called you know like um the you know x the x with an exponent right what what we want to what you want to do is you want to have a weight with symbols specify these and say that um yeah this is when sempai is is needed right so and this is when this method is only it's not only gonna have an input map which we can call cubit qubit map a qubit map so input map replace with cubic map but it's also going to have a um which can be empty you shouldn't yeah skates must apply at least to one operation but and then you can have a params map so a map that kind of maps parameters right and what this is going to do is the rule actually should declare some parameters probably if so right yeah so so the rule should say look i've got these patterns which is these parameters which is basically array of symbols right and how the sim pi look like so first of all how do we install zinpai how do we get simpa installed in here so is it just uh senpai just want to install this it's just as just like um peep and stuff like that anaconda keyed other methods uh can i not just install these you know with with peep yeah pip install senpai okay so senpai and we should add this in the requirements right senpai and i don't really know what version 1.7.1 let it be this way so same pie is needed and then basically here we need to import senpai same play as sp and so what we do here is probably we just have to declare symbols right so we have to go and say uh dear tutorial just the basic exercises introduction symbols uh symbols x okay yeah so it's basically so you can just okay symbols x yeah yeah yeah so symbols just symbols right can i do that i can just say and the symbols then we'll just do a you know you basically declare them with symbols right so you say you know look it just you know just one symbol in this case which is um you know the exponent and um could be could be a right sp symbols a and then what we say or what we want to say here is that we were apply a this is an exponent of these right i mean exponents would be easy because it's just apply it's like applying the gate twice right [Music] because now i'm a bit afraid i don't want to get into algebra you know what i mean that's the problem but i mean but i mean i have to like there's no way if i say i want to apply half of the eggs gate right that i um that i that i go and you know even if it's not that even if it's just a rotation let's just do a rotation like i just got to rotate the thing like there's no sense i mean you know actually i mean there is right i could compute or i could you know i could think of in general what this expression is right that you turn um to keep it into but it's not that it's not somehow it's not straightforward like what it what it means right like when what does it mean if you have like a quarter of a rotation a quarter of an x gate well it rotates 45 degrees so you have in linear algebra you know that just maps to a certain matrix then you multiply the stuff and there you go and you obtain what you can see here that says you know transform zero into this and one into that which is ideally what what i'd like to have like intuitively defined with a rule that i can give it a parameter and this parameter is then you know when i'm it's then matched here right um because i can then reduce the rules and i don't have to compute them every time the same another story will be when we um when we deal with the the parameters in the matching side like these symbols won't need to be declared because the symbols are just basically populated on sort of on the match path uh paths so once we're passing too much things that we're populating the symbols per hybrid and all that stuff so but this is uh because this brings me it just brings me to an essential question which is like if i don't rely on linear algebra do i need to rely on the state vector formalism at all i mean even for one cubit right but i got to have a model there at some point and if it's not if it's not if it's not a simple state vector then what it is then then what is it like i don't know i think that's something it's sort of a um it's it's one of these things where i can't i can't go beyond that right so but that's that's definitely a problem with the rule definition which is that um when you when you've got matrix the gates defined and modeled as mattresses it's easy it's easy to think about the square root of a gate to think about the power of a gate um when you have them as rules like yeah you could say apply these these rules three times right but you cannot say apply these rules a quarter of the time um and we're not operating we're just replacing stuff so but anyway you would so what you would be doing here right is if your match is zero cat then you would do something with it right so you would still so you would still take a and not multiply like i don't know just multiply just to say some stupid thing right just operate that with zero right like you're you know you you could think of you know just doing something like these which it just makes no sense right but like you have that you have that basically symbolic expression then then because you're um you're matching the symbols then you can do that but it's a symbol are symbols needed for this at all are symbols neither like these at all um because i could just as well i mean definitely symbols i needed for that for the sort of the more arithmetic kind of stuff but like here i could just well define um a list of keys basically a list of keys you know like i could just say so this is at least this is basically a dictionary where you know like you've got something like this by default right and so what you will have here is you'll just have like params yeah but that's not something you can do can you can you just say can you do something like this like does something like this compute at all let's try if i if i just you know initialize an object and i call it like you know um result basically none haram parm basically you know um not really so part basically non and then result is and then result is basically self dot self param times two right and if i now say a param equals two and then i say print probably that doesn't work self is not defined yeah oh yeah of course not but can i just do like this that's not defined because i'm defining it so that doesn't work that's this kind of stuff doesn't work um view edit delete cells that doesn't work so we'll have to keep this keep the symbols i think that that's what makes sense we have a symbol declared here and then we have a symbol that is used in here right um but also i'm not so sure i'm not so sure if this is something you can do either hmm because that might that might as well just be the reason that kind of requires me to have a class to have these um properly implemented because and it's defined here right so if i import sim senpai as sp and i do a clear output thing here can i do something like this let's try come on oh god something's wrong here in terms of what rules yeah the output ah this is also not gonna work but let's just say replace with zero reverse with a okay so just i'm just testing um if this way of defining rules works that i define the self itself contain so within the object i define the the parameters and symbols and then i use them in the rules whether that is recognized at all sp is not defined it's empire's sp oh oops oops oops does it work no name a is not defined okay so it doesn't work um so i'll need to go with objects i'll need to go with objects um and i'll need to go with objects and then even the rules will somewhat be different right because i cannot declare symbols so i have to declare the symbols basically upfront i have to declare the symbols a bit upfront it's just this is really just technical detail right like how to implement that but so out of all these i yeah i will have to go with i will have to actually go with uh with a class and then well you know it's like i don't really have to but i have to because then it's more self-contained otherwise i gotta define symbols for everything in here make sure there's no clashes and as far as i'd like to keep things sort of flat and and simple in terms of you know flexible and just have like sets of rules and stuff like that i think i need to go with the um with a class and then um this class is gonna have a cube maps have primaries map and um it's gonna have rules so we will just pass the um so we will just basically pass the operation here which kind of this can be you know the way that the operation is initialized right so yeah cool it doesn't feel super productive but at least it's bit of a kind of from a design perspective um some some more clarity awesome we'll leave it here see you in the next stream |
next I want to talk to you a bit more about super positions and the whole purpose of this play list is not just to give you an introduction without the maths but also it's really the actual goal is really to kind of give you a tool set so you can then you know go ahead with your own pace and your own learning and this is not intended to be significant hero material or whatever but it's more like you know a list of things that you should know a list of things they should be you should keep in mind and maybe that gives you the toolset but then later on you know read read quantum circuits that you find in papers or in articles sort of get a bit more you know be able to get a bit more hands-on and break down things by yourself and then kind of learn further so I think this is an important topic so this our arithmetic circuit let me just clean this and let's talk a little bit about superpositions what do we know already about superpositions and there might be more to it but this is so far what I know right so you've got here five cubits so with you can get into super position by using harem arcades you didn't have to use a har market for all the qubits you just get smaller super positions but basically the idea is you get you you get equal probability of measuring measuring those that that particular state now an important thing to consider to keep in mind is that maybe it's gonna be easier if I reduce if I reduce the circuit district abuse to explain what I want to explain right now so so here you've got like eight possible eight possible values if you represent with three beats right so um because we are in an equal superposition in the sense um what's important to note though is that so far we've been talking about the probabilities of measuring things but really what happens if you just measure 1q be like how does this affect the other qubits right because that's really not intuitive like it's really not intuitively you don't really from a classical perspective here like yeah sure you measure one the other to stay in an equal superposition I guess which in this case happens to be true I think yeah because if it's an equal because you have an equal superposition of everything rightness is an impossible position but um but maybe let's do this so let me explain you this with an equal position and then with an on with a not you know like not a full superposition so you will see the difference the thing is now if you measure one or a zero basically constrains the space of probabilities or the space of possible solutions you have for these other two qubits correct because if you measure that's here means it's gonna write into the into the bit zero of this classical register here but imagine you measure a one this means that now because you know it's a one the rest of your system right so this is definitely not existing anymore this is not existing anymore I mean let me mark them if you measure one right so this doesn't exist anymore this in exists anymore this is because they they have a zero in their in their lowest beat state and you've just measured the one so you know that the rest of the possible values are the ones that you have left so here's still of course that still means the the you know zero zero zero one was your own one but the trick is partial measurement becomes useful when you it becomes tricky and useful at some time when you don't have that types those types of superposition so for example what if we what if we had something like this I don't know just to give you an example right so now if you measure one here you know that the rest of your state is going to be in a zero one one one type of superposition right so you know what I mean right I mean it's the thing is when you measure part of your system you're literally chained you're literally affecting what are the probabilities of the rest of your system and we talked a little bit about entanglement right but this is also kind of related to the same concept if you've got an entangled state and you measure one qubit being zero then you know the other one is going to be zero right because you know they always agree like that's sort of for one particular type of entanglement so that's the that's the idea and actually you will see that a lot of circuits out there a lot of actually there's a lot of stuff being done also conditionally here this means you can do something like I don't remember how you do that here I think you must add I think you must like for example this you could you can do a conditional operation right as in apply that I'm not I'm not so sure okay I guess it says if C equals zero then applied on Amarth that's basically what it's saying but C is like the entire register okay so you can anyway you can apply basically conditional operations based on your measurements as in other circuits where they do this a bit differently where they basically say if so if for example if you go here and and then if you measure I think they do something like if you measure I don't even know how to draw this but like something like that right so and then and then like an egg scape so and they say if you if you measure like one applied the XK if not and apply so conditional conditional operations based on measurements that's also thing because basically as I said they modified the rest of your system so that's you know those are important things you should know about summer positions this one one important thing I mean we've we've seen that you can do arithmetic with it and that also you can measure superposition partially in a superposition doesn't have to be that's the most common one is like the the equal superposition because it's the most the one that that's used the most but superposition is any as any state in which more than one value is is probable and it isn't it doesn't even have to be code it can be like ten percent something and ninety percent something else so that's that's it about superpositions |
cool um oh god i look like it just came back from the martha or something i don't know anyway that's a life messy hair um what are we doing today so we're kind of a bit going farther with the rethinking of the model right and i um if you've been following some of the voice notes that i put in twitter and if not i recommend you to do that so you can actually go and check them with these hashtag voice notes there you go so these are just all the notes that i you know kind of record voice notes when i'm just out there on the uh in the real world and i kind of have a bit of a niche to think about some of the stuff and then i've also got some text notes which are just basically specific tweets of just my thinking right but nevertheless so what i've been doing i've been doing a bit of playing with this stuff here so with the model right and i kind of came up with the so let me see if i can show the tweeter one here i think this is this one yeah so so basically what i was trying to say here is that um like how would you how would you basically build these how would you model the state after the control knot in this case right um this is working well yeah there you go right so well you kind of have you know it's if you if this is zero then you kind of see that that the probability of the qubits being zero um of each key being zero is higher than the probability of them being one it's not 50 50 right you can see this here but still still there they're entangled uh whereas if it's one you know for sure it's one one so you notice that there's no there's no entanglement and i was kind of thinking that um one way to see these i want to model this would be by using something like that right where you would say so each color is a cupid and then you have a so a hyper edge first of all at the top level high price tells you that whenever the target is one then we know that that this is um that this hyper edge is that that's the that's how that's the state the system is in right the other two are one right and the way you would read these features is say well like you know it's kind of like you ask questions right if i know this bit of information then um what do i know about the other qubits right so if i know that this is a zero um like i know it's not a one so i know that it that they are both both up here they are not like this hyper edge is not where the system is so we are not this means it's a bit that this i have to formalize maybe this is the way this interpreter but but this would mean that you know for sure that they are both not one one right it can still be that one of them is one but they're not both one one um right but the same if you just know that that cubit green is one like that doesn't tell you anything it it you still have a bit of a mixture here right and we can test that uh we can check that quickly real quick and quirk so so this is these if i know that this is one and i wanna and i and i do that here oh no zero sorry if i know this is one then like then i don't like you see like i i i don't know anything about each qubit separately that's what you're you still have a mixture you're not you you can't like the hyper just don't tell you anything here right and so you so i think that seems to be a good way to think about this now the the the whole thing is how do we operationalize these right oh thanks cue study and now i see that actually the things popped up in the screen i have to edit the i have to check i didn't know you can actually overlay the comments because i'm using the the the better thing oh there's a draw mode oh i can i can i can draw stuff in here ah interesting cool okay that's nice yeah but i have to draw on the actual stream which is not always optimal but i guess it's good cool um so what did i want to do so i was kind of you know basically thinking about how do you operationalize that in terms of rules right and we said you know you kind of have these nested rules uh like the cc the ccna rule would be something like you know match a one and then so much one for for for q qubit zero then match a one for a q one and if so then kind of do the um zero one one zero kind of rule matching for uh this is already qv2 right so you kind of have this double nestedness which is what creates the um the hyper edges because you initially have these right and this is in the zero state and so this is a an outcome so you kind of it's it's it's it's an lcom match so you would separate you know you would separate the two and still have these right um then you you go here and you'll go here and kind of separate that too but because it's nested you gotta you gotta you gotta nest it with the with the outcome match right so i mean now there's a normal match so if if we're here right so now there's an outcome match and so you kind of would do the same here but now you would you would wrap these two like these right um essentially that's kind of what what the nestedness is telling you right and then um and then you have this and then you have the zero which kind of like you get the rule that says go from zero to one so you take this zero and you put into one because it's an lcom kind of match you're not getting rid of the zero and because it's nested then you wrap these around another time and that's how you end up with that picture right so this is sort of what i have to program i guess um that i think that's that's that's really what i want to what i want to do now it's not only that right so i'm missing a part of this which is what do i do when um so the the and what i've been thinking about now is like there's something missing in here right which is what happens when like obviously these operation right won't won't do anything like it won't turn your cubit into the same qubit right right away the only thing this can happen is because you already have an initial state like this one and now if i apply a a ccx gate again um well now i have a i have a full match right so now i have like a full match everywhere it's like one that matches with a one and i have a full match for this rule here which turns this one into a zero and in this case that will actually replace it because it's full match right that's gonna be tricky to um implement so it's gonna replace it because it's a full match so it's gonna so i'm gonna do like this but now like you know now now this hyper edge makes no this group here makes no sense anymore because as i said right the there's no information that you're explicitly storing there like regardless of the states you know that you know that this is always in the state zero so like the fact that it's the same it means that so here algorithmically i can go and check okay so check that and then so i should check all those are zero and in this case because it's all zeros right then i would just get rid of these so this would lead me into um you know merge the zeros but what do i do with this thing that is wrapped now into hyperedge um that's interesting right because how do how do you even interpret that if i do that that literally just undoes it right well you know in theory that kind of resolves everything that is in the hyper edge this means that that high pressure itself doesn't make any sense in and of itself it's just it was here as an element of that hyperextract doesn't exist anymore and so i think just something that goes away as well right it's it's something that goes away as well and now uh i guess the rule here will be if you've got these things in in the same hyper edge you know you kind of have to re-combine them again right and so you're back in in there just but just because these happened that opens up a lot of tricky questions right we know what happens if this this match is like across hyperedges [Music] but this we can play with the with with quirk and and figure it out right like what happens if i'm here but now i do a like this right nice this is going to be fun because then you have like an outcome like a full match a full match hmm that's interesting right because now you still have an uncertainty here so it's not your you you will still yeah that's for next time i'm trying to figure out about the the the face kickback so that's the that's the face kickback part i still knowing how to operate how we evolve this is going to be complicated um the next so the the thing with the phase kickback will be the following right so what if i have this initial state right and now you can see that this is not entangled at all it stays as minus but then there is these whole things here they are both entangled right so so so so so but if i take a look at these i see the entitlement is because this is this is what happened right so the face kickback here so this is the one one element that was to have like no fan overall at the phase and now it has a relative face so so this is what's happened is that the kick that the face has been kicked back and so that's kind of like so the whole resolution is going to be one complicated beast in this whole thing thank you because let's say that's where we start from now so we we start from from disk from the the zero plus one right i tried a lot more than that in the voice notes but i'm doing it here again because there is complicated so now you have kind of the same the same process and now come here and i'll come here and then you're building this guy up here and then you're like oh so now uh minus one and now it comes in the operation it says well i'm gonna i'm gonna do the the not gate and so you know i end up with like a uh minus zero plus one but i'm not gonna make a copy of these because it's the same right so there's no no no thing no entanglement no whatever um and so what i can call what i can just do is like i'm gonna so i'm gonna i'm gonna get rid of these but i have a global phase right it's called basically like this is not equal mathematically it's equal physically but not mathematically so i'm gonna take this global phase of minus minus one and i i i'll just bump it up one level of the rule let's use just the rule right so the rule is this level less than this so we'll just bump it up one level so we'll put it here and now yeah and i think that should work yeah okay but now now comes the problem which is how come these resolves and that doesn't you know what i mean like i i here i was just so happy they're saying well because i resolve these but in this case it's like it never happened but those are equivalent so i cannot and and i cannot just now say well because they stay entangled right they stay as mixed states the face got kicked back but it did it and it's not the thing is and the thing is the up thing is that it's not that um when one is one we know the other one is one either right like like if if i'm not here and i say like that's not true i cannot just say if this is one and i plot blocks here here then then this is minus yeah of course then we know this hmm and if it's zero it's plus okay okay that's weird why hm no why if these guys zero they should be minus no oh god i didn't think about this that is actually complicated i know that if so if i know this is zero i know this is a plus and the reason i know this is because these so this guy is zero right it's either these with these right it's no no these are these or these because the minus is only in the in in the one one case oh so how do i know how how would i know these what happens here so what really happens here is i have that global face here and i can so i just keep it out here but i could have kicked it out somewhere else either right so um how to deal with these because i i oof that i mean that is how so how would i model that that is com that this is a super weird edge case right i know that this is i know that if this zero then this is plus like how how on earth would i do that the only reason the only way that i would do that that would be able to do that is if this minus would really somewhat belong to the edge right god because what that would mean is that [Applause] what that would mean is that the [Music] the minus here only appears oh god i don't know i have to admit that i that caught me by surprise like but even my ultimation wouldn't work there i think would it that's the thing is like this minus is in a way only valid for the case where these two are one one that's why and the same that's why i know that this is one right that is going to be a minus cool i guessed it but if i know this is a zero then this bus be a plus that is funky so the trick here is the okay so the fact that they are both mixed makes sense right but uh like this means that i don't these are not sort of i consolidate consolidated back to these uh god it might be just because of the relative phase you know oh god it is a way to express that and we know there is if we but it's a different type of correlation is it is it a different type of correlation it's still entanglement but it's not anymore a pure state correlation but it's a phase correlation is that a thing face mediated correlations that is the thing i can't believe so i've learned something new or not let's see in phase mediated correlations in quantum field theory whatever ah so first we have traditional wave function at the very core in the 50s version express style is a system in a singlet state involves aromatic correlations that can exist between the two spins even if they have separated the states might contain small mismatches in the faces between the two spins no that's not that's not what i mean no phase correlations because that is a phase correlation right in a way that is a phase correlation so so what's happening here is there's no correlation between the states per se i mean all possible values are are there same probability of measuring everything but the phase plays a role in answering the question if i would have a way do i have a way to say for example you know no no like what if it is minus one no yeah well that is that that's it is it so if this is minus one then this is one whereas you know if this is minus one then this is one okay i i huh so so that that actually shows me there's more to these that i have to her model because i'm not saying you see that's the difference between this picture right and this picture here be right before that so here like i have these right but now like whether this is 0 or 1 doesn't change anything because there's no phase correlation crap god why is this happening this doesn't happen in two qubits right this definitely doesn't happen in two qubits so what would happen if i would start with a minus state here right so say say that i start with a minus state here so what do we have now is that we have that after this um ccx gate if we know that if it's a zero then then this is a plus y because these minuses cancel right yeah so indeed okay so so yeah so that is that is what so i can't kick it i can't kick it up to any specific note i have to kick it out in the correlation that is the trick like because there's a difference between saying look i i'm starting at zero minus one and zero plus one god zero minus one and then you know by virtue of of the mixture like this is gonna kind of stay like these right but then there's a minus one associated to these if i know that this is a one then that's going to be a minus boom because i know i've observed that value but then i still have this guy let's go down here yeah and if i know that this is a one that's going to be a plus right because this and this will cancel yeah that makes sense that makes sense so they must stay there so okay so so i could kind of have the rule that says if if the hyperedge has a relative phase it means there's a relative phase correlation it means i can't get rid of it why can't i hear though because it was part of that it was part of that thing up here right it was a nested hybrid edge but if this high patch would have had a relative phase then i wouldn't have been able to do that it feels so hacky it feels it definitely feels too hacky but it is what it is um so that so this means i got to keep track of this means i have to keep track of them of whether the edges are nested right whether it's got apparent edge or not that's nasty because i need to be able to access this stuff um hmm i'm still thinking about you know things beyond control because you know beyond ck you control arcades and stuff like that um but i guess i guess it's a matter of finding the compositions and then doing this kind of stuff right it's just controlled operations i think that's the key to implement that it's the key to implement and the other stuff just it's just because that relative face gets kicked up here at the correlation level nasty that is nasty zero it also has to be plus how does it work now no it's minus look at this if it's zero it's minus why is that why is that yeah okay it's yeah of course it's that because you're it's not because you ignore the the phase correlation this phase correlation only appears when the states are 1 1 right so so in this case you you just ignore that and just take you just take that that's why it's important to make a difference in the previous implementation i was not so sure what was the difference between having relative phases at the hyper edge level and having relative phases at the node level and now this is clear at least now this is the this is clear clear clear clear um the relative phase is at the hyper h level it's another kind of correlation it's it's it's a phase correlation it's sort of a phase state correlation relative phase state correlation that's why this non-relative phase is the same as not having one because you're like yeah it's it's just still sort of the concept of like cool so now i've done these so these are the same right um so these are the same so i don't know exactly what is it that you destroy here it's like okay so you don't you don't draw that or is the same so you so there's no need for that anymore i don't know if you have to take them out or you have to take the zeros out or take the zeros in or just delete the hyper edge i don't know what's the best approach and then you do that recursively then you would say what about the other hyper edge or maybe maybe the trick here is that because there's no hyperedge in here right because you can't you can't just relate the the these guys right um then that's why this this makes no sense because it doesn't have the counterpart now that's fine but like you really want the bell state to be like to be built like these right you must have the belt state built like that so if you have a zero plus one and a zero okay isn't that good you know the the rule is if one then do the the whole like you know zero to one one to zero i think that's the not gate so you is an l com so you so you mix that and then you say okay so the zero stays here and now i match this one and take this to one and i create that hi that hyper edge that's that's the you know that's the delicate thing here is i'm kind of making some assumptions some things don't seem to work well with if i get it to work for two kiwis i i don't get a door for three and vice versa because it's the combinatorial part that messes up with all these so like here that makes a lot of sense but here it doesn't make one it doesn't make sense at all because i want these to be like that i want there to be a hyper edge here because that's really what this is telling us is that if no yeah well he hasn't know right to be fair yes and no because it's it's something that it actually can be answered like that because you're like okay so if you have this state let's say you have this state right and i'm telling you that i know this guy is zero then you know that it's you know it can't be one because then this would have to be one right so so it's zero so you can answer the question whereas in this particular case i think it's fine to have it like that because if i tell you this is zero you know this can be zero so what you know is that these two can't be together one one okay now so it's cool it's cool i don't have to have it like that and that's probably that was the main mistake that's fine that's that's correct for bell state that's actually correct for a bell state because it allows me to answer these questions regardless there's a symmetry here right so i could just like which is as well have the hybrid surrounding the zeros that i wouldn't care about these like that could also just be uh yeah whatever uh okay cool that's fine what i have i discovered today that i need that and now okay but so so let's let's put us in this situation right so how i think i think maybe the whole thing would be do i get rid of the high pressure or not logic has to be more complex i think it's just too naive to just compare and that's it it works for the two case but it doesn't work generally here here i don't want to get rid of the edge because knowing the state of of another of of of these it tells me something different it tells me information about the state of the other cubit well here right let's say yeah and here you know the trick here is that knowing the state of of these right knowing the state of these doesn't tell me anything about the state of these two right exactly so this just goes away in this case knowing this state it doesn't tell me anything about the set of these two but i can't get rid of this because it is because there's this additional piece of information there in the in in that relation that i can't that is telling me that if if these are two yeah so hyper edges don't hyper edges don't tell you necessarily the full story right like it's just because that piece of information that that stays there i think one thing that i have to kind of get out of my brain is the fact that if i see two states circled three-state circle that i that that they know that they must belong together that's not that that's not what this model tells you that's sort of a that's that's a misunderstanding of these like in this case if i know this of course it doesn't tell me anything but it's still still still this information is there if this wouldn't be there then i then then it was this was just this would just cancel so the thing is like and here so so okay so here's the trick right so this means that whether an edge or the hyper edge stays or not it's not something that can be decided by the hyperedge alone it's something so you probably have to ask two questions first question is like does this hybridge have additional information does it have like a relative phase if so then i can't get rid of it um that's my current assumption if not if not if not what if not do i is it really that they passed kind of a span across all the cubes in the system that's also not true right because maybe i just the next question is this one right so what if i what if i do these um what if i do these so this model this is where the model breaks man so what if i do these so this is this oh no whatever that's in the zero state that's zero and i now i do like this right so that is actually this damn here right that's where it breaks because these doesn't tell me anything about these two right it's not enough it's not enough is there something else there i'm i'm missing a piece of information to fully model these because i cannot with this model i can't distinguish between these and and this no what sorry what no what am i doing what was that case what am i doing you know they go away yeah okay no it's an intermediate state it's not that i can't distinguish that's not right but it's just an intermediate state that it's confusing me because i get rid of the zero get into the hyper edge and then i i i'm left with these so how do i know that i can turn these into that and that this is not this state and that sorry that this is not the um you know this state right it's this kind the the the resolution has got to be defined to be better anyway it's almost there i'm almost there i feel like i'm kind of like almost there it's just i have to kind of get get i just have to do that again that exercise again the hyper edge does not mean the high pressure does not necessarily have the the full answer it's not telling you that's the type of states that you like that if you've got this one there you know the other one is there there's this this nuance to these this nuance to these especially with multiple qubits like knowing that this is zero one doesn't tell you about whether this is zero one but it tells you that it's both okay it's both one one so that definitely tells you oh god uh almost there see you next time i don't know if i can make it tomorrow i'll do my best bye |
this is what are we doing today today is a bit of a different topic than my typical um live research today i'm gonna be taking a look at an idea that i've been um that have been sort of like entertaining for a while which is like i've been i've been i've been doing all this kind of you know playing with the idea of like raw videos and role you know kind of like role research like how kind of how do you you know um capture that like honest and oh guys give me a second someone's at the door yes now cool so uh oh sorry i gotta switch the scene back there you go cool so um what i was gonna say i was gonna say that um yeah i've been playing with with this idea of doing all this kind of like raw video and stuff like that right i think i mean ultimately why is this useful um it's a good exercise i think it's a good thing it's it's an interesting experience for myself as well but then i also kind of realized that you know it might be useful something else it might be useful um to actually kind of train an ai in the future like kind of use that as training data right as in like raw data of like some you know kind of showing machine um what learning and so the research truly uh looks like i mean i don't know it's actually it's it's interesting right because one one could think about like what is it that makes something um conscious smart human whatever it is right but like how do we learn this stuff i mean and if you take a look at things like this gpd3 story and all this kind of stuff you you do end up thinking like you know maybe all that all that really is there is like it's a language model right in a way like it's it's a sort of a symbolic way of understanding the world i mean that's what we do even when we think right it's just in our head to steal language but it's in our heads and um you know watching keats grow is and explore the world is a it's a super interesting experience to kind of that actually kind of in in a way reinforces that that vo on on consciousness um just seeing how like a child just actually tries to understand things but just kind of like imitating uh you know what they what what they what they perceive what they kind of see right it's like it's it's really interesting i mean there's of course everything that goes into like you know learning how to build our how to kind of like operate our body but then sort of the the interaction with the world is like the language and how do you think and stuff like that it's just super interesting but anyway it's it's one of the things where it's like well you know how do we how do we learn how do we learn to to learn right like we we learn this by um essentially seeing others do that essentially kind of uh searching for four things and and when we think about okay how do we how do i learn about a complex um how do i learn about a complex problem how do i learn about something well you look for someone to teach it to you you look you're i mean that's kind of one way to do things right so you kind of um you're looking to be trained in a way right so you're you're looking for some material that you can just go through that you can process whether it's text video whatever it is so that you so that you can basically adapt your model right and so this is like th this is this is this is what learning is in a way but then you kind of sometimes sometimes think about like cool so what what you know what is then self learning when you know you're not really i mean essentially there's nothing that's not really like self learning a lot self learning this is interesting right how how would you how do you come up with your own conclusions or how do you kind of do research is interesting because in a way it's like you're not really it's like unsupervised learning in a way right so um or the classical traditional learning is like um you're basically you're basically doing in a supervised learning in a way right so i mean let me just am i am i saying i'm not saying or not um let me just uh can i just start any cognitive one it's just like supervised learn i think this is what when you're providing data with this to a system and you're actually telling no no ah and you're actually telling the system it also known as supervised machine learning is the subcategory it is defined by its use of label data set so you're actually basically telling the system what is the correct answer just many many many times and that is just that is how a lot of the you know human teaching works as well right you just assume what the teacher is telling you or what the material that you're looking at is telling you you're just assuming that assume that the labels are correct and so you're you just you know you're just gonna go through these but then unsupervised learning is when you have that um you you have that kind of you know just raw data there's like no you know it there's no guidance in terms of whether something is correct or or or incorrect you just kind of uh so the hope is that through mimicry which is an important mode of learning people the machine is forced to build compact internal representations of its world and then generate the imaginative content from it inc okay within a stacked exhibit self-organization that captures patterns as probability densities or combinations of now feature preferences the levels of supervision spectrum so it it this is essentially i guess something that you could use from a like a as a dichotomy between sort of research and and actually learning research is when you're kind of you know you've got some basics some base knowledge but then you're trying to learn something new on your or discover something new on your own you're not just like you don't you you you know you just don't stumble upon these on the world you just you try to understand make sense of make sense of data do experiments do you know whatever research stuff search for things right and so in a way um you know but even even if you're relying on you know existing things like let's say i want to learn quantum mechanics but i want to learn about some so i want to learn about quantum mechanics it's kind of like you have you have knowledge there that you can use for super for sort of learning stuff in a supervised way right um but still like you i guess one way of doing that is a bit of that kind of mixture of you're getting things and then you're trying to discover things you're um your yourself in an unsupervised way i don't know how to explain that i have the feeling that there's this kind of you know dual mode of of of learning you know in in a way you kind of have you know you have the staff that's the body of knowledge uh that exists out there and wikipedia is a great example but then you kind of have that like okay so let's try to discover or let's try to answer this question without actually finding you know well it's actually understanding it right like how do you you know build build the understanding right like you know give an example it's like how do how do i learn uh what a fourier transform is well i can go and look you know look at the answer but it's when it comes to this kind of topics the answer is there's no just unique answer like one liner or something right it's just there's there's a bunch of things around a concept that you kind of have to discover if you want to understand the first principles of it and that you can then apply later on so it's actually you know this is what i'm exploring with the whole uncertain systems right so how do how do we learn this kind of stuff how do like how does it look like this process kind of exploring this a little bit and then you know with a very specific instantiation of the problem which is quantum mechanics um but in general i guess that applies to everything so one of the things that i um that i've been entertaining in my hand in the past for the past you know months is why you know if if we just kind of build a huge body of this kind of data will be you know you build a great data set to kind of teach a machine how to do that how to actually um how to actually how to actually teach itself in a sort of in this kind of mixed mode like supervised unsupervised way right and and and so you know i thought why not like encourage other people to do that and that would be a great idea sort of a bit of a little little web 3d project where you kind of have a contract a sort of smart contract or something and you would upload say a video or a document or a tweet whatever it is like whatever it is that you've used to capture that raw process uh around a specific and it doesn't have to be self-contained like it just it's gonna show you exploring stuff right or it's gonna show how do you explore you know um whatever it is that you're exploring and capture that and then basically you know give to the contract and then the contract will kind of give you um you know sort of coins in exchange or something like that like whatever it is like a made-up you know knowledge currency or something like that and uh you know of course that you know the the the idea is to use the this kind of like uh blockchain web three economical model where you know these coins would allow uh you know what kind of build up intrinsic value in a way uh would allow you to do other stuff i don't know and i just kind of wanted to play with these especially because i've recently discovered replete as i was building as i was building or playing with you know some of the um senpai stuff and i kind of uh came across these uh the other day where basically apparently there's a apparently there's a really easy way to build smart contracts and test them and play with these within replay so i think it would be great because i'm a lazy person and that is basically um you know a lot of the stuff that is difficult about web3 is how to get all that infrastructure ready in terms of like how do you build a contract how to deploy stuff and i'm talking about some playing with some very sort of the prototype would be very simple i i don't even care about like upload it's more like i i'd assume you know you have you know you kind of connect your wallet and then you basically give the thing a um uh an ip fs link an interplanetary file system link uh and um so you know i kind of assume you've uploaded whatever it is already in the ipfs uh peer peer peer-to-peer network and you just give it to the contract and then the contract gives you some kind of currency in exchange of course the question is uh what how do you how do you make sure so that's just a folder what is these and and then you know of course i think these are just examples i i hadn't seen that before can i just delete these folders oh actually um i i did try something a while ago but i i forgot that this is it was in here what is that is that an example and solidity but yeah so basically i mean there's still the problem of like how would you validate that i'm not just uploading like a cat video or something so you'd still need a human you'd still need a human to do that in a way so you depend on someone else you know we need to incentivize someone else to actually maybe validate your yeah that seems to be examples token simple token wallet simple wallet nft so there's some examples um i'll give you a second i'll be back in a second sorry i had to run to the door again um oh god i should exercise a bit more anyway uh essentially that's kind of what i um what i had in mind so basically build that contract and uh yeah kind of encourage people to just you know um upload the video so upload the content upload the stuff get the coins uh as a reward and uh you know by sort of hoping that by doing these we encourage more people especially professional researchers for example to you know kind of upload their own you know their own ways of doing things they're their own ways of of doing research yeah oh i'm gonna go for some water and then we'll get started with the coating i mean we'll see my goal for these is try to spend the next 20 minutes half an hour or so and trying to kind of understand how could i how can i do that in replit should i just go and create something so literally starter and i'll just call it um probably call it ancies do you call us like aunties or something and this coin or um whatever i don't know uh [Music] as always naming is the the most complicated since when is ansys columnar i'll just go answer this coin and then see what happens can i get something to drink and be back in a second cool so what do we have here um yeah it feels i think that's exactly the same i had in the uh in the other folder did i delete that uh i just want to make sure i have stuff clean so um i have these folders here you can just delete these and then delete this yeah and can i just delete folder stops existing okay ans is going cool um we have here tools so welcome to the world theorem or three i'm sorry here's better easier to what read that whatever um so press the run button uh you should only need to do this once and might take like 15 seconds to install okay so this will install the installs everything replica is really cool but i have to still get to you get used to like how how does this actually work you hit the the run button and it actually what is it that it's that it's running ready contract simple storage files those are all the dependencies interesting whoa whoa it actually opens up some kind of ui that's cool reloading page oh ah she can connect a wallet deployed contracts interesting okay um yeah but i want to see i want to kind of open the this thing here how can i how can i okay actually okay it just runs it like that no this is the room i one okay so um you should only need to do this once and might install the run tools and uh start up the contract develop the the contract deployment ui and compile your contract soul file okay so that uh-huh well so this will automatically compile whenever you edit it and all your contracts inside of this file will be available to deploy from the ui pressing control s will reload the ui we have pre-installed packages from open zeppelin contracts to install other solidity packages that are distributed in npm make sure you install them using the package installer in the sidebar i think that's here cool examples we include a few example uh contracts in the examples folder this will not be automatically deployed or accessible in the ui but you can copy paste them into your main contracts all um import them they're there for you to reference feature work okay cool lsp support i don't know what lsp is is it linting integration with um hard hat for local in rebel testing um an actual solidity route to quick prototyping uh testing functions are lines feedback okay cool seems uh yeah but how do you how do you deploy contracts do i actually have to connect my wallet um like how do i test without like how can i test in a sort of a dummy network or something like that because that would be definitely the the best right i don't want to start deploying things on ethereum and pay all the money for nothing i'll get one youth for testing so there's actual there's some actual testing stuff that you can do ethereum ethereal mainnet ah let's see what happens so what if i connect the wallet sorry guys uh why it's in german i have uh i don't want to what i don't want to do is uh oh because i'm in no i'm not an incredible let me just move that away from the screen here see if i can just uh reconnect because i um i already have a wallet but [Music] so give me a second okay because i have to figure this out um you know just in case i mean i i in this wallet i just have probably like 0.018 or something so not a big deal but maybe i'll just i'll just put the beer back screen just in case and uh because i i when i installed that i did not connect my wallet again so i'll do that in a second i'm still here hopefully you can hear me but what i need to do is i need to actually find the fine you know i have stored my keyphrase in a super ultra secret way in place [Music] it's a secret i can't remember where here maybe come on or ah or wade maybe maybe they are here so let's see if that if i imported the right uh have i imported the right it looks like yeah okay cool so now so i have 0.02 eth in this wallet seventy dollars currently that is and and i have another wallet but i don't know what's in there anyway back in here so what if i connect the wallet now what will happen will it okay so that's the account right and i want to connect it with this account connect it okay cool so 0.02 e to have here that's my address okay so i can switch to a test network and i can use this one this is pretty cool replicas net so replica gives me a test net like this is nice so i can get a need for testing okay so with these i can actually deploy contracts that's cool okay um i now now that i have the wallet connected i can choose the contract and actually deploy it so what do we have here so we have contract simple storage and then we have a math test um and we have examples what have i done here examples nft token i think it's token is what i want right because that i want basically i want to kind of have uh so this okay so um so that's an it's an ethereal ethereum example um and so basically i hope you guys can read that because i don't want to zoom in too much uh because then my stuff gets messed up in here but maybe i'll just i'll just i don't know i can't can't i hide these no i can't um yeah maybe i'll actually exactly i'll do that and close it here so i i have these oh and i have to connect the wallet again let me connect that's not coinbase i know i'm convinced i want to connect that one and i want to switch that switch to test network i'm okay i'm not in main net right because i don't have a need let me just refresh that okay whatever i have an ease so i'm i'm because i don't have an eth in real life so i assume that i'm on the test that's that's wrong i'm assuming it's probably a bug but anyway um cool so this way i can just kind of you know get rid of these for a second and focus on that so this is okay so an address is comparable to an email address okay so blah blah blah so i have so there's a token supply i mean again this token wouldn't really have a supply right because in a way it's it's just you want to um reward people for uploading the stuff but i mean it feels it feels somehow uncomfortable like if all it if all you need is if all you need is um a video to get that like a coin without supply doesn't look like it doesn't sound like a great idea but let's ignore that for a second so here it's basically you map the balances um okay so you get the addresses and you see how many uh how many how many yeah basically and then the constructor okay so the deploying the owner is is the person who deploys the contract and then it say set the token balance of the owner to the total token supply so the owner has all the supply and then sends the amount of tokens from any caller to any address so basically this method um basically uh okay so requires blah blah and then what it does is it takes the amount from here and adds it here um so what this does is i guess it's just a contract where i can't i can send or anyone actually anyone can send any money to anyone right there's no i could of course require that the sender is the owner so only only the owner can actually send the staff um but that's not uh i guess that's not the point but that's a good basics a good basis and then wallet what is the what is the wallet doing so you have an owner lock deposit log withdrawal hmm withdrawal okay so only the owner of this wallet can withdraw so the the sender must equal the owner uh and then the balance must be there recipient transfer what that is what is what is these doing uh there are events events allow for logging of activity on the blockchain software applications can listen for events in order to react to contract state changes ah interesting what is address payable recipient boo there's a lot of stuff that i didn't know about that payable solidity this modifier allows the function to receive ether okay there's definitely quite some stuff that i need to learn um yeah but that will be and an nft simple nft uh whoa whoa whoa what is that ownable token counter okay so whatever disease i don't know what is that is it some sort of kind of standard or something it's the non-financial token standard okay so it reduces the standard for nft in other words it's the type of talk this type of token is is unique and can have different value than another token from the same smart contract maybe due to its age rarity or even something else like it's visual but visual yes i actually have a variable called token id so for any contract the pair contract address must be globally unique that said a dab can have a converter that uses the token id as input and outputs an image of something cool like zombies weapon skills or amazing kitties okay yeah okay so that's just an nft standard the nft standard and uh mintik is essentially creating an nft so who opens it what is the tokens uri which is a link to an image hosted on ipfs and then um okay so that is already cool so that's already all built into the standards so there's sort of a mean function that puts it into the blockchain you said that's pretty easy okay so there's a lot of stuff that's been done already cool but what do we need i mean let's think about these what do we because i'm like so i am uh torn between two ideas right so one is that experience that you're uploading you know you turn it into an nft because essentially that that's what it that's what is it it's a um it's sort of a unique experience right so that's uh that's what it actually is but then but then what's the value right so you you you mean that and then why would someone buy this well someone will buy that as in you know you you could buy nfts just because but yeah the thing is you know it would make sense that you kind of like turn special moments in nfts like for example let's imagine that you know a researcher kind of um goes and like you know disco sort of creates a new theory of everything that it turns out to be you know that that one right so the actual thing um and then captures data into a video and then you know then if these the video but then for this we have already nfts right so why would you um you know what would you encoura you want to encourage people to kind of upload all the raw data not just like oh that is kind of what led to that moment right that kind of aha moment so to say so you want to encourage people to just well um do that so in a way you want a token right you want to give people i i really think you want a token because you want to um you want to basically give give people something in exchange for it for for the video right um so i would i i would start with these simple storage store data okay so actually the contract has memory you know you just can have memory and they can store things and blah blah right but let's just ignore these let's just have let's just use that contract as a basis and say um well it's the ansys token and so yeah well it is the owner uh you know yeah you have you have the owner uh the owner it's gonna be me because i'm gonna deploy the contract so um i don't know about talking supply what i do want to have is a mapping between so it's the balances and then i also want to keep a i basically also wanna store do i wanna do i wanna keep ownership in terms of like track of ownership of like who uploaded certain things probably not like nobody cares um so in a way you want to have i probably would have just like removed right store data ah that's okay so i can declare whatever i want can i declare okay so can i just say um well i wanna keep track of the balances do i want to keep yes right because essentially that's that's the coin otherwise how how does this work though really like you did the smart contract needs to keep track on on who's how much of a token someone has then how the wallets actually work like do wallets have wallets have the log deposit lock withdrawal how do you get like how do you get a wallet to how to get a how do you get a token to be recognized um how how does dokin get recognized by um how does a token because i think the wallets really just do these i think i think that's the fun thing is like the wallets actually just lock deposits and withdrawals send ether from the function color to the simple wallet contract your own token um i guess there's a bunch of services that allow you to do things like that make my own okay so awesome okay um open metamask and click on the add token button select the custom token option and paste the contracts address in the first field okay so basically we'll face the token symbol and decimals automatically click on next and your token will be added to the wallet it will be available under the asset section in metamask okay so first of all but this is okay so there's a standard for these i guess what this means is it ensures there are certain functions that i need to implement there we go uh using contracts you can easily create our own contract which will be used to track gold an internal currency in a hypothetical game so uh okay so you you import stuff like that okay but i guess if i want to import things do i have to this is importing stuff like that and do i have to packages do i have to actually install it like that doesn't find it or what shouldn't find it automatically no okay um i often use the inheritance and we're using for both basic standard condition and the name symbol and decimals okay so this basically ensures that it has we'll we'll actually go with these so we will we will just go with these actually it's going to be ansys token oh actually i i do i do i have to um does this token need this um so these are the functions is the uh this thing must be like three letters i guess so symbol or can it be whatever i don't know why people make so much of a fuss out of these this is just very basic programming to be honest there's nothing really special about these other than like the mental model i guess simple is there any restrictions to the symbol um i'll call this the ansys token and and basically the name is anses and then this is going to be like what i mean isn't it is there any length symbol okay so there's different no there's no restriction it seems like because to be honest i'll just call it anses right um i mean initial supply so i have to give it an initial supply okay but i guess the supply can increase right it would be good to know how that actually works to be honest um but okay what is it what is it doing so you deploy the player's balance ah so there are different supply mechanisms okay let's do that uh it's funny because um what is that what is open zeppelin what is opensampling i guess you can just create okay it's it looks like a place where you can just create your contracts and stuff like that complete security products to build manage and inspect all the software long versions theorem projects okay yeah that might be an interesting thing to consider at some point but where was i so in this guide you'll learn how to create an erc20 token with a custom supply mechanism we'll showcase two automatic ways to use open separate contracts um okay but i don't wanna specifically use open zeppelin staff in here ah cool yeah sure of course those will be the miners right so the people actually providing the videos because it's like you're you're you're kind of you know working yeah that's actually cool okay modular is the mechanism um but i'm afraid this is open zeppelin specific so this is from ethereum.org so maybe that's better as the technical standard is a smart contraction blockchain for fungible token implementations defines a common list of rules that all fungible ethereum tokens should adhere to con consequently these tokens time empowers developers to full-time security predict how new tokens will function within the large ethereum system this simplifies these developers tasks because they can proceed with their work knowing that each and every new project will need to be redone every time a new token is released cool so um okay so these are the functions so total supply balance off okay so yeah that's and that's what wallets are using probably right so that's what wallets are using to kind of query the balance of of someone allowance transfer approve transfer from transfer approval total supply returns the amount of tokens in existence this function is a getter and does not modify the state of the contract keep in mind that there are no floats in solidity therefore most tokens adopt 18 decimals and will return the total supply um and other results as followed for one token okay not every token has 18 decimals and this is something you really need to watch for when dealing with tokens returns the amount of tokens owned by an address this getter returns a remaining number of tokens the spender will be allowed to spend on behalf of the owner this function is a getter and does not modify functions moves the amount from one address to another um events this event is emitted when the amount of tokens is sent from the front office to address so what's about code to base your from show ah okay so that is yeah okay ah so open zeppelin has a good implementation as well to supply okay so actually this has a total supply decimals 10 ether why is an ether it's just an example i guess and opens up link has another symbol total supply balance off okay so everything is well i mean i can just copy paste that and to be honest you just need to add probably uh okay so supply mechanism has to be added um in a drive to contract in a drive contract using mint um for generate mechanism c okay so that will probably just take me where i came from but i like that so i can implement a supply mechanism using mint and i think i think i'll just use that i think i'll just use that copy so i think i'll just use that basically and say cool so 0.2 but whatever it is so um string private symbol okay it's like a constructor the default value of uh 18 to select different value all are immutable they can only be set once during construction okay and how do i call the constructor what i call the constructor when i deployed or returns the name of the content of the symbol decimals total supply total supply uh where is mint in here though function mint so it creates amount tokens and assigns them to an account increasing the total okay so that increases the total supply that's okay that's interesting so it means the transfer with from set to the zero address okay nobody is your address ah so that's okay the zero account the zero address is my address i guess um and so it increases total supply okay ah okay so this means that i am then you know i can then just add more supply um okay but it actually does have these so what is it that it's uh missing in here ah what am i doing so there is a am i stupid that's the link probably how the mechanisms okay [Music] okay i probably want to read about these calmly and and then do that uh yeah but i think i'll probably go with something like that so it seems like i thought i thought these was already included in here i mean you have the mean function that adds total supply burn basically burn supply approve so what is this doing this is uh okay the thing with the spending whatever i have to go through this okay but that's a good probably good basis um what is this is this just that um connect the wallet okay so there's parsley resources not found oh so i actually have to import that probably i just have to import yeah i have to import this i guess okay so i'll get that sorted out next time but yeah let's do that essentially and i guess i guess all i'm missing here is to um to basically uh do i need to do any yeah i need to keep track right i want to keep track of uh sort of the uploaded videos right i need to keep track of that um and then um because essentially that's basically the yeah that's basically the you know i'm gonna i'm gonna be giving i'm gonna be giving yeah to be honest i don't know do i need to keep track of it in this contract because maybe that's just the meeting it's it's it's basically the supply it's it's basically the the the supply mechanism do i want do i um do i want to kind of have a supply that i just kind of give out or do i just want to say look every time someone basically uploads a video i'm gonna mint so i'm gonna uh i mean and i can keep it to videos i guess but i i because i don't know is there a way to define the value uh of of a specific thing it can be a long it can be short it can be like you know the amount of i guess it's just raw data right it doesn't matter what the size is because it does not determine the value of it necessarily um so maybe there's just a standard way to do that and then um you know i would just kind of do the upload but i should probably keep the list somewhere um because i then essentially i won i want another set of users to then basically say cool you know go in and check is that really uh is that really is that really or isn't it like is that really a valid video it's just like a cat video right um and uh how to make sure though that that's not you know that's that the approval mechanism it's going to be it's going to be tricky to keep it decentralized um but yeah if that's the case then then you get the coins and then uh yeah you go ahead what can you do these coins i don't know i'll have to think about these but i think that's an interesting idea basically yeah i don't know let's see and then of course more people to do that right um i kind of collect that dot um the database of uh of of raw data that uh could can potentially be used to train some sort of artificial intelligence in the future i hope so i guess cool then see you next time |
so we are live it looks like this is already live let's yeah it's 9 o'clock sharp um ok so I see three concurrent viewers for that's good ok so I hopefully can see my screen as I have announced in Twitter yesterday I kind of I want to basically repurpose a bit this session and I want to at the same time kind of you know call it holiday call it a day so to say with this project right with the with a kiss kid textbook so and you know the reason being that I kind of feel like it's not really engaging me the way that I would expect and the way that it would allow me to actually give you a good life experience on these and so you know this is mainly because of the direction that I want to take in the next weeks and months with the channel and the time that I have basically have to spend on this so this is kind of the one main one main thing the other thing would be you know I will at least use that session today to give sort of take a look back at what we've done in the book maybe go through the things there's a lot of a lot of the I've seen just now like that that they've added the face kick back chapter in here and so there's a bunch of things that we can take a look at really quickly sort of in in a summary way kind of try to summarize what we've done and then take a sort of a quick view on what what we haven't covered in the book and and that's kind of the sections five and six and then the demo seven but it's going to be a rather short video so it's gonna be really unless you guys have questions and you know we have sort of a started discussion in the child or something it's going to be a relatively short video I just don't want to linger too much in those topics but basically so I wanted to do these two things and maybe the first thing that we should start right away with is taking a look at the textbook and how it has changed and you can see already I mean the the first page the preface is really already the preface or I think it's Browns breakfast it it has changed already a lot it's just new content I'd say and what I'll probably do is I'll probably go I'll probably go through each of the chapters quickly each other pages quickly and then take a look at and see if this basically okay so look at this so they've actually added a bit of guidance in here in terms of the structure of the book so we okay so we basically get all the way down to the algorithms that say the basic algorithms exactly so and then you've got still these two options and they're still the hardware option that's at the end I have to fill in this may be my first piece of feedback here I have the feeling that this book can be can become huge as one of the one of the maybe one of the things that kind of makes me feel like not super engage with it anymore is the fact that it's it's just it feels like it feels a bit rushed and and it's not because the content is rushed but it's because it tries to cover a lot of topics at the same time in one single book so they're saying and it's your combination I don't even know how to say this but it's a weird combination of the first three chapters of the book it kind of almost like doesn't feel like you need kiss kit you know you know what I mean like you you do have the introduction was really good the rest of the stuff you use it for is just building circuits and it maybe does not I mean which is kind of kiss kids it's it's what kiss keys right it's just like the basic the core of kiss kid would be to you know help you build a circuit and interact with the real backends that sign and then I'm assuming that in chapters four and five you really start to get some of those building things that you haven't kissed get right like vqe and Kyo and stuff like that you've had you have like in other in other parts of KISS game but it's them it's one of those things where it feels really it feels really stretched it feels like it doesn't invite the reader to sort of you know kind of staying for a while deep dive into some topics now maybe another way to put it would be it kind of feels like it's more of a skeleton at the moment rather than like a fully fledged and fully well you know developed and thought through book this it's getting there I think for example so I just took a look let me just jump to the face kickback thing for example which is a new addition to the book since I don't know some some time but basically this is this is the kind of stuff that I personally love so I haven't gone through all the details but basically you've got the entire chapters just focus on exploring the cenote Gate and face came back from that perspective and so you know and up until before like section existed this was just barely in like mentioned somewhere in in another chapter but here you really you know take the time to go deep into these and okay analyze what the cenote is doing and you know basically take a look at different I mean that's that's that's really I think really well done because it's really you know at a deep sort of it helps you fully understand how the things work maybe not deep inside in the hardware but how the things work at a small scale right and so it's not like hey here you've got the algorithm it works here's the mathematical proof I'm done right so that's kind of the stuff that I don't like much and here's a bit more like you know here you have the state and here you have these in here and then what's happening here right in this case is that when you do not your your your your swamping the faces in here right so it acts all solve the amplitudes of 0 1 & 1 1 so this would solve implicitly the the phases as well right so you've got the state vector in here and so it really takes you step by step to just one element that then you know it's it's something that's applied in many many algorithms so that's a good example of the direction that I would like to see the book going going towards order I think it would be useful oh and even for the TV I see it's that's that's cool and then I mean probably there's some exercises about doing it with another cake or something like that so that's uh yeah that's a good example that's I'd say that's a good example um to do all the stuff today so this has changed a lot he's keeping your algebra introduction to I said if you have any questions feel free to ask if not I'm just gonna go through these fairly quickly and there's some air in another thing that it's interesting is that it despite the fact that like the whole USP of this thing and of IBM and Quantico the IBM kind of experience and whatnot it's um it's really the real backends right I mean it's that's kind of what makes it different from everyone else in the industry right now is that I I be mq provides you access to actual real hardware and while that's really cool it just doesn't feel like I think it doesn't feel you know it's it's something it feels really redundant that in every section you're like oh that's disability that's a real machine that's similar there's a real machine and it doesn't really feel like I realize I haven't used it at all and maybe when you go through the actual quantum hardware using microwave pulses part right is the lowest level it's the lowest level possible this is the lowest level and that's actually I have to meet pretty cool come on we're again with internet connection problems I want I'm not here color includes an open open polls so I have skimmed through that in the past not on a video but literally the idea here is you can basically define the the pulses that are sent to the qubits right so you can actually manipulate this one machine and you can manipulate that that's kind of available to be manipulated down at that level which for educational purposes is really interesting because you can see how how the whole thing is really you know it gives you a bit of a an extra insight on how the whole thing is kind of built in its built up but and for research is probably pretty useful because you can calibrate that also something about accessing higher energy states and we're not here so it really kind of feels like you can find you the machine to your needs [Music] but like era it still feels like for the newcomer and someone who's actually coming from a computer science background and doesn't really do any active research in these it's not like doing something fancy with quantum simulation yet and whatnot this part in here maybe could be somewhere else or maybe my point is it's III I think we've got to a point where it feels like it's you know everything packed in there like you try to pack everything that the IBM kind of experience offers which I mean again might might be fine for some people but it it just over one I think it overwhelmed me a little bit [Music] but it's probably the only part idea where the actual real machines make more sense I mean make sense at all cuz otherwise you're just going with the simulator it doesn't make any sense right when you find you into posters and whatnot so what else there's something cubed staying it's know so we were here at the and the introduction part the atoms of computation I'm gonna go anywhere with this speed I split information to think this hasn't been [Music] the strata some minor updates here and there right but that's of course for sure the case but so representing Cupid states oh this is this is something new in here it feels click here to expand aha I said there is indeed ok service but this is another thing there's a lot of different there's a lot of playing around with different formats and different things and it almost feels like each section has been done by someone else in the completely in a completely independent style which is kind of good but at some point I think if you want to make you feel like a solid consistent work there's a bit of a bit of a bit of a look and feel that you probably want to standardize I guess but that's really not a major major issue so classical versus quantum bits in quantum physics we use state vectors to describe the state of our system this is different from classical physics where we generally just use numbers so example we say say we want to describe the position of car along a track this is a classical system so we'll use the number X alternatively with instance a collection of numbers in a vector called state vector each element is in the vector contains the probability of finding the car in a certain place ok interesting qubit notation this is also new remind my tradition right so there's some reminders to previous sections it's like the it's funny because it's kind of also you know the the label the label that that you're treating some of the concepts in here versus what you can probably find in vqe is they sort of a huge gap I'd say so that's also another explain keep it's whiskey so this is okay okay so this is a bit more I think that that's also other rows of measurement okay very important now I think that's good I think that's good that's one of those other good additions like like the face key back so it really takes you down on global face oh so you talk about the global thingies okay no I think that's nice the observer effect information about the probability of us finding the key billion is it but once we have measured to give it we know with certainty that what state of the cutie is for example of a magic given State I know about quantum simulators using a real quantum computer we cannot see the states of our carrots meet computation as this will destroy them the behavior is not ideal for learning so guess good price if you're going to simulators well the Bloch sphere visually representing a qubit yeah no I think it's good I think it's good stuff yeah III I think that's definitely keep it stay single qubit States single qubit state yes okay what is the I Gate City identity or I guess so because they're walking you through different Kiski gates measuring different bases so this again this whole thing with measuring in different days ins why there is a measurement in a harmonic afterward stone yeah anyways so oh there's some exercises in between rotation gates IITs and the t gates I don't remember what this section was goings I was going that deep always be more superficial before multiple qubits and entanglement quick exercises they've done I think I think some of these exercises are extra they were not here before which is good actually I think I think exercises definitely helps mmm exercise do help because it depends on the exercise if you're mixing exercise I hate proof proof these or prove that and then kind of something that to be more practical it's also it's again it's a question of the audience right tangled states Jessica Parker took a look at Pro universality I think we went we didn't even do this last time more circuit identities and I think here we talked about tougher gate and the control rotations at off the gate so oh oh yeah okay so the the problems the icon here basis yeah those were all the hello kiss keep game there's something else in here beginning with beats puzzle one know what computers are made of beats and it gives with c-plus thing didn't do it China update see what happens is this is this sort of inspired on the hello quantum on the hello cotton game from probably inspired to be on this hello quantum game from Dakota ku this game here which is sort of a puzzle based okay different puzzle different exercises interesting and then you jump into the algorithms part that's all we spend most of the sessions recently going through the different different algorithms if that would again it's getting really lucky that can't even move my mouse come on live broadcasting is not my thing I already see one teleportation jutsu that's the same I think super-dense coding Deutsch user or however it is pronounced the problem yeah I know I don't think much has changed in here the gift he definitely changed I really that was a good that's good there's a good change so I don't know I don't wanna I don't honestly want to steal more of your time but I have the impression again like I and you probably feel that but it's it's one of those things where I don't really feel like that's cool yeah that was cool animation it's it's even difficult for me design is thinking about it yesterday quite a lot like how could I approach the feedback in general it I don't know it just feels it feels good but it doesn't feel good it's it's a it's a weird mixture I want to take a look at the guide wanna take a look back at what we did and what's kind of coming ahead while simulating molecules using vqe some of the maths in here variational forms okay parameter optimization structure conversational forms now it feels it feels you know it kind of feels everything is just you know kind of packed in there and it's really working progress but it's good I mean it's not that's that's not bad necessarily it's it's a work it's it's sort of an alive thing kind of you know you're kind of evolving the book as people go through it and give you feedback and that's I think that's that's that's really good the error correction part I did in another video I did cover this in a video because I needed it for for a review that I was doing on a paper and I found it I found it I found it goodness well I find it really nice well you're basically we're basically walking the readers through a way you can correct some of the or you can correct them measurement errors right by basically building a model of those and then applying the model to your simulation so that shows me more of what's kind of you know what's more powerful about Kiske or what's kind of building in kiski that you might not find in other in other languages maybe I don't know and the quantum volume part I also did yeah I don't know I'll probably I think I think I'm probably going to wrap it up all honestly if I have to if I have to summarize the book in kind of like a sentence or two I'd say this feels like is coming from there's a really good initiative but it doesn't feel like it's everything really well glued together or it doesn't really feel like you know this is something that you're you know you go through and you're like okay I got I got a good feeling off of all the topics that we've got to cover or not but it's again it's I think that's normal so it just doesn't give me that it doesn't it gives me sort of a weird feeling at the moment that I'm not able to express either in words that probably make sense to you anyway so yeah so I say I'm gonna I'm gonna end up here unless any anyone is kind of you know has any questions or anything right now in the chat I think I will basically what's this estimating pioneers in Quang Tri situation okay so that was what was released in pi day about estimating pine I don't know I'll definitely be doing some of those parts at some point because of all the videos and other things that I other projects that I'm working on and I'll probably need maybe to go through some of those concepts but I I think I prefer if this is kind of more driven by by the need to understand a bit more you know whatever challenge I'm trying to I'm trying to go through then rather just the textbook in itself so that's that that's the main reason that I that I'm really stopping the sessions that it feels like just doing it for doing it it's not kind of it's not kind of taking anymore with me a qaq I am gonna try optimization powers isn't killing I to do that's definitely something that I want to touch in again at some point for example anyway if there's no other questions to see this some people watching but if there's no other questions in here then I'm just gonna just gonna wrap it up it's in fun and I think the the textbook has a lot of potential and the content that's in there it's it's it's really good but it's still as I said it kind of feels like there's something off to it that you know if I basically managed to put some you know some of my thinking straight then I'll definitely share share this with you in one form or another so good having said that I think it's time to go and stay tuned for more in the channel and I hope you enjoy their next and the sessions are you know following up with Sam and other people in intellectual creating sessions yeah cool then have a good day and have fun you |
Thanks. I really enjoy these videos. |
cool so the next step is to basically make sure that we drag and drop properly and we actually clear that stuff here so I'm gonna I think I'm gonna comment out the clipping for now because that's that's fine it just it doesn't seem to work well because it's clipping all over the place um so I think I'm just gonna comment out this stuff oh no can I is there a shortcut to comment something out uh oh wait no but it's fine I'll figure out later I know I can just do this probably and I'll be fine um and then at least we don't get the clipping uh but still uh still we're trying something nice in here so um what we want to do is uh obviously uh you know this is this is the kind of the issue right so um the idea would be well I guess uh uh can I give so Mr Ghost Rider can I give a position object uh we'll probably do it like this and then does the initial position and then I guess what we um what we need to do probably is uh is essentially say um so so this this will be this would be a last position um and so what we want to do is we want to clear we want to clear last position and then uh these these should be um [Music] I guess exactly I guess that makes sense this is a suggestion so this becomes the last position now and we we gave it here yeah I mean maybe whoops can I how can I do multi-line select uh what have I done what is this doing ah I guess I'll just do position I should learn the shortcuts at some point it makes it easier yes so this is position and then um we clear the position take the position and then uh you know basically this is a new position let's see this is working well no it's not working well um so I'm updating the position and painting the position right um so uh can you help me here you help me understand why uh no not moving um also dragster let's go I have implemented that it's now let's move but that's what I have already done position so I've got this position uh then it then it's like that oh my God oh my God so I found it before uh Ghost Rider um yeah but that's not working either so it's not because I have the new position and I wanted to basically um with and hide um so draw the square so essentially and yeah so I mean that's already implemented but I can't seem to drag the square to the able to run the square um variables you're using make sure that the variables you're using to store the mouse coordinates such as drag start I set the right positions when the mouse is clicked additionally make sure the track Starfire is set to True um that's not true uh oh wait a second what are these okay so check if the mouse is over the square uh yeah that is the no that is fine right because this is just the initial event and I think um and I think if it's just like this is debugging this is true debugging no but it's not clicking actually oh I might wanna I might want to just say position x uh foreign truck start X it's the offset okay yeah yeah I mean it definitely shouldn't be 30 but it's like position X plus uh position wave I guess um I guess that's the point right then here uh it's probably going to be basically y position y trying to start the Y position Y and then hide thing that should be fine yeah now it works cool test um I mean it works the the at least it goes in here so but I still can't drag it right so uh so basically that when you do miles down you calculate these these check that it's inside and then sets these variables to true and then a drag element clear position uh yeah that's actually wrong uh of course that's definitely wrong why why have I why would you why have it done like that thought that okay there we go that's awesome man okay cool um but I still leading the lines um it's definitely building the lines which is not nice uh how can I how can I make sure the lines are not being cleared maybe I just have to repaint the whole scene right clean the canvas uh no that's already there what I mean is uh uh yeah probably what I probably need to do is uh can uh you refactor uh the horizontal line creation into a function so cool so basically there is something that uh yeah basically we'll do it like this and then uh and then I guess I guess after each of these then you kind of repaint basically control horizontal lines so I guess there's no way to avoid that right right yeah nice look that looks really nice perfect so um yeah that's nice now I just need to be able to drop it where it is and and that's it and then we can work on the clip um I know that uh I guess we need a function Mouse app how um the code for the mouse app function let's see if actually understands what is it supposed to do that's funny because I I'm not oh that's pretty neat actually yeah it's actually fairly simple oh man that is actually it does feel like cheating in a way oh yeah but I know what's happening so I need to do the same here for the mouse app Mouse up and then Mouse up uh yes and uh another thing that is happening is that I'm not painting the line I'm not doing an initial line paint which I should um here should probably refactor all that into like something that just Paints the whole um the whole circuit you know uh and there you go there you go nice it's like how long have I been recording now 11 minutes like this is this would have taken me to sort of take me like I'm telling you I'm not kidding like two hours uh to actually get that done like these now I just need a clipping part um because we definitely want this now to just move we need we need we want this to be able to always the movement to be constrained uh within the lines okay um so we want the movement to the constraint with thin the lines so you know what I'm gonna I'm gonna uh I'm gonna delete the uh I don't like that I'm gonna delete the clipping um and I'm gonna ask Ghost Rider uh to help me there so how can I constrain the movement of the square to be uh along with the lines it's pretty cool how much context actually gets from these so you can constrain the movement of the square I don't know about that [Music] um okay let's see okay so let's see what what has it uh I'm curious whether that sort of like typing effect it's um it's a design decision whereas like uh because it definitely like if it will be more like boom that's the answer you know it will feel a bit more like an actual Channel experience I know it's cool that you see that uh happening um but I'm not so sure if that's so useful to see the the things being printed out um by adding some work out so this code should check if the mouse is within the bounds of the canvas then check which line the mouse is the nearest to that's a crazy man um and then move the square to the position so what is what has it done here so um if you compare these so basically clear feel uh repaint the lines through horizontal lines okay so what is it doing um I mean I I to be honest I think that actually that's not correct here and I I think that's something that actually shouldn't be part of the function so it hasn't really understood that but sorry the score is All Is Over The Line check which line [Music] um and then actually it does this no but I mean that's that's no that's fine I think that's uh this is definitely wrong right I should definitely it keeps clearing the original position that's funny um okay so let's just copy this part let's just copy this part and um and paste it in here so this I'm sure I'm sure this needs to be these and I probably can optimize that code somewhere else to make it less so you clear you clear the position and then you just I'm assuming let's see if this competition or calculation has been done correctly oh that's odd okay that's very odd it's painting somehow all the lines uh uh so it seems like the code is hitting uh everything below the mouse position that's not what I meant um offset left what is this 500 okay so the the lines are painted like in these intervals and then there's a 500 here so that's why we'll need to turn these things into constants I guess but so so you can replace the following line with this line uh if it's an actual condition right so check if the square is Over The Line uh now it's uh yeah okay I think this seems to make sense because it's actually considering the y coordinate I I haven't I'm not I'm not going to check the detail but maybe I should because the thing is it that might actually improve the experience okay at least it doesn't paint it but it does not doesn't uh feel and then what does it do it it actually painted it seems like it does it it does it I think yeah I think the the solution is um so but I think what's happening here is what I wanna it's still basically under um uh it's still basically I think I think that doesn't make a difference because uh it seems like it's starting to paint it always okay so it keeps the X but then it forces a specific why is that correct uh but then this this whole thing is here I think that is the problem if I comment this out um I think because that that keeps painting the whole thing where the mouse is I think that's the okay no okay it's gone um uh what I wanted to achieve is it looks better but uh there's a square and a steel pretty much following the mouse and stand it off looking to the closest line that's what I that's what I wanted to do so now it's going to suggest me to adjust that um how so it keeps this stuff and then it's telling me to change that's fine uh so that's what's missing right so kind of clipping these [Music] but you see that's kind of the same the issue that I mentioned like it seems like it is not finishing that but okay I'm assuming that's that's what it what it wants me to do is um it just wants me to add I need to find a better way I think I can maybe I should just put Ghostwriter I'm not so sure where it's better it's a fairly big screen but like it keeps bothering me to the you know the formatting of the code no matter how I do it like if I I'm focusing on the on the on the code or on the Ghost Rider chat um check if the mouse so we'll basically kind of goes here so so let's see still it seems to work seems to work but it seems to work initially because it does clip it like if I'm let's say I'll leave these is close so it's in between if I click here it's clipping into the bottom one if I click here um [Music] some mouse line distance yeah but where is that calculated that's not calculated anywhere have I oh sorry guys this needs to be here obviously um so that's what happens when you don't copy the whole thing yeah but it's still not doing the same but if I'm here and I okay but I think we're going on the right I think we're going the right track so it's just a matter of fine-tuning these I wanna I want these things to clip and then the next step would be to make sure that they also kind of clip or the dislike some certain horizontal positions to you know to achieve this effect here right that there's the same kind of like that is not a continuous thing but you you kind of force in a way to have this layout where things are always like well aligned so see you in the next session |
so over i decided i i decided i'm gonna i'm gonna jump into uh i'm gonna put that aside the dewy stuff for a second uh i'm gonna learn about the tensor networks quantum inspired optimizer that they talk about in here as well i don't know i don't know why i just it's the other one that performs the best if you take a look at the results i think it was page seven so no that was an outlook no here tables and so you have the tensor networks and the d-wave hybrid those are the one the ones that perform the best now it says the profits computed by the different methods those are the parameters uh so the d wave is a clear wiener i think already if you make the jump between the xl solution and the xxl one um it's quite impressive to be honest the vqe and vk constraint didn't make it past the air we can certainly make it past the m um yes i'm curious they seem to perform the same or really close even for science m is better right like it's it's computing more profits for size l as well that is actually a big difference like there's like a you know 20 points here um but then for this size yeah it's interesting i'm curious curious you know interesting to figure out why um and what are these the run times yeah well that is actually in seconds 171 seconds and like a hundred thousand seconds what is that like 100 100 000 seconds seconds two hours that's like 27 hours wow okay cool um but for smaller problems it's not that bad right like 120 seconds and here it's like 26 000 seconds versus 52 seconds but the solution is it's quite better because it's uh more profits so i want to i want to understand maybe kind of you know what is it the uh that they do and then i'll i'm assuming i'm gonna have to dig into uh into the concept uh what am i doing zoom out reset oh that actually zooms out the page i thought it was another kind of zoom anyway uh where where do i find this though next steps this is the so where do i find up to 40 who is five what is this i talk about the cube all the time uh gonna find that a native application of eqe was very rather limited for this problem our tn's code is able to handle its tn tensor networks cubic variables using a macro pro without problems even though the performance can be improved in a number of ways we believe that a highly optimized code in c plus plus fully parallelized on an hpc cluster should be able to handle really large problems very efficiently [Music] but do they explain here what are they doing let me see next steps more constraints what i'm looking for is i'm looking for a description oh tensor networks there you are so let's first kind of go off of the paper and and then we'll dig into the theory that's needed or if any theory is needed for these what is here okay perfect so tns are representations of complex quantum states based on their local entanglement structure okay uh i like that does this mean because i've been playing recently with this idea of um representing like a quantum state as in just a collection of the individual qubit states and a set of maybe edges or connections between the qubits that basically specify how the entanglement is defined or how the entanglement is sent you know to give you to give you an example what i what i what i mean is i'll open pain pain so what i mean by this is oh wait a second that's not it tells me it's not connected crap oh whatever okay sorry what i what i mean is i'm gonna be able to paint today awesome uh yeah whatever so if yeah if if i have like for example if i do like let's picture try to picture the bell state okay so you have two qubits um and they're both separate they're both like in the zero state now the first key the top qubit um you apply a harmar gate so it gets into the zero plus one state plus state and then you apply a control not gain what this would do is it would basically um this your state representation would now be a zero plus one and a zero right but then the zero plus one basically um gets split into two branches one has the zero of of of the zero component of the state as a root and then it connects to the second qubit that has also a value of zero this means if the first qubit is zero the second is also going to be zero and the second part of this of the state of of your top qubit was used to be one so now this one is connected to uh to the qubit one so to this to the next qubit but that now has the value one so you're saying basically you've got two branches one it's a zero and a zero and another one is one one and those are the two values that you can have and i don't know if this is what uh so we're in page six if i what are these references uh again was 18 and 19 18 and 19. oh that is uh annals of physics 349 and nature reviews physics one where is this going to take me i don't know tensor networks for complex quantum systems okay except for cookies but that's under that's behind the pay wall uh they are quantum representation of many body states based on their entanglement structure symmetric tensor network states enable more efficient simulation methods and description of pharmac systems lattice gotch theories topological order because my point is that will be really efficient to simulate a system like that as long as you don't need uh to know the state the full state vector at any point in time right because basically what this means is you only need to keep track of the qubit so that doesn't have that exponential component to it and then and then the entanglement which is also going to be something that is a function of your you know control gates that you have somehow like you're not going to have really complex entanglement structure but you don't have like you're not having a full state representation in there right if you have a normal matrix in your linear algebra the standard linear algebra model you're constantly keeping a you know the full state of the full quantum state which basically gets bigger and bigger exponentially and so um let's go ahead and click file to take for instance the system of n qubits any wave function of the system can be described inefficiently just by giving its two to the power of n coefficients in the computational basis as such these coefficients can be understood okay so as such these coefficients can be understood as a tensor with n indices where each index takes two possible values say zero and one okay so so this is a tensor it's a nine dimensional tensor or no no ah okay sorry no no it's a it's a it's a tensor with n indices it's a vector okay it's a vector basically right yeah we could then think of replacing this huge nasty tensor by i like the word nasty in the paper here by a network of interconnected tenses with less coefficients see figure 1 for an example the coefficient of the quantum state of n qubits is a tensor with exponentially many coefficients in the system size the inner structure of the tensor is that of a tensor network which is a network of tenses connected by ancillary indices that take into account the structure and amount of entanglement in the quantum state we represent these here using diagrams where shapes correspond to tenses lines to indices and lines connecting shapes to contracted summed common indices the tensor network on the right hand side is an example of matrix product state so these okay not so sure understand this but uh this construction defines a dn and it depends on uh like its polynomial complexity uh parameters only assuming that the rank of the interconnecting indices is upper bounded by a parameter d which is called bond dimension similarly interconnecting indices in the network are also called bond indices and provide the structure of the many body entanglement in the quantum state any d bigger than one provides an entangled quantum state okay i don't know exactly if this is what i'm what i was talking about though as is well known in physics tns are a natural tool to solve optimization problems people have been using them as a nonsense to approximate low energy eigen states of hamiltonians and many algorithms have been invented to this aim the idea here is that by mapping optimization problems to a hamiltonian eigenvalue to hamilton and eigenvalue problems as done in quantum annealing we can then use the huge machinery of tm techniques and algorithms to solve the position problem at hand okay so it's good so that's cool what it means is yeah basically the same hamiltonian you calculate for the cubo would kind of be the starting point for a tn algorithm our case we implemented an optimization strategy over the so-called matrix product states nps the assembly of states has been tested already in a variety of algorithms for many physical examples moreover in order to improve the performance we also tailored our optimization to the specifics of our problem um as a first step so data preparations the first step we benchmarked our different algorithms for the optimization problem using random data starting real uh oh sorry that is actually already uh that's not part of the tense of tensor network so what this is already um section four right uh where it's like data preparation so this is what they talk about like the reduction of the dimensionality and all that stuff and then results okay but they don't go into the into the details of of what if you use for the tensor networks but we can see if we can try to figure this out they used okay they used what they call the matrix product states whatever that is let's try to see [Music] matrix product states product states we'll also google for uh introduction to tensor networks that's in our products a practical introduction uh you know you're screwed when your introduction and tutorials are actually published in archive in in r in the archive like in the r key for however this is pronounced and it's not like a it's not like a regular blog post oh is this from roman as well it's really nice it's all over the place uh is it long 51 pages cool um this is a partly non-technical introduction to selected topics on tensor network methods based on several lectures and introductory seminars given on the subject it should be a good place for newcomers to get familiarized with some of the key ideas in the field especially regarding the numerics after very general introduction we motivate the concept of tensor network and provide several examples we then move on to explain some basics about matrix product states and projected entangled pair states selected details and some of the associated number and numerical methods for 1d and 2d quantum memory systems are also discussed so this looks like a good place to start to be honest [Music] a bit of background white tensor network sensor theory mps okay so it gets there fairly quick but this is this is a concept that seems to be really just tailored to quantum computing or quantum quantum mechanics at least right i thought it was a more more of a generic uh generic idea so a matrix product state is a pure quantum state of many particles within the following form it's the sum of traces of amount of what complex square mattresses in this is as i go over states in the computational basis for qubits in the si01 for queueings in the tier level system is zero one d minus one the computational basis yeah but that's the that is really takes the whole computational basis so that actually the size of these is really exponential right one method to obtain an mps representation of quantum state is to use the schmidt decomposition n minus one times alternatively the quantum circuit which present antibody state is known one could first try to obtain a matrix product operator representation of the circuit the local tenses on the matrix product operator will be four index tenses uh the local nps sensor is obtained by contracting one physical index oh so the elements are actually kit w state the superposition of all the computational basis states of hamming weight one i don't know what is this aklt model um matrix product state okay uh i don't know for states that are translationally symmetric we can choose okay i think i better just i think i should just uh go through these let's zoom in a little bit more introduction during the last years the field of tensor networks has left an explosion of results in several directions this is especially true in the study of quantum antibody systems both theoretically and numerically but also in directions which could not be envisaged some time ago um such as its relation to the holographic principle and the ada cft correspondence in quantum gravity nowadays tensor networks is rapidly evolving as a field and is embracing an interdisciplinary and motivated community of researchers this weapon tends to be an introduction to selected topics on the ever-expanding field mostly focusing on some practical applications of the matrix product states and projected entangled pair states it is mainly based on several introductory seminars and lectures that the author has given on the topic and the aim is that the inexperienced reader can so that's me can start getting familiarized with some of the usual concepts in the field let us clarify now though that we do not plan to cover all the results and techniques in the market um but rather go uh but rather to present some insightful information in a more or less comprehensible way sometimes also trying to be intuitive then it fits the channel together with further references for the interested reader in this sense the paper is not intended to be a complete review of the topic but rather a useful manual for the beginner cool uh several sections a bit of background on the topic motivation introduction to basic tensor network theory such as contractions and diagrammatic notation and its relation to quantum many body wave functions then some generality generalities about the matrix product states and blah blah blah a several strategies to compute expectation values and effective environments for mps and p p e peps can i call it peps both for finite systems as well as systems in the thermodynamic limit generalities and okay let's go through the background understanding quantum many body systems is probably the most challenging problem in condensed matter physics for instance the mechanic the mechanisms behind high dc super conductivity are still a mystery to a great extent despite many efforts all the important condensed matter phenomena beyond london's paradigm of phase transitions have also proven very difficult to understand i don't really understand any of these examples these are topologically orbit phases quantum spin liquids phases of matter that do not break any symmetry okay i don't know the standard approach to understand these systems is based on proposing simplified models that are believed to reproduce the relevant interactions responsible for the observed physics hybrid tj models once a model is proposed and with the exception of some lucky cases where these models are exactly solvable one needs to rely on faithful numerical methods to determine the properties as far as numerical simulations algorithms concern tensor networks methods have become increasingly popular in recent years to simulate strongly correlated systems okay that's interesting so it uh strongly correlated systems in these methods the wave function of the system is described by a network of interconnected tensors intuitively this is like a decomposition in terms of lego pieces um and where entanglement plays the role of the glue amongst the pieces yeah but that's exactly that's what i'm that's what i was playing okay so that's what i was let me see if i can get let me see if i can get the pen to work give me a second okay i'll pause the recording for a second i can't see the video i'm sorry um also i i promise the quality of the video will get better at least the lagging and it is going to be better i figured out what the problem is but i forgot to set it up before i start this video so you will have to stick with these uh with these a bit more um nevertheless more precisely tn techniques offer efficient descriptions of quantum many body states that are based on the entanglement content of the wave function mathematically the amount of structure of entanglement is a consequence of the chosen network pattern and the number of parameters in the tensors so dna okay i like the i like the analogy so you put a bunch of dna strings together sequences together you have a person you put a bunch of tenses together and then you have quantum state so it seems that tensor is the fundamental building block of the quantum state okay um the most famous example of a tn method is probably the density matrix renormalization group the mrg introduced by steve white in 1992 one could say that this method has been the technique for of reference for the last 20 years to simulate one the quantum line lattice systems however many important breakthroughs coming from quantum information science have underpinned the emergence of many other algorithms based on the ends it is actually quite easy to get lost in the super names of all these methods t ebd folding algorithms peps tensor randomization groups tens entanglement normalization groups oh my god yeah that's a bunch of them a nice property of tn methods is their flexibility for instance one can study a variety of systems in different dimensions of finite or or infinite uh sizes with different boundary conditions symmetries as well as systems of bosons fermi fermions and frustrated frustrated spins different types of phase transitions have also been started in this context moreover these methods are also now finding important applications in the context of quantum chemistry blah blah blah possibly developing algorithms for infinite size systems is quite relevant because it allows to estimate the properties of the system directly in the thermodynamic limit and without the burden of finite size scaling effects um okay why tensor networks considering the wide variety of numerical methods for strongly correlated systems that are available on many wonder one may wonder about the necessity of tn methods at all this is a good question for which there is no unique answer and what follows will give some of the reasons why these methods are important and necessary let me i don't know if i can thinking about stopping these and then gluing it all together i'm gonna stick for these for this video without with the quality i'm sorry um i don't wanna i don't wanna have to quote things together and and then pause process the video gonna take too much um good all the existing numerical techniques have their own limitations to name a few the exact diagonalization of the quantum hamiltonian is restricted to systems of small size thus far away from the thermodynamic limit where quantum phase transitions appear serious expansion techniques rely on perturbation theory calculations mean field theory fails to incorporate faithfully the effect of quantum correlations in the system quantum monte carlo algorithms suffer from science from the same problem which restricts their application to um fermionic and frustrated quantum spin systems the depth of these staff made it so crazy like how can someone just write about all these stuff tn methods are not free from limitations either but as we shall see there their main limitation is very different the amount and structure of the entanglement in quantum many body states this is this new limitation is in a computational method extends the range of models that can be simulated with a classical computer in new and unprecedented directions um new language for condensed matter physics so tn methods represent quantum states in terms of network networks of interconnected tensors which can i not maybe can i skip to can i skip to to to the actual introduction of these and then go back to that to to to the background it's going so slow that's another theory yeah but maybe it's maybe it's worth going through these i i i don't wanna i feel like i don't wanna skip anything essential uh where was i it just crawled so slowly um this way of describing quantum states so dn methods represent quantum states in terms of network of interconnected tenses which in turn capture the relevant entanglement properties properties of a system this way of describing quantum states is radically different from the usual approach where one just gives the coefficients of a wave function in some given basis when dealing with t and state we will see that instead of thinking about complicated equations we'll be drawing things in our diagrams um financial languages describe quantum states of matter including those beyond the traditional endo pictures and just a quantum speed okay so what is all these about this is a new language for condensed matter physics matrix product state and projected entangled pair state pips for three by three ladies with open boundary conditions entanglement induces geometry imagine that you're given a quantum antibody system this is an interesting title what does this mean that entanglement adds definitely some sort of structure in the state specifying its coefficients in a given local basis does not okay so imagine a quantum antibodies wave function specifying its coefficients in a given local basis does not give any intuition about the structure of the entanglement between its constituents yes i totally agree with this it is unexpected that the structure is different it is expected that this structure is different depending on the dimensionality of the system this should be different for 1d systems 2d systems and so on but it should also depend on more subtle issues like the criticality of the state and its correlation length i have no idea what these things mean yet naive representations of quantum states do not pos possess any explicit information about these properties it is desirable thus to find a way to represent a represented quantum state where this information is explicit and easily accessible as we shall see a tien has uh this information directly available in its description in terms of a network of quantum correlations in a way we can think of tn states as quantum states uh given in some entanglement representation different representations are better suited for different types of states one d2d critical and the network of correlations makes explicit the effective lattice geometry in which the states actually leaves the state actually leaves we will be more precise with these in section 4.2 and this level this is just a nice property but in fact by pushing this idea to the limit and turning it around a number of works having proposed that geometry and curvature and hence gravity could emerge naturally from the pattern of entanglement present in quantum states that's interesting so that entanglement would actually the geometry that the entanglement is creating would eventually then give place to to gravity here we will not discuss further this fascinating idea but let us simply mention that it becomes apparent in the language of tn uh is precisely the correct one to pursue this kind of connection hillbid space is far too large the main is the main reason why tns are a key description of quantum metabolic states for a system of ends means the dimension is 2 to the power of n which is exponentially large therefore representing quantum state of a metabolic system just by giving the coefficients of the wave function and some local bases in an efficient representation the hillywood space of quantum mechanical systems is a really big place with an incredibly large number of quantum states um in order to give a quantitative idea let us put some numbers if n is close to the if n is roughly 10 to the power of 23 the order of the avogadro number then the number of bases states is 10 to the power of 10 to the power of 23 which is exponentially larger than the number of atoms in the observable universe okay luckily enough for us not a quantum state in the hillywood space of a mini body system are equal some are more relevant than others to be specific many important hamiltonians in nature are such that the interactions between different particles tend to be local nearest to the next to nearest neighbors and locality of interactions turns out to have important consequences in particular one can prove that low energy eigenstates of gapped hamiltonians with local interactions obey the so-called aerial law for the entanglement entropy no idea what this means c figure 3 does make it easier this means that the entanglement entropy of a region of a space tends to scale for large and north regions as the size of the boundary of the region and not as the volume and this is a very remarkable property because does this mean that this that in in smaller spots you have higher entanglement like that entanglement is more local and this is a very remarkable property because the quantum state peaked at random from a mini body hilbert space will most likely have an entanglement entropy between subregions that will scale like the volume and not like the area in other words low energy states of realistic hamiltonians are not just any state in the hilbert space they are heavily constrained by locality so that they must obey the entanglement area law should go back to this at some point by turning around the above consideration one finds a dramatic consequence it means that not any quantum state in the hillwood space can be a low energy state of a gapped local hamiltonian only those satisfying the area law think so the the many fault containing these states is just a tiny corner of the gigantic kilogram space um so this is the corner of relevant states so here's where the good news come it is the family of the tensor network states the one that targets this most relevant corner of states okay moreover recall that renormalization group methods for many body systems and to precisely identify and keep track of the relevant degrees of freedom to describe the systems as it looks just natural to devise rg methods that deal with this relevant corner of quantum states in fact the consequence of consequences of having such an immense silhouette space are more or even more dramatic for instance one can also prove that by evolving quantum anybody system a quantum antibody state a time with a local hamiltonian the manifold of states that can be reached in this time is also exponentially small the manifold of states that can be reached and this time is exponentially small so the vast majority of the hillbid space is reachable only after a time evolution that would take an exponential amount of time this means that given some initial quantum state i like the graphic um most of the hilbert space is enrichable in practice to have a better idea of what this means let's put again some numbers 10 to the power of 23 particles by evolving some quantum state with a local hamiltonian reaching most of the states in the hillywood space would take 10 to the power of 10 to the power of 23 seconds which is and the age of the universe is 10 to the power of 17 seconds this means that we should wait around the exponential of 1 million times the edge of the universe to reach most of the states available in the hillwood space okay maybe that was not that relevant to go through all this but it was nice um this is why the hillary space of a quantum antibody system is sometimes referred to as a convenient illusion it is convenient from a mathematical perspective but it is an illusion because no one will ever see most of it tensor network theory let us now introduce some mathematical concepts in what follows we will define what a tensor network state is and how this can be described in terms of tn diagrams we'll also introduce the tn representation of quantum states and explain the examples of matrix product states for 1d systems tensors tensor networks and tensor neural diagrams a tensor is a multi-dimensional array of complex numbers good the rank of a tensor is the number of indices this rank zero tensor is a scalar rank one tenses is a vector and the rank two tends as a matrix an index contraction is the sum over an index contraction is the sum over all the possible values of the repeated indices of a set of tensors for instance the matrix product uh let's process this so a i guess alpha better and and i guess those are the [Music] indices is the contraction of index beta which amounts to the sum over its d possible values one can also have more complicated contractions such as this one i'm not sure i understand this so you have vector alpha matrix alpha beta alpha better meaning being the two dimensions the matrix product so you're multiplying two matrices and you're doing a summation in there as part of the matrix no is the contraction of index better which amounts to the sum over its d possible values maybe i need to see an example for this to make sense where for simplicity we assume the contracted indices and can take different values as seen in these examples the contraction of indices produces new tensors in the same way that the product of two matrices produces a new matrix uh so contraction is just a word for product in here indices are not instances that are not contracted are called the open indices a tensor network is a set of tensors where some or all of its indices are contracted according to some pattern i guess what this is go where this is going is that that the relationship between the tensors will so this contraction kind of gives you the pattern to then really built up the the full state i think in the more classic sense uh contracting the indices of a tn is called for simplicity contracting with the n the above two equations are examples of tn in equation one the tn is equivalent to a matrix product and produces a new matrix with two indices in equation two the tn corresponds to contracting indices uh intenses a b c and e to produce a new rank for tensor f with open indices blah blah blah blah in general the contraction of at the end with some open indices keys there is as a result another tensor and in the case of not having any opening this is the result its color this is the case of the scalar product of two vectors so this is just means feels like a fancy generalization for the product and and kind of calling the product the product being a tensor network so you've got tenses and they are related to to each other and this relationship is the product that's what it that's what i seem to understand it's kind of like a bit more abstract than it's like a an abstract general an abstract an abstraction of the product and so you have a vector product that that that you know leads to a scalar so what what i see is a complex number ranks here tensor a more intrinsic example could be this one here yeah okay where only this is our contracted and the can and the result is again a complex number f um okay so once this point is reached it is convenient to introduce a diagrammatic notation for tenses and tns in terms of tens on our diagrams in these diagrams tensors are represented by shapes and indices and the tenses are represented by lines emerging from the shapes ah okay a tn is represented by a set of shapes interconnected by lines okay so these lines will basically then you will connect them so you're knowing what indices you're contracting and what indices are free the lines connecting tenses between each other correspond to contracted indices yeah whereas lines that do not go from one tensor to another correspond to opening this is in the tensor network tensor network diagrams so scholar scala a vector a matrix a rank three tensor tensor using tn diagrams it is much easier to handle calculations with the n for instance the contractions in equations one two three four can be represented by the diagrams in figure six also tricky calculations like the trace of the product of six matrices can be represented by diagrams as in figure seven the trace of the product of six mattresses from the tn diagram the cyclic property of the trace becomes evident this is a simple example of why tns tn diagrams are really useful unlike plane equations so you have a is a matrix product b is construction of two tensors contraction of no uh contraction of four tensors with four open indices yeah so the open indices are going to tell you how many dimensions you're going to have left in your uh in your tensor so here you're going to have two dimensions left it's a matrix product here and a half non-left it's a vector product and the contraction of four tenses without opening this is a trace of the product of six mattresses what is the trace of the product of mattresses traces the is the the the trace is as defined to be the sum of elements in the main diagonal and it's invariant tn diagram is allowed to handle with complicated expressions in individual way in this manner many properties become apparent such as the cyclic property or the trace of a matrix product in fact you could compare the language of tn diagrams to that of fema diagrams in quantum field theory surely it is much more intuitive and visual to think in terms of drawings instead of long equations hence from now on we should only use diagrams to represent tensors in the ends there is an an important property of the ends uh of a t of the end that we would like to stress now namely that the total number of operations that must be done in order to obtain the final result of a tn contraction depends heavily on the order in which indices in the tns are contracted see for instance figure 8 both cases correspond to the same overall tn contraction but in one case the number of operations is d4 in another one is d5 this is quite a quite relevant since tm methods seen since in tn methods one has to deal with many contractions and the aim is to make these as efficiently as possible so figure eight what with figure eight ah interesting so if you contract those first if you contract those first and you can then you're left with these but how is that five uh one because this has like one two three four five i get it because this tensor here is got only two dimensions and those tensors have three dimensions three yeah so if you do this first you're only you have like one two three no how is this counted one two three four only involved and in here you have one two three four five six i'm not so sure i guess if you sum the overall is like six minus one is five and you have one two one two three four five and one is four i think it's gotta do with these to minimize the computational cost of a tn construction so osi that's quite relevant for this finding the optimal order of indices to be contracted will turn out to be a crucial step especially when it comes to programming computer codes to implement the methods to minimize the computational cost of a tn construction one must optimize with different possible orderings of pairwise contractions and find the optimal case mathematically this is a very difficult problem though in practical cases this can be done usually by simple inspection cool okay let's break the wave function into small pieces that's where i want to go um see if this is matching okay so let us now explain the tn representation of quantum antibody and of quantum many body states for these we can see the quantum antibody system of n particles the degrees of freedom of each one of these particles can be described by p different states hence we're considering systems of np-level particles for instance for a quantum antibody system such as the spin half heisenberg model we have p equals two so that each particle is a two level system or qubit for a given system of this kind the wave function that describes its physical properties can be written as the sum of blah blah blah blah yeah yeah so it's that it's it's a tensor product of yeah of course of all the different qubits right and qubits yeah yeah once an individual basis for the states of each particle has been chosen in the above equation are ah so those are different bases uh c i 1 i 2 to i n are p to the power of n complex numbers independent up to a normalization condition yeah tensor product of individual quantum states for each one of the particles in the many body system tensor product of individual quantum states yeah it's a tensor product of the individual qubit states that's the point right so if you have a two qubit system it's the tensor product of the qb two qubits that gives you the four dimensions uh of the state uh now notice now that the p to the power of n numbers that describe the wave function can be understood as the coefficients of a tensor c with n indices tensor c with n indices so you've got a tensor okay and then this is at the qubit so you've got a tensor that has as many dimensions as qubits where each of the indices can take up to p different values exactly 0 1 in the case in this case p is like the equals 2. so this is a tensor of rank n with p to the power of n coefficients tensor of rank and um this really readily implies that the number of parameters that describe the wave function is exponentially large in the system size uh let me try to see if i understand as well so with ending this is where each of the indices can take out to be different values okay it because it's a bit more abstract i'm not too sure what this is telling me is this the density matrix what is this sort of trying to represent or is it the or is it that because this n indices and n is the number of cubiets and each qubit can have different values like zero one right meaning not each cubit but like each okay i don't know uh tensor with rank n so with two qubits you have a four with two qubits you have a four elements matrix two by two to represent the state yeah okay and the parameters that describe the wave function uh specifying the values of each of one of the coefficients of a tendency is therefore a computationally inefficient description of the quantum state of the many body system one of the aims of tn states is to reduce the complexity in the representation of states by providing an accurate description of the expected entanglement properties of the state this is achieved by replacing the peak tensor by a tn of smaller tensors by tnf tenses with smaller rank for some examples in the diagrammatic representation check figure 9 this approach amounts to the composing the peak density and hence the state phi into fundamental dna blocks namely a tn made of tenses of some smaller rank which is much easier to handle i want to see an example the final representation of phi of psi in terms of tm typically depends on a polynomial number of parameters thus being a computationally efficient description of the quantum state of the many body system to be precise the total number of parameters and taught in the tensor network tension over the composition of tensors c in terms of a and nps with periodic boundary conditions b a paps with open boundary conditions and c and arbitrary tensor network where mt is the number of parameters for intensity in the tn and hence the number of tensors for atn to be practical and tension must be sub-exponential i want to get to an example to give a simple example okay to give a simple example consider the tn in figure 9.8 it's an example of a matrix product state uh we'll be discussing this next section okay now that's not what i want to do i want an example of a state and it's the end i want to see the tn of like a two qubit system for example part of the magic of the tn description is that it shows that these p p to the power of n coefficients are not independent but rather they are obtained from the contraction of a given t n and therefore have a structure nevertheless this efficient representation of a quantum antibody state does not come for free the replacement of attends to c by a t by a t and involves the appearance of extra degrees of freedom in the system which are responsible for gluing the different dna blocks together these new degrees of freedom are represented by the connecting indices amongst the tensors in the tn the connecting indices turn out to have an important physical meaning they represent the structure of a made body entanglement in the quantum state and the number of different values that each one of these indices can take is a quantitative is it the connecting in the system and the number of different values that each one of these indices can take is a quantitative measure of the amount of quantum correlations in the wave function they're called bond or ancillary indices and their number of possible values are referred to as bond dimensions the maximum of these values which is called both d is the one dimension of the tensor network to understand better how entanglement relates to the bond indices let us give an example imagine that you're given a tn state with one dimension d for all the indices inside just one figure 10. uh this is an example of at the end called projected entangled pair state which also will also be further analyzing the forthcoming sections calculate the entanglement entropy of a block of linear length l i don't want to do this i want to i want to see an example generalities mps and ppeps generalities can i see a tensor tension network an example quantum state in a nutshell it's probably not in a natural images i'm quite curious because it really feels like that's the same idea i kind of stumbled upon and i just wanted to understand what this is it like what this is what i meant there's some fancy images in here quite a competition for god's sake can i just uh in a nutshell it's literally the nutshell quantum legos peps nps dtn from tensors to networks in in the tensor diagram notation the tense is labeled shape such as a box or a triangle with zero or more open output legs and arms pointing up and zero or more open input legs didn't didn't crack only once like made a crack i think he once made like a post about this or they already tweeted about these what are tensor networks and which is the relationship they have with quantum computing yes generalization of matrix multiplication graph where the nodes are tensors and quantum circuits are kind of tensor networks the gates are mattresses specific kind of tensor and the qubits are series of constructions lines connecting the tensors in some cases even the vertical lines in this circuit diagram the c notes control two circuits correspond to some sort of contraction though circumnotation has stayed straight far enough in tensor network notation that is not always the case yeah but that's not the answer can you okay i just want an example that's not relevant doesn't say anything with tensor networks in here no i remember him drawing some stuff like that but anyway i'm looking for an example of a quantum state and it stands a network representation let's set the determinant given the to cubit pure quantum state its concurrence is the absolute value of the following tensor network expression oh wait a second quantum circuit okay so a simple quantum circuit that can generate entangled belt states it consists of two tensors a hallmark gate and a control knob gate denoted by the symbol inside the dashed region the signal in the hammer gates are defined as these where the additional or the addition in the c naught is module two the really should verify that acting on the quantum state the zero circuit yields to the bell state blah blah and acting on one one yields to this state oh so this is a type of this is a tensor network that's what this is saying that the circuit is actually a tensor network copy and xor tensor as one came with the synogate itself as a contraction of two order three tensors interesting bending and crossing wires okay so that goes like transpose partial trace okay so this is already more advanced notation okay so so it's not what i thought it's not exactly what i thought it's basically i mean it is just a way to say you've got a bunch of tensors which are like can be your gates but i'd love to see a great example quantum circuits for epsilon states and cups plastic this is the tensor okay that's way too much i think matrix product states some properties i wanted to i'm probably skipping too much ahead uh so i shouldn't really towards data science quantum circuits with tensor networks okay networks quantum circuits we'll discuss the open source software quimbe for simulating quantum circuits one of the things that makes queens interesting is the ability to perform computations with tensor networks if you're not familiar with teson networks we'll talk a little bit about them you can also check out one of my other articles on black hole machine learning history of tension optics and physics into using machine learning in ai and if you're curious and enjoy digging into into a little code you can also check out my okay queen tutorial so what is crimp quantum circuits are diagrams that closely resemble the staff notation musicians use indicate measurement quantum composer tensor networks suppose you have a quantum circuit that can be simulated in a classical computer efficiently um when we are simulating some quantum circuits efficiently using tensor networks that's now is a graphical representation of tensor which can be thought of as multi-dimensional arrays of numbers around the utensils just a scalar blah blah blah yeah also actually google has a tensor network library interesting google has recently released a library that runs with its well-known tensorflow as back-end while having a library built onto potential flow is a huge plus due to the popularity of the wide spread of tensorflow it says not the most user friendly at present i use different alternative queen opens the software of performing quantum circuit simulation and constructing tensor networks okay excellent implementation what makes queen even better is it uses network x a python package for the creation of manipulation of structure and dynamics the function of complex networks so i might have to install that to maybe try it out for the project as well and machine learning they can be using machine learning as layers in deep neural networks these bridges are computing classical machine learning and new field of quantum machine learning which is quantificational circuits oh come on that's a shitty article that doesn't tell me anything network x is a python package for the creation of manipulation and study of the structure dynamics and functions of complex networks i assume complex networks is the networks of complex numbers but that be they can be quite that can be something interesting actually because because i actually might use these for uh for the stuff that i have in mind in terms of simulating uh or the building a simulator let's quit doing normalized that's not singularity research what is this can someone just give me an example please network x uh black hole machine learning okay so those are different tenses tencent network library tencent networks crimp because there's some fancy stuff looking like here um tensor network with tensorflow or pytorch speed up that's a network that's a network tensor network so so this is entanglement in a tensor network how is entanglement represented with a within a tensor network any family of tensor network states specific entanglement structure given by a graph of maximum entangled states along the edges that identify the indices of the tensors to be contracted so tencent august provide description of a strongly correlated quantum systems based on an underlying entirely structure given by a graph of entangled states along the edges that identified the indices of the local tenses to be contracted considering a more general setting where entangled states and edges are replaced by multi-part apartheid entangled states on faces it allows us to employ the geometric properties of multipartite entanglement to obtain representations in terms of its prepositions of tensor network states with smaller effective dimensions leading to computational savings matrix product states maybe maybe i maybe i really have to go a bit deeper into the mps part of things and i'll maybe understand it's a generalization of of matrix product so i don't know how so is this basically saying you can just you know translate the circuit into a tensile network into a tensor network and then you can find there's some numerical techniques to find ways to calculate to actually calculate the contractions in a way that is computationally efficient but i don't understand how like how is the entanglement there like if i would see an example of the bell state but i think i had it there almost what is an nps let's see what is an nps it's the most famous example of tn states this is because it is behind some very powerful methods to simulate 1d quantum one-dimensional quantum body systems most prominently the density matrix randomization group but it is also behind the or the well-known methods of just time evolving block destination mps are tn states that correspond to a one-dimensional array of tensors such as the one one-dimensional array of tensors such as the ones in figure 11 in an mpa in an mps in a mps there's one ten separate side in the many body system the connecting bond indices that glue the tenses together can take up to d values and the opening this is correspond to the physical degrees of freedom of the local hillbid spaces which could take up to b values ah wait a second so two examples of mps the first one corresponds to nps with open boundary conditions and the second one within place with periodic boundary conditions as example i don't know if this is basically what i mean tends to network bell stayed deep quantum ai so um tensor networks initiative and let's show um all tenses are fine mps could be different yeah i don't know some properties of mps that's not that's doesn't seem to leak to anything that i can currently understand and i'm starting to have too many open open tabs [Music] tensor network stance networks it seems to be a b pin so it's generalization of the product but but what is he that i can like how does it show the entanglement for example like i'd like to see this and now then i would just you know probably finish the episode finish the video let me close all these so tensor network um isn't isn't this isn't this the same that isn't this like similar to what i saw from like uh what is what is it the the ah well it wasn't twitter was it it wasn't twitter uh oh sorry i even have it i have it in the discord give me a second i have it somewhere that discord uh because i shared the article from i forgot her name and her twitter handle you have to change the encoding stuff this is socially what's new so math i think it was here yeah yeah this is what i mean did she mention tensor networks in here oh yeah an actual mathematical picture is found in the tensor tensor network diagram representing representation of icpd okay just like these okay but then this is not so this is not exactly uh mattresses is tensor network diagrams so this is a matrix node okay so maybe is it is it the long post no oh i saw this image okay i saw this in the in the results so that's the okay so that would that probably have been a better introduction uh which is a matrix yeah no but i think actually i think i get the concept right so it's just a graphical way to play with mattresses or with tensors and then how you're multiplying them and how you're gluing their pitches together some matrix multiplication is a contraction drawing isometric embeddings as triangles an isometric embedding is a linear map from a space v into a space average of larger dimension that preserves the length of vectors such as such a map satisfies blah blah blah uh but you certainly can't squish all of w in onto little v then expect to undo the damage after completing v back in w okay so this might be something to explore as well symmetric not symmetric the transposable matrix can be represented by by reflecting its picture matrix its transpose so the symmetry of a symmetric matrix is preserved in diagram uh-huh interesting okay smallest base launcher space smaller space matrix factorization has nice features unitary mattresses and hence isometries enhance triangles the matrix d is a diagonal which i like to represent by a diamond ensured matrix factorization is the composition of a single node and i guess that's where entanglement but so the way that i understand these is uh so this is just a so it's a generalization of the matrix of tensor multiplication and but i still don't really understand how does these represent or show you the entanglement more than like what you could see in a circuit where you basically see the control not gates and you know that the chances that there are chances there's entanglement between these two right but yeah i get it it's not what they're saying the point is if you see a state written in in a way that it's like uh it's just this just the the the wave function you don't see the entanglement whereas in a circuit you're going to see the entanglement more because you can see where the control knob gates are but i i want to go even even farther than that and say um that doesn't tell you doesn't really tell you a lot about an agent about the nature of the entanglement in particular right like without knowing really what are the states like if you have maybe i'll just open the note the note pad and it's going to make it easier i got to fix the pen i got to fix the anyway so if we have qubit zero in the state zero plus one let me just zoom this in a bit more and we have qby in the state zero plus one right so in the first step you have cubic zeros in the state zero cubit one is in the state as the state one and now you have these after applying uh after applying like um you know automarts right like uh to watermarks or how about a layer applied and now comes like uh a control knob from from zero to one uh and so with these sorry no no no no this is still zero so it's a haunted mark for cubic zero and so now you have a con a control not like that's applied like these and so what you have is that q so basically what you have now is you kind of have two branches you have one branch that is like these right and one branch that looks like these so and and here we're gonna have q0 and q1 like they're not independent anymore uh and so that tells you a bit more about the entanglement that's in here right if i apply another uh another control so i apply another control not gate the same way right um well what it basically what it does is it takes that and it turns this into a zero because it doesn't touch this branch but now the both ends of the branches are the same so you literally glue them and so what you just have is you have one branch left which basically means there's no branch and so you end up with your zero plus one state and the zero state in you know in q1 and so it is q zero there's no branch anymore i don't know um that's kind of what i had in mind in terms of you're simulating so you're simulating a circuit by uh by keeping track of the states of each qubit so you would have a state of each qubit right so you would have a um it's it's a it's basically the cost of these is n right so you have n qubits and then that's n is like you're keeping it's maybe not an n okay but like one qubit can't really get more complex it's literally one qubit can be just maximum two complex numbers um so you're still at the realm of n and then you and then you still have entanglement but let me let me see what would be the cost of representing that because you would have basically the amount of entanglement would depend the amount of branches would be something that depends directly on the amount of controlled operations you have in your circuit um and i don't really know what is the how much this can grow i have a sense i have something that tells me it's not exponential the amount of entanglement you can have um it's it's it's it rather grows uh linearly with n um because the more qubits i mean you have to keep it's how much how much you can entangle them you can just entanglement like you can have two branches right if you have three qubits then [Music] yeah okay so this also grows with the number of qubits you have two three key bits then you can basically entangle and tangle them so you can have for each of the well you can entanglement entangle them in three like possible ways right because they can tangle qb1 with cubic zero with one and one with two and zero with two it's basically there are three possible um correlations you can build all the although one can argue your like what would happen if you have a uh let's say i have three cubits right and then you apply like a harmon to the qubit zero and uh how to mark to keep it one so you kind of have zero plus one z plus one and then you apply a c naught between the controls are zero and one and the target is two so that would leave you with uh basically what is the what are the branches you'd have here right you'd have how would you say how to how would you interpret that you would have well if cubit zero is zero like you well you definitely have a branch that says if qb1 like is one then uh how can you represent that this cubit one is one like can you represent that with branches as well or not cubic zero is one keep it one like it can be so that is definitely a branch right and now you have that is cubitty's affected here is zero um well i think is this really creating for this is really creating four branches actually because it's essentially telling you that now wait a second so that would be zero that would be zero and we're having like so we're having q0 q1 and q2 okay and then one zero b zero so those are all like those are all share right they're shared so you would basically have in here as well so you'd have actually these two can be summarized into one branch where you'd have if it's in the plus state i'm not so sure if you can do that then that's in the zero state you could probably summarize that this way it doesn't zero state that's in so zero so you've turned these two rules into one rule and now you can say and now you can say how can you do here so you could also sort of unify these ones and say could you like zero plus one one one what's the point in doing so though this this makes no sense this makes no sense because it almost feels like i'd have to define the entanglement as a combination of it's the it's it's you know both must be one it i i almost feel like i really almost feel like for more than two qubits this whole thing i recently discovered the quantum quantum bias networks works is really the right representation so where basically you've got nodes right and then these nodes have uh basically probability tables or truth tables that are might depend on other nodes so in this case you would have a network so in this case you'd have a network uh that's april 2020 that's refresh but i'll keep that for later i think that is stochastic systems then genetic description bayesian invasion i don't know how it's pronounced knows that maybe you have two more states multiple qubits the marginal probabilities associated with the root notes notation gates knowledge about a certain domain a bayesian name works represent as a directed acyclic graph with nodes and ages the probabilistic dependence between nodes so you would kind of have so what you would have is you would have a uh a table that basically tells you i should do this so you would have a network you have a graph that is like you know uh q0 q1 right and so the the probabilities for those will be completely independent or not really they'll be actually dependent on q2 as well right because there is entanglement but like the table for q2 would be something like like these right and so you would say the prop like the probability that it's 0 is like 100 right whereas one is like zero percent kind of and so you kind of copy these and here it will be maybe the other way around uh whereas the table for say you know and this is basically q zero q1 and the the the version for for q0 for example and that being q1 and q2 it would mean that uh what would that be so we know that this is going to be 1 100 if they're both like that but it's funny because this case should not be possible at all so this should be both zero percent and here we know and here actually well we know it's going to be 100 percent zero because otherwise this would be one if we know q1 is one but here in this case you would have 50 and 50 to be honest right so that dictates the probabilities now that's not a okay but that's not really uh more efficient at all because it requires that you keep that that amount of data per node and so which basically means that like if you it's it's two to the power of n minus one that's that's the cost of these representations so that that is not more efficient that is definitely not more efficient graphically it's maybe nicer but it's not but that's it's impedance related that's something that i've got in mind for the building a simulator kind of thing with the branches the thing is i don't know how how will those branches look like i have to think about this be more how can i how can these branches look like with two with more than two qubits but that's another that's that i should i should let's put that in the draw for for a second and let's let's finish this so tensor products is just a way of saying no no we're not going to represent states like that we're going to do is we're going to basically i mean the quantum circuit is denser network right uh to some extent it just tells you those are the matrices and that's how you want to multiply them and so tensor network methods kind of might give you a more efficient way to compute those multiplications i i if i understand well uh what is methods tell you and so but i don't know what techniques and how how do you decompose the hamiltonian you give it a hamiltonian it's mission problems i'm not doing i can valid problems is done so in our case we implemented an optimization strategy over the so-called matrix product states the assembly states has been tested already so they've used matrix product states but i need to understand maybe what is matrix product states then like it's written in this form complex square matrices you need to focus maybe then on these and play with google tensa network and what can you do with these a tensor no romper for tensorflow uh jax python numpy okay so mattresses tensor network diagrams crash course intensive networks video that is there's a video okay in a nutshell practical introduction yeah so cancer night so what i'm trying okay it's an hour 44 minutes nice mini crash course there's no works but we could watch these we could watch these definitely i mean i think i get i get the essence but i'm not so sure install tensor network so you create the nodes and then optimize contractions i guess that's that's the point it flatten parallel edges over contracting them in order to avoid trace edges we have contract between contract parallel split node node edges names named axis okay so that will be the next there'll be something that we can do i mean i really want to do that as well i i really want i really want to go deep with these i'm learning a lot of new cool stuff so i'm happy awesome cool |
so we are lies we are alive let's see what can let's see if we can share these or how do I do that share copy let's go Twitter now the chopped open here and just give me a second because I'm having some network network issues I hope you can hear me well I'll just just post these cool perfect ok so let's see let's see IBM a lot of experience of things that's where you can start with a challenge I think so this is getting me not enough video to maintain a smooth streaming so since it's lagging again can I just sign in so signed in hi hi hi hi the guy told me these ok four thousand dollars I can tell you how to the quantum research and implementation how to learn quantum computing for your project and I was telling you that I think this expensive thousand dollars for to the quantum research who told you that yeah I mean it is expensive I guess depending on what you want to do I don't know let's get going you know guys we're trying to do here so we're trying to do is we're trying to I'm gonna try to do the IBM quantum challenge but kind of like speed run it okay I don't know where this is gonna lead I don't know what I'm gonna complete these I just I haven't been using this for a while and it looks pretty neat I have to say so I don't know I don't know how this is gonna end up this just two days left I want to try to do all the exercises or as much as I can right now in a run let me see if the line is going to be better like these a guy on fever thousand dollars I can tell you how do the quantum research an implementation I don't know just talk to me later we can I can offer you a better price for that don't worry no no feel free I guess let's see um say how does it look like we're just gonna take a little bit of time here waiting for people to join if there's more people interested in this in the stream and this live stream is not that's gonna be available anyway later on so I'm not gonna be paying too much attention to the chat while doing that because I want to spin a speedrun it so but from time to time I'll take a look cuz I'm afraid my network it's not always the best the IBM quantum challenge so this is sort of a I think basically the idea this is to celebrate what is it that I've been celebrating with these that's an impressive number though it's crazy so all four X's evolve for you complete individually in any order you would like as I can jump to exercise for if I won okay no I'll we'll just go we'll just go one by one okay so we'll just go step by step yeah perfect if if expressing I need any issues with the live stream just let me know but otherwise I'll just go ahead I p.m. quantum challenge okay let's get started I'm assuming that all runs within the platform so let's just let's just go and I got a speed round okay so I'm gonna go fast I'm gonna go fast I'm not gonna go and I'm gonna read the stuff I'm just gonna try to get the exercises done and that's probably not gonna work really well so we'll see I just just do it a bit differently I I don't wanna I don't want to just you know see then read for you because this is a live stream and okay let's get started so purple and traffic is key so this is a bit of an inch of oh by the way I'll probably should open Piske textbook because I think that's gonna be a helpful reference for these so good so I'm assuming that's the notebook so I'm just gonna does this work with Ethan enter I think so okay so we're just gonna skip these nice fancy realizations what is it here so what are we doing we're manipulating qubits nice nice nice okay so what is these press shift return to return to run this code cell click on the gate that you want to apply for cubic meaning what is the mini composure okay small exercises so I have run out okay so now it's loading okay so I can I can what is this okay I'm just be running me so I'm not gonna play with these right now although that looks cool to be honest okay so they cease encoding here is it just me yeah can I clear that select the cube eight I don't how that works q is zero okay so you press here it's a mini okay it's a mini composure like they have been composure okay you can see the stuff going on here okay so we've got the operations okay let's start by performing a bean flip next in quantum algorithm implement yeah I came in its I don't know so you want to implement something into research and the guy is offering $1,000 for consulting basically that's not something that you should paint that amount of money for I think what is he going to tell you $4,000 I mean okay so it's like the goal is to reach the state one from okay so that's easy so that's that's this guy that's a qcx euro yeah so we'll do QC x this euro so what does this do so how do I know whether okay well done cool now we go want to go to the plus state so we know that we do these with QC hora margin:0 cool then now to the minus state so this is trickier right so we first want to do QC x zero and then one of the QC hora more it's in your own so we're speedrunning these people good done what are we doing okay so what is this so this is either so this is yeah so one of the other ones right since basically can I do can it can I do what gates can I do can I do am I allowed to use whatever I want here this the okay said SDGs because I'm gonna use the escape for that or the stck it's kind of it's my I think it's my network I don't know if it's where was I where was I was I was I was I was I in here I did this okay so is this QC basically we're going to superposition and then we do I'd say SG let's try a CT on of its answer I since you are looking for here correct okay so now it's getting more complicated so they're multi okay multi qubit gates ah okay knowledge I won't pick him that yeah don't picking that man don't pay him $1,000 for these don't do that I can actually chat with you guys I didn't know that don't do that don't do that okay cool so what do we have here quantum circuits multi qubit gates okay we have C xcz and swap control target mini composer do we do these do we need to do this no we can start right away so construct the Bell State okay I know this I know this I know this I know this this is basically QC so we put the first qubit into super position okay so keep it super position and then we do a control knot between 0 & 1 is that correct no okay syntax error what did I say harm or gain yeah cool well done now the - so sort of that other Bell State oh crap how do you how do these is it that took us will matter later okay is it that we need what do we need so I guess is this what happens when you just intuitively - basically these then you do a Hanuman on zero right and then you do a control not is this correct it's just correct so you do that no how do you how do you make that Bell snake man that's basic so how do we do that what if I get into anti-correlated so the mask is agreed so the mast is agreeing so so maybe this is this no how do I do that that's so basic right so you should go so okay okay give me a second so how can I go from this one to that one so I get to the Bell State and if I just then say turn you can start the Bell State come on how you construct that so did you do this with a do you do this with a man it's the first exercise and that's so basic how can you construct that state so it's the I guess it's the opposite of come on work come to help me I need some visual help for this so if I if I do this okay if I do this is there a way that I can so with the Zent here I change the sign and with a and with a control na and with a side of that how do how to construct that state so oh so what is it something like this yeah here we go yeah yeah yeah now achieved okay so it's that the one day year so now there should be about fine I think yeah okay so basically the reason these works is because okay let's not let's not get into why this works okay we're speedrunning this so so what did I say so it's a hormone okay so this is these this these and then we're basically are missing a Z on the QB one right that's what that's what that's what that's what these yeah okay correct cool I don't know if that's the best way to do that but anyway so what are we doing now here so I stuffed my right circuit probably measurement doesn't work okay so what happens if we do this so this will again okay so we're given the quantum circuits current function below softest States on the first and the second cubed QC sort the states that should I just should just swap 0 1 that's that's but it's what you're just telling me to swap or what so correct okay right the prom from scratch that creates this date okay so this should be easy cuz I should yeah think so so we need three cubits here okay so we need three cubits so we need two ways we needed to Q C equals quantum circuit and valid typing fast quantum circuit three cubits and oh damn everything from scratch so I actually have to add the classical classical bits in here as well and so I need to do you see so what is that so this is probably a Hanuman on 0 QC c X between 0 and 1 then it's a QC kata mark on one and then it's like a QC CX between 1 and 2 I think that's the way you create this thing right so you basically do these like that no that's not how do you create that how do you create that state so I want to get rid of those guys now I just want these Indies so maybe we're just doing that almost how do I do that I mean I can cheat and look it up but I don't want to cheat and look it up on three cubits okay so if I need to do something here and then basically what I need to do is I need to kind of copy that right so what is here now what do I need a permit measures and returns the counts returns the counts so this is the oddness this this your counts run circuit so this is what we need to do so we need to so I need to copy that and where am I I live here so so we need to do these but this is gonna be at the end so there's gonna be at the end of the circuit okay print count so that really does the printing of the counts so the histogram so the cat ran circuit you see so modify that so it uses the number of shots okay I'll just not mess with names each IP iti teeth and stupid so it's gonna be 2,000 okay so it's gonna run it 2,000 times but I'm missing now how do you create that how do you do that how do you do these and I'm not gonna make to make it to the four exercises that should be simple um how do you create that stay so let's let's be let's be smart about it okay so oh yeah okay that's so easy I'm sixty P yeah so down in the Harmar and then you just correlate both like these okay okay it's the nerves of them of the life okay of just being life so we're gonna do QC Hana market on 0 QC control X between 0 & 1 & QC control X between 0 just like that just a measure go to the bottom of things so we're measuring the three cubits I believe there's a better way to do that more compact and whatnot but that should be right so we're running the circuit shots we'll call it just not the right the function and that's happens well done okay so that gives me number I mean again so in the following text I'm assuming this is not just you know if you want to do this by yourself Don yeah so if you want to do this by yourself I guess that this is spoiler I don't think that code works I think that's kind of generate it just for my specific execution I would assume so otherwise that would be a bad design of the whole thing but that would allow me to pass exercise 1 is that correct yes cool next exercise so so who is ethiop incubate hi I'm gonna take in the challenge but keep in mind this treatment that's trimming it has the potential to show solutions to orders can you maybe unless until the challenge ends yeah so this is gonna be again don't watch this if you want to do this by yourself I'm not gonna yeah I guess I'll enlist it that's no problem so I'll once we're done with live stream or basically I'll basically publish that I guess it's it ends in Friday or Saturday or something like that so then I'll I'll publish so they play back and for all this sure no problem I don't know can you can you Unleashed some stuff that its life or like the stuff that I'm doing right now I don't know if you can just I don't know a key can you just do that so they go to studio well I guess I can check here I guess I can edit the live stream okay it's unlisted anyone with the link can view okay so I changed that it's now unlisted so so this means that you that should not appear in YouTube just for people searching so only if you got the link and I mean I'm assuming if you want to do that and you know the link is about this so you're not gonna just go into it good having said that exercise tool what do we have here get started so okay this is gonna be what is it about so noise error correction okay perfect I say there's a whole oh that changed quite a lot so there's some error correction in here measurement error correction there she goes measurement error mitigation so it says excellent connections I hope that it all goes well because that's not doing by the way exercise - measurement error mitigation again we're trying to speedrun these so I know I shouldn't I should probably not do this I would love to read this stuff but I guess it's not fun right that looks like Grover's algorithm by the way a lot of plots in here okay Dada mitigation yeah we know we know what is this about okay okay so we're gonna play with the one real device are we really okay so 4x measurements so this is this part of the kiske books it's basically you just get a model of that okay we're speedrunning course be running crispy running so let's initialize this stuff what does this do by the way it loads my account if you remember if you remember they having you know a feel your app group and project information to provider so what I do premium devices okay no no no I'm not I'm not member of the network so I'm not gonna get any premium whatever devices in here but that I guess works out of the box so okay we're gonna find it back in the least busy let's see what this tells us we're running some experiments on a real device okay but it still it's still thinking um let's let's use the time to read a little bit about it then so run the next sound so what was here before we're creating a calibration matrix that you can apply to too noisy results provided by us to infer the noise free results so we're basically trying to run some kind of okay cool so I picked up the back end that's nice okay so this is the connectivity then the air gates and all this stuff what is the back end by the way Burlington Burlington I think is the one that I use for for the misc friendly grower algorithm thing so when I try out something against this machine try to understand what the error rates are and then apply that to an actual result to kind of then-candidate use the error free results I'm guessing that's what's gonna happen so weird oh so what is these by the way so experiments okay so this is what it's kind of gonna generate the stuff for us right it might be in queue for awhile you can expand that into jobs window that just appeared in the top left corner to monitor your submitted jobs make sure to keep your job ID in case you're in case around all the jobs in the meantime it's that the job that's the job idea I guess nothing appeared here to monitor anything though I know that you might be in queue for a while okay so this is so how do I know whether this is done or not complete mass guilty the first noisy count so this is so what is this doing I I basically didn't read and I kind of realize I don't know what I'm doing but it's running something against the Machine transpile mask a leaps voice man scallops okay state labels complete man's cow ranch Nike whips circle a circle a will end cow what is this okay that's something from so he skied that's probably the textbook or in the documentation so I think that's all fine right with it with a connection you're alive because I'm I seem to be having some connectivity issues okay there's a list of measurement calibration circuits I I remember that so that's a the list that's a list of circuits that it's I remember that that's explained here it returns it a bunch of circuits that that are supposed to give you yeah okay so they're supposed to kind of what were they supposed to go you're supposed to be good at at kind of detecting the noise yeah and then it kind of oh yeah that's what we're building this matrix and here's what we're building so is it it's done running okay it's no wait a second it doesn't seem it's done running I don't see though any monitor in here retrieves job you can then easily access the results for jobs finished by running these oh here we go okay so there's this okay I don't know what that is but it just I guess that's the result in collision but again see the correct outcomes on the diagonal well the incorrect ones are off diagonal so all these other stuff in here okay cool but proper and there no it counts for four different circuits none of the measurement error mitigation is the best particular correction exactly because that depends on that particular device apply the mission field they know to get the mitigated data and given the mitigated data choose which error free outcome would be most likely so okay so what are we getting we're getting the noisy counts of what get counts backhand what are these noise accounts for though me to get it counts so we apply the to the mass filter we applied the noisy counts me to get it noise it counts I don't get it what okay here so noisy counts is the rent is it and the blue is communicated the other way around mitigators red but what does it get counts back in well you're getting the councillor on the back in get the noisy counts of of what okay it's not clear to me but let's see if we can do the exercise regardless that so which are the MA which are the following histograms most likely resembles the error for accounts of the same circuit okay it's definitely it's definitely the first one it's definitely a right which are the following key stands mostly the error free accounts of the same circuit so those are okay we're running the same so the nice accounts of the same circuits we used to kind of get noisy or whatever the matrix is right Zor so it's a it's a we're supposed to speed run these people and we're not doing it answer a I'm gonna know at the end whether my answers are correct that's kind of like a questionnaire consider a second set of noisy counts that's from the second circuit okay and so we apply these Y mitigate accounts one so we need to actually do this thing here now sorry so we need to apply we need to calculate what is the mass filter though it's the filter it's what we're actually generating out of these thing right it just shouldn't it be the actual argument of so we should do is plot noise accounts one and so we're doing these and these correct I think so and now we are getting which is following service Muslims the Africans in the same circuit wait a second say a but it's not what I'm wait a second plot those accounts me to get it counts one Nesfield airplane noisy counts one Ryan something is suspicious in here so [Music] we're supposed to was the first to appear on these people mmm so I'm gonna risk it and oh it a second yeah the blue ones are the so the blue ones are the actual because the the red one sees the the blue ones is ones we were looking at and truth be told it's nothing in here I mean that's uh it's just one little guy in the middle so so that should be you know rather d yeah okay that should be kind of rather de it's 0.38 point six one and here we've got point three three point six yeah it's rather d okay so let's say it's D next the third okay so we're doing now the same okay that's easy copy paste so again it still looks like like I said looks like just a bunch of because all those are zeroes is like point nine point zero nine point one point one three so it looks like I'd say rather rather be right so we're go what numbers do we have so we have four is that seven yeah for seven right so we've got like four seven yeah I I think it's I think it's I think what did I say B did I say bi it's B it's B and finally consider the fourth set of noisy cameras and we're doing these like that copy paste okay so it's a bit of a roller coaster I mean I've been ups and downs I'd say I'd say it's being again I would say it's because those are alike I think the peaks are too high right the peaks are too high so can I just can I just plot these people I am why am i plotting what am i plotting what I just distracting me but I think it's the Pixar point three the peaks are also increasing right so the peaks are also increasing so this should be what did I say bee bee we're supposed to speedrun this I don't know how long have we been streaming already 35 minutes that's too long B for two exercises okay did I okay is it correct yes it's correct so that's the text which probably only works for me I hope so so don't use it please just do the exercise yourself cool what come on congratulations submit the following text ah so it's wrong okay okay so sis it's wrong DPP okay so okay what what what is wrong here let's go back to the first one so have I done something wrong it's go to the first one first one is a I said and it's not a that's that's the problem the problem is this is this is distracting me yeah come on that's point four in point five that's rather that's rather that's see let's see man that's C okay but now I should be able to submit that and try it again because the other one should be fine so those are the answers I'm so stupid oh cool got it perfect yes you should be careful someone sheets and or so yeah I don't think I thought I don't think they can cheat you know I think those coats although truth be told I think the one I just put is the the actual answers I thought this would be designed in a way that the code is so the the code for the submission is generated for my specific execution and so you know there's no way people can just cheat I mean come on it's not it's not your it's you're not gonna gain it's just you know it's just that it's just for fun right exercise 3 okay so we are now 37 minutes on cool this is third exercise about BB 84 trade I've never done DS the cryptography protocol developed 1984 one of most famous IBM errs Poppa buckets let's get started exercise 2 was okay yeah I was not really I'd say a challenge it was more like copy paste first one was a challenge the first exercise okay so that's a funny picture so what are we doing here do we need to read all these do we need rid of these probably probably probably read all these because oh my god that's gonna get complicated okay it's not really long but it's basically I guess encoding stuff this Bob machine protocol decrypting stuff Morse code kiss key it's a code here to decode Alice's Morse code and we've got a dictionary here okay that's gonna take that's gonna be that's gonna need to be more reading we can't probably speedrun these stream health is it all going good excellent perfect I hope so because there's a couple of errors in here but I mean I think this trip should be going on well analytics so what is it saying here yeah let's go ahead so what's the the goal of the pp 84 for us to create a secret key between two parties not as involved that can be I'm bad at cryptography people and I can then be used by both parties to encrypt and decrypt I hit a message organists with different steps of the protocol socratic secret key to the crypt our encrypted message sad you see okay so what is the BPA default protocol in the first step Alice chooses two random strings K and B that each consists of n bits her bit string K contains the actual beats Jones to encode form well b8 determines the basis in which she will color beats for B I equals 0 if the if bit is 0 she encodes the if keep it in the standard zero one basis while he's watching cuz in the plus/minus basis okay okay cool okay okay okay okay okay it's interesting so that's kind of the basis of in which entities gonna be encoded bitstream a random bit string so case what we want to encode okay so after encoding her n qubits Alice sends this qubits to Bob Bob also choose random beats drink concern and it said the minds in which basis he's going to perform measurements he serves the outcomes of the measurements k together with the corresponding basis beads be in a table next Elizabeth called her three bases beats whenever the base is different Rob measured in a different basis than alleys and okay so I guess then bob has to thank you and couldn't so he gets each outcome with probability 50% Alice and Bob therefore discard all keep its corresponding to these bass beats if the basis equals however it's the same they prepare the measure to keep it in the same basis so unless someone eavesdrop Bob will get the key bitten Allison coded these outcomes then compose the key okay yeah I kind of a think it's simple enough so you've got K you've got B and so you're encoding it and then you know this is B and so this is so these are the ones you keep and that's the key but how is that the key then 0 1 1 0 and and how is it used though to decrypt so the key produced in this example B 0 1 1 0 to make sure to make sure that the key secret and correct Alice Bob would sacrifice some of the key beats to check that no one eavesdrop if someone had measured keep it on the curing the way this could have changed the state of I cubed with and with probability 1/4 Bob's on a skin will be different by checking and beats the probability to not notice an e-stop rate decreases okay it says if they check and off beats and they are all the same they can assume that no one has eavesdrop and the key is secret however to keep things simple will not perform this test in this exercise ok good so obvious will keep us all beats all the key will be used ok message encryption once the secret is distributed Alice can encrypt her message by using the so-called one-time pad technique she she simply adds the key beats on top of the so so I'm it's just taking a look at this channel analytics from time to time because it's it was a bit of a hiccup we think with but I hope help nothing's lost so Alice can encrypt her message by by using the so called one-time pad technique she simply adds the key beats on top of her secret message beats that she was to send using the example above her key is 0 1 1 0 if her secret message bit string is M 1 1 0 then the encrypted message or be ok but that's just a one-time that's not this is not quantum right this is just a one-time pad encryption technique so you're just doing an XOR I guess that's what you're doing Bob can then decrypt a message by adding his key on that exactly because the XOR okay so what's so what's quantum is the generation of the key the challenge in the following four tasks you play the role of Bob you will create such a secret key with Alice and use it to decrypt encrypted message from her four useful tips to complete this exercise we'll so I was punished for communicating with other participants and asking questions please take a look at the following repository you will also find a copy of the exercises so feel free to edit and experiment with it with notebooks ok so those are the exercises and stuff like that cool nice stuff how to simulate this protocol so in this anyway it just steps what do we have in here so we have Bob spaces for cubed Index this is Alice creating the cubed DS cubed Alice prepare cubed Bob measure cubed we collect all the circuits and put them in an array now let's get all the circuits for each cube in for each cubed okay so Alex prepare and combine the results use a task Bob measure cubed insert your code here to measure Alice's beads how to simulate the product linkies kid you will spy the procedure Bob takes in the function increase exactly step tool of the parallel to get your bit strings so what is what is how this prepare is this this is somewhere oh this is imported already from the challenge this is a discredit so I just need a measure stuff right in so your code here to measure Alice's beats ball paces cubed index cubed circuit that's a cubed index mmm num cube it's okay but where are my qubits all cubed circuits no those are the circuits in am qubits okay that's just the index so what I do is Bob measure qubit finishing the protocol below okay so if so what I have to do is I have to i should be careful if someone cheese and so I read that right so issue I should be okay so what you should do is you should basically if for that cubed index I need to check what is what is the power basis it's it's a binary string your patience okay so if that basically I needn't first know what that what's the basis right so the basis all my bases would be Bob bases cubed index right and I'll say if my base is equals zero then I then I so the then I think the base is just just 0 1 right yeah and if not is plus minus so then what I would then in this case I just have to I'll do it the other way if it's 1 what I need to do is I need to do cubed circuit hot amarte cubed index if not if not I don't need to so if my basis is here than I just don't do anything because then the measurement is just gonna happen so the measurement is added here with the results right measurement for measurement no sorry I do need to add the measurement measurement because they just append the circuits and execute the circuits so I'm assuming that I need to do that so what I need to do is then I just need to regardless that qubit circuit measure cubed index give it index I guess I'm assuming there's gonna be in the circuit where is the circuit created all cubed circuits I need to create a circuit actually okay Dwight it gives okay so Alice prepares this cubed circuit and gives it a pop to my function and then okay and then these things are all appended so but they must be circuits okay so but I have the circuit here okay so so I'm assuming Alice Alice is creating it and is creating it correctly so this means is I'm assuming that there's the right amount of classical pizzas well they're to read and so this should just be these I don't know let's see after you perform your measurements Alice announces her basis beat so first of all what happens if I do that still running okay a mistake so index error where's the index error circuit ages you're off at least index out of range what index out of range how can this be an index out of range so what am I doing wrong in here so if my base is equals one so what if I just do cubed circular H cubed index so this is basically this is where the but the cubed index is given to me in your right is it that I'm not about measure cubed do I need to return the circuit no it's just doing it's just it's just doing by reference okay it's cubed circuit but it's appending this circuit okay so how does this circuit look like maybe let's just print one so let's just do I'll just do just two for one sir if cubed index I don't want to print if I just want to know how this looks like so if if the killing is one take cubed circuit draw output SML P NP l I think it's nice okay that breaks him before this out of range sorry I should all just ignore this for the moment and this just in case this is causing issues let's see oh so this doesn't pretty my circuit or what I just want to take a look at what the circuit looks like can I take a look at how this is being built these circuit alleys prepares okay so I guess I can take a look at these stuff in the Kiske in the in the github repo right so I can search so in exercise 3 encoding alleys I know this is just those are just the exercises I cannot take a look at where's this may fortune extry yeah but that's not okay that's not a nice red ball right that's not this red wire because otherwise that's probably in here somewhere x3 so Ali's prepare cubed x emesis maybe it's always zero the index or what because it's a 1 but then what's the point of what's the point of the cubit index what is the cubit index used for so basically the solution is then probably just just doing 0 and you're 0 right oh maybe cubit index is needed for these that could be the right thing because because then we're gonna have to combine maybe I don't know let's see let's see - true - no houses now working ranch I think is gonna be what is keep it indexed and that's just to get the basis okay probably that's the only reason come on let it work now they work now not the right beats pops beats whoops not the right beats what does it mean that I'm doing it wrong or it means that it's just table spaces check beats yeah not the right beats so if my bass is equals 1 right we do a lot of market because we want to measure ah wait a second so if Ali's because if we're pending the circuits okay so if Ali's has a okay so if Alice says have a zero right she's not she's just gonna but if Ali's encodes it this way if Alison codes it this way I don't need to I'm already in that basis so in theory I don't have to do that to apply the harm our game but I'll be weird because that's what so that's basically what's going on here right so Ali's so if Alice's has a one it encodes it like these if then Bob has a one as well he wants to read it in the same basis and so he performs a measurement yeah under harm of you know he has to play the harmonica heroine bases yeah and the plus/minus basis all right yeah no no that's correct so what I'm doing you some what what am I doing wrong I'm not understanding the way this is the way it is to set up so so what is doing so what is what is Ali's prepare cubed now Ali's it's not prepare cubed is it prepared now let's prepare cubed so this returns a circuit X innocence for excellences so alice is prepared in Jesus's so for qubit index in X illnesses Alice Alice just prepares and just as an X just just a thousand x Kate have I read the protocol wrong so if so that's that's so now what we're one and I'll be a zero because it's it's encoded like that okay so this is just encoding it like these okay so this is just basically creating in the same case for index 0 in this case right it's like it's a hard-coded thing here okay yeah okay and so and so what I what I want to do is I want to do this I want to basically so alex is Alex prepares the circuit for this cubed and I need a completed and yeah I've got my bases and the cubed index so if so if my basis is a 1 I want to apply a measurement on the plus/minus basis I need to play a harmonic 8 but if my basis is not once it's 0 then I just want to play a measurement right so but it's not and I I'm supposed to complete maybe I'm not supposed to merger no supposed to thought measure it finish the protocol below Bob matrices keep using the specific set of bases which is a viable basis for you supply the pursuit of all things okay this create will be create disc itself rich give you also call it together in the rank up also we have supplied the code to do this fine Oh simulator to say you need perform in your randomly string unit reproduce the queue and perform measurements in the basic responding keeping beat string and return the outcome as a beat string in this for instance awesomes obviously the reverse order so so am I supposed to so am I supposed to reverse something so melded keys click return self comes always in a reverse order so if accuse you if the cubed 0 is the stager and keep it 1 into 0 it will return so you can check whether you're okay check these strings so I knew reversed I need to reverse that come on it was just a test to check whether I read this stuff been irreversibly - - reverse its drink is this a beets lettuce beets by the way returned beets beets is literally a string how can i - over the string how do you do that that should be easy right that should be simple simple and easy let's try this but that I should probably reassign that right so I should probably say it's equal speech days let's see see that works No then I don't know what to I I don't know what I'm doing wrong I'm that stupid come on it's exercise three already minor results cubed index invention carrots beets for measurement result get counts cubit index this is really work though just let me just quick test okay ABC and now I print my print test CPA sure principally okay so that does the right thing that that's the right thing but something's something I don't like something about the and so these are all circuits okay so these are all one single qubit circuits and we're measuring so but there's something I don't know I'm just gonna try some just some stupid stuff I thought it's the so when the cuban e0 when that when the ball base is zero still running now of course not what am i doing wrong that just should be simple it's really simple to it certain 'if so for my sake psilocin coats the stuff and I want to read it right so we're just creating the key I mean we're just really we're just creating the key we're not encoding anything yet so we're just doing these so my missing am I missing the last step Bob measure or is the last step already always the last I've already done to get your beat string so first I'm getting my beat string okay this is just a bit string so it's not the checking or anything it's this anyhow how do I know I don't know that's this work as expected as in like it's it's a per reference so we're appending this cubed circa or should I just maybe cube it circuit that's why I wanted tool what if I just do this and I just can I just say for example print cubed circuit turn qubits so I'll return I want to try to print some of these stuff because I'm just getting lost right now so I'm gonna return these I shouldn't mess with the code I know and so I just basically wanted beads draw output MPL just keep it circuits a first one okay it's gonna work is it gonna work nicely or it's gonna pretty nicely still running it's a draw this circuit oh sorry for eating sushi it's it's when I print because the my basis starts with a 1 so I'm suing the first circuit circuit is gonna have a harem art and a measurement yeah so X autumn worth in a measurement right and alleys for in index zero has an X how much can I try to print so so I'll call that special so the test circuit so this circuit equals these guys I just want to print this and see what happens for only if it's it's um this is the buy this is so bad just do these can I do this I wasn't gonna complain it spice I'm gonna complain mmm so I wanna know what what it does their error test is not defined on it we find it here probably just need to do here's a test sequels quantum circuit take this to this right let's see Alex paces onam circuits not be fine to other goodies how do you declare quantum circuit this way I haven't imported all the stuff that's needed for that can I just import I just imported everything you poor kid kid from kiss kid comport everything come on it cannot be that complicated it cannot be that complicated what am I doing wrong I'm just I just feel stuck so I might just jump to exercise 4 in I'm about to do so because I'm it's empty so you're saying this is not cold or what hmm or it's that Alice encodes nothing their number that's not true Q indexes euro that's not troll my friend they should have an X at least they should have an X so if giving index equals zero Tessie test C equals this circuit hmm I don't know if that makes sense way I mean probably she just kind of just reload the whole thing every start the colonel just make sure I have that I have the clean coat okay so I'm gonna close these I'm gonna start with exercise three you get started with a notebook fresh start fresh start so good constant simulator oh crap actually remembers all that stuff huh okay let's three more for this debugging code cuz just bothers me what else did I add these goals so this returns the beats this returns the beats bring the beats check beats but I mean that's the simplest thing ever if the bot if the bass is one that's what the whole thing here is telling you right so also just around the b-string you know thank you right switch basis he's kinda prefer measurements he stores the outcome of these measurements together with the corresponding basis beats in a table so if the B from Bob has one then he's gonna measure so he's gonna measure a zero right because he's nibbling on my gate and a measurement so he's gonna ply just he was gonna he is not gonna know cuz it's - so he's gonna yeah it's not the right beats why not the right beats sorry clearly the first circuit it's got an X and it's got a Hatem art and it's got a measurement and so it should measure a 1 that's correct it's not the right beats maybe she should move on check meats what does check beats doing but at least no I should actually get right beats I can't believe these I can't believe these I must be getting something really come on is there so there's some conversations going on there I guess questions the circuit is being prepared and so Bob needs to just ah unless what I need to do is a needle take maybe that's there so I don't need to take the cubed index I need to take the ball basis length ball basis so the length of bar basis but - QE but that give me the same result and if I just do that here right or not necessarily maybe that's that stuff index error lenz fall phases - cute index and cuban index starts with zero right so the first element we saw - one like this thing so know if your index is zero I want to - one yeah so we are getting we're getting that bought basis from we're starting from that from the tail of no then discussion or comments or stuff in here nothing come on what am I missing so it's bothering me so much around the base drinking you provide with the random basal speed straight involve heartbeats that uses c84 perform measurements in the basis corresponding to the given beat string and return the outcome as a big string in the form so first we won the beat string so first we won so ok so on this is the bit string from cubed 0 no it's gotta be here now that kiss gate returns outcomes always in the reverse order but this is just a cute thing and then we're execute all circuits wait a second not execute you're setting in a bunch of okay you're sending in a bunch of results that you're combine the results beats so cute index in range ma'am cubits attach the measurement for the measurement in results get counts keep it index zero that's sure the monikers were just measuring one cubed to be honest I've got my bases and so that should be that should be cubed index right so if cubed index we're adding at em Haram are two key be zero because this just one cubic I don't know I feel I feel completely lost and I'm like 20 minutes somebody's already fault measure QE that palpable hmm because we want to measure on that basis so we need to apply the harmonic gate because if it's a zero then it's gonna turn into plasma term zero and then we're gonna apply the measurement that's that's what bothers me it's just a measurement what I'm what I'm applying it's just a measurement Alice sends the qubits to Bob so islets Alice encodes it with it a plus plus minus and or 0 1 then sends it to Bob and then Bob basically stores the outcomes of his measurements together with the corresponding basis be unattainable so that's what I'm doing why am I so blind why am I so blind supply Bob speeds Bob speeds Bob speeds not the right beats but you're always prepared how's it gonna check if we're just doing the shots in here but that's just all I should do is this that's all I should do so if alleys if Alex prepares it prepares keeping the plastic and so then adds a Hana Margate then and if I have a one in there it's gonna add a harmonica and it's gonna measure so so what's the deal with it what's the problem and we're just getting get the counts for that cubed index and it's literally just these like if I if I print here it is and then print print beats for measurement in so I want to see how this beats drink Rose is gonna show it the me or is gonna show me all the attractions that's not rocket science zero zero zero so it's it's growing like these so it's always growing by one bead so there's nothing to do with the order in here my beats not the right beats friend cubed index come on add it adding any honda minds at all we must be oh we're not no oh okay so what is the problem here Balt basis so it's never that's not possible so the problem is in is oh oh I hope that's not the problem I hope that's not a problem I hope that's not the problem I lost half an hour for these yeah it's a shrink cool so yeah I should be a better programmer that's the issue okay cool so now it works I just now someone just problem I spend the last half an hour just because of that okay um anyway that works now cool no problem let's move on great so good you're good you're good to go okay and how is this printing them circuits I'm just so bad at this okay so let's execute step 3 other BB 84 protocol to extract the key just a couple hours basis beats be to your base this means give me the pretty step and I start the key by keeping the only outcome so we have okay your code here to extract keys so I guess this is so we've got what we what do we have to check now we have to check whether we've gotta check whether it's not keeping only the outcomes when your base is where the same you can also have all these spaces okay so I've got my key basically it's sort of start like these and then we'll just go and say for let's say if I know how to eat rate me so for thing I can do something like can I go sort of a double iteration I think I could do like a double iteration or something that but I'm assuming the length is gonna be the same so I can just also just two ways a very simple while okay so while I is still smaller length of Alec spaces and is it like these or is it just that what's the Python let's just end i small like the length off walk basis basically what we do is if the base was the same then we keep the result that's what we do right so we keep the beats okay so if Ali's bases I equals I then he was equal mmm I think that should do the trick sorry um I should actually increase I you know at some point sorry for that can I stop this somehow interrupt the kernel and I just do like that I think so because otherwise it's just gonna run forever I plus plus that's this work or I should just do like these so far so good you're at the key right okay cool um using this key Alice can now send you her private encrypted message while for full security key would only be used once sir 50 or 50 feet long four times in a row to encrypt her 200 feet long message so now we needed to crypt this okay so and this is the pad thing right so I needed with the XOR thing or is is it an XOR so using this key Alice can now send you her private encrypted message while for full security a key would only be used once she will now use her fifty feet long key four times in a row to encrypt hurt 200 feet long message her encrypted message is okay so in chunks of 50 I gotta do this four times so so that's M and so now I need to basically so decrypted it's empty and now kind of so how was how was this done was this this pad thingy the one-time pad so if the if the bit string is M then the encrypted message is now and so the decrypted message is I did the see and and do I think that's the some module tool right but that's literally an XOR I think because 0 1 X 1 1 0 it will be 1 1 1 0 2 0 it will be 0 you know that's X or Python it so I can have a Python XOR M XOR hmm utilize exclusive or no I didn't get a logical eggs or two barrels of - so I think that's an XOR so basically I would just go and say so this four times okay so but basically what's the key length is basically the land key right so this is this is the key key key key key yeah okay so keys key so while key land is smaller healing is smaller than and then keep going so although this four times and at the end we'll just add the key length so and basically what we're doing now is we're we need to get the portion of so now you get another loop in here so I equals I guess so here as a starting position but then at the end so I would be survived if the key length is 50 first I wanna go from 0 to 49 right so and then at the end I want to just say I ski lame so just try it in the right in the right position so what I want to do here is while is more than keel and basically yeah I'm gonna I'm gonna make sure that I do like I just I plus one and so good and so when I second I tend to get lost with these stupid things so we needed to annex or write so basically decrypted decrypted inequal decrypted and here were doing this and now it's kind of the XOR so where was that how to get an ex or so another fine on all objects for our strings yeah okay I guess so I guess I'll do a trick so we turned those into bullets so it turned basically we say return key key I and M I and then that is basically should I cast that into a string or that just gonna be this is gonna work I don't know do I need a cast into a string or something - string cuz I want to 0 or 1 right so I don't one cast polenta string [Music] can I just do that seeing a protic into that we'll see so that's what I do right let me check I'm in creating chromatic that start with 0 the length and you see meaning so increment is I'm decrypting that and it's like the Crypt I finished in the first batch then I say now start a keel and killing this updated until the length of the aim is done let's see can only concatenate string not ball to string ah so I saw maybe I just all this out of index out of range how's it possible key oh sorry no no no no yeah I am he's always he's always 50 right so this should always be I should always be yeah I guess we can just I can do the dirty way we just have an idea just add another index and I just you know I guess killings I could also just do I modular 50 I think or something like that I'm really bad programmer as I can see yes so and I you know just say J equals J plus 1 because basically I it's gonna be updated constantly so I is gonna be the reference for the message but the key needs to be between 0 and 50 here right yeah cuz I gonna keep that and it's to reset it to zero every time what I'm processable URL great is it me maybe or I just don't have internet no I do what is the problem print decrypt it huh okay come on I guess I can just do can I just do look integer last integer okay and I just do a modulo two I think that's how you do it right and then that is a string integer yeah yeah yeah I can't go cool not the right beats grade it's why [Music] I mean doesn't matter whether it's like you know so I've got em killin and so I basically all the way to war send chunks of 50 her to beat long message so M is the message to decrypt the message so I go and I go and I say okay the first chunk is so I'll kill em great so what am i doing yeah okay not okay well killing is smaller than the length of the message then start j with zero and then and then i and so an i is zero and so while i is more than keel and then basically these decrypt message and increment I and increment J so we're gonna increment I until the key lands and I is gonna be the key lines key length is getting update it and J so is it that this is not right or what well it's not key it's beats no what am i doing that's the key it's key right yeah I do or this is the product module tool what is these what am i doing oh no I just open the edit mode or something so can I just really start these not in edit mode come on but Papa Papa I should probably should probably do all the probably run these and run this and then exactly so what is that if a second is the string is m110 the grip message will be but so the message key so 0 plus 1 is 1 1 plus 1 is 2 which is your modulo tool 1 plus 0 is 1 and 0 point 0 0 so this is it's it's M it's a bit why it's the bitwise a plus right mm and so the message is the key on that encrypted message so yeah so why isn't that working so why isn't that working you got the key say about the key ride and now I just go and so this is to be correct so I take the key J turned into an integer turn into an integer and and do the sum module tool so just to be sure that's how it works because I'm just already trained one plus one modular tool what does this print zero yeah so zero plus one module tool prints one okay cool and so just because I'm panicking already saw ever just do these routine if I just do all each one plus int one yeah a zero so those this this thing should work right then it's just maybe I'm not maybe it's here I should put the message backwards maybe that's here why I should put the message backwards that's where all the time this whole thing is being it's kind of suggesting oh is that how do you - there's a string it's a shame it's my coding skills are so bad while the length of the key is smaller than the length of the message zero and then which is basically now saying you know well well I Jake it's Jake gets reset it to zero here so we're starting from the same key right that's what that's what alice is doing so doing a 50 feet long key four times in a row to encrypt her 200 feet long message yeah come on otherwise I'll just move I'll just move to exercise for that that's again another another of my lack of programming skills I'd say or just not I'm just not seeing some stupid some stupid thing in here it's key sister yeah so decrypted so a decrypting string by string bit by bit I plus 1 J plus 1 I get a day to kill to kill n and then kill him gets updated to kind of double X ice it's gonna be free service oh great so I think that's the issue No Lana the key the key killing gets updated by the way the key but let's 50 that's not the right kid I Peter and that's also thank you sound of where is the index no that's cause there's killing that's correct that's killing and then I update killing to be killing Plus to enter the key [Music] and cuz eyes been J gets reset this year or so maybe I should just print I see I'm just missing some zero on it through an antenna oops okay so I'm missing fifty or equal here we go probably that makes it that yeah one our first convert the chromatic involving Morse code in in the second step use the provided dictionary to decode the Morse code copy and paste the solution string on the idea kind of complete the exercise come on okay come on we're doing this we're doing this um so the Morse codes about what is the yeah that's a map converted the cryptic message from above into Morse code okay of course is an actual message encoded in the code to encode later is a least use divorce her as she could only send zeros and ones and keep her message assurance puzzles she encoded a dot and so one and a dashes of one one an intra character gap between each dog and the - so an inter Carter gap between each dog and - with a character a zero a short gap a short gap that separates letters a zero zero a median got the separates wards I guess the first we should spleen no again I double-click in here markdown okay whatever so first I should do a medium get a first I should split though the the message so every time I find three consecutive zeros it's a medium gap so I should just replace okay so I can just you can just replace right I think I can replace that so or can I just split I think it's called splice or something like that right so Python because then I need every word or not replace a Python replace should work Python string replace replace alt new in sooth was okay string replace so I should just what is going on here the amount of like a speak so she just come on I should just go in and say string the place so let me print first is okay so I'm gonna print this string replace not this is what I should do with the cryptid rip let just I want to say is this big just doesn't work this way so there's no spaces gonna just print the cryptid it's just one war disease because I don't see three consecutive zeros anywhere okay so but we should baby basically do these rights a medium got a short gap that separates latest as a zero zero so that's your parade that that basically it's this is what we want to get the latest solution okay so this would be yeah so those are the letters and then and then now we want to do an intro Carter gap between each dot and dash as this euro so is the way I'm doing this okay that's not clean it's not nice but like a deal so that separates sort of the the Morse codes in between exactly and then I just have one one one so so the ones and the so the ones is a dot and the one one is okay so now I just should kind of replace the dots no the first at two ones with a dash and then the one one with a dot and so that should give give me this whole thing in here okay and now I should just I guess here I could use probably can I use splice because I if I turn that into an array I think the thing it's I think it's I think it's called supplies I think this is lacking a lot I'm sorry guys if this is liking a lot or slice it's called slice no data no data there's no data suggestions from is healthy I'm sorry for these guys exactly and what so this is I'm not doing what am i doing parsing split okay it's split okay so we're doing we're doing here a so we can do at the end so split message is final M split split by these and sort of I print your if I print these I should print the array okay it's a pretty array and now it's basically yeah I just basically say so final final final final and I'll say or each whatever see for waiting for which average element in played M final final append basically we're gonna use the maps are gonna say Morse code to dictionary so Morse code and I think that directly because we don't know what the index we want that and so I wanna do at the end is gonna print print print final final no oh crap there's a key error so short gap yeah well why am i why am i using these oh that's bad why am I using these things so I wanted to have later separators but I don't need to have letter separators - why so if I print final I'm here what is later separators for though a short gap that separates letters a short gap that separates letters so I should actually separate that should split that right because those are letters so should do s plead equals so I want to split by these so fan-out print so I say for each element for an Escalade and I want to go through these right because I wanna I wanna I want to now do split and I'll say for each insulating ee to this I'm stupid I'm stupid so I need to just do final final goes here man I'm just backwards at these you know and then I just go and do just backwards at these and then I just do a print so I don't split here or anything I just go and say just Morse code just Morse code and baby no okay so one of the ones here is by the way an inter Carter gap between each dot and dash was in a Carter why so just need replace them with nothing an inter contact cap between each dot and dash with Cena Carter but it's just a Carter is a right so I don't want that so I never replace these with these so I get rid of those zeros what's wrong now here Oh going god it's almost one and a half stream so that cannot be that complicated so I have to fit a final message I get the final message I split the zero zeros because I wanna so there's no medium gap so I'm going to separate to meet the zero zeros and then because there's no zero zero zero rights if I do these preen final em there's no final but if I do these there is right so okay so this separates the staff then and each good then I take the final M and I split it in F split and so then I said for each element in s plate replaces euro with these no this is yeah this is the last thing to do replace the one one with - the one with dot and the zeros get rid of them and so and then at the end what you get is this is an inner curve or dot dash dot but that's here ah it's in the values it's not a you don't do these it's a so - look up for index even value - reverse map so I get the universe map and get the universe map so first get a new verse map of these and then I took this on the inverse map okay so this is the this is the string that needs to be submitted in here and we are finally done okay guys I'll just go out for a second just to take some something to dream quickly and I'll see I'll take a look at our exercise for cool bye complaints yeah I don't know if I'm gonna finish exercise for because pretty late over here and so it's still going on by the way is still alive I don't think so right I guess playbacks yeah okay cool um the fourth is supposed to be a puzzle even for the quantumatic spirits so don't if you cannot solve it single keeper rotations you know gates consider the case as a good decomposition okay so what is it about closing this close this close this close this close nice load I mean I think I'm gonna just read that and stop and I might you know try to complete it later on so this is just a quantum circuit check circuit so what do we have managing yourself can efficiently it's really well so a set of chronic gets a to be nurse over even the following game Atari what is this new shape 16:16 which what circuit would make such a complicated unitary oh come on it's some symmetry or is it random we just updated kiske with introduction of a quantum circle library these are gives us user access to a rich set of well studied circuit families instances of which can be used as benchmarks quantum polymers building blocks in building more complex circuits others order the data today so only using isn't only single cube rotations in against panic one circuit approximates in turn you buy entry fee after the following error ok we're living this for not vague um yeah I don't know if I'm gonna make it before this a week but can I just download that download ass I don't know if that's something I can do later on as well off the challenge I love tool to be honest see if I can do that but that's definitely tricky so this can be seen so I just killing factor that nemetrix is only initial normal I stayed one can sure that it's not response all just you know when you submit a circuit we remove the global face and Sigler so you're sure sure sure it say ok isn't much higher flight than two kids will look at the number Cena we will look at the number of Cena gates and then I where if you generate the cost of it the composition okay so this is even then from a cost perspective so kind of you know kind of like the competition so only a few three mcx gates exist considered correctly soft if your cost is smaller than 1,600 indicating with our principles okay cool yeah I'm not gonna do this now it's too late definitely for this and but it was fun it was fun basically wasting my time would just be Python Python issues but it was funny otherwise I hope you enjoyed the livestream and you know you guys subscribe remember to subscribe to the if you like it and if you're generous and wanna support me you can also support me on patreon patreon and you know send me hacks semi-nice emails nice messages as well we'll try to get exercise for done before the end of the week but we'll see got two days left nevertheless I was yeah it was fun so have a nice day a nice evening I guess we'll just sit here |
Won't you do same for currently going on IBM Quantum Challenge?, Hey I won't be doing it this time as I prefer to invest the time pushing other projects further. This challenge is just too long and doesn't feel like the right place to spend time on currently + not sure it will give me the right learning "ROI" ;)<br><br>The problem set is public though so maybe I'll go through some of them later if time permits :) |
Hello,<br>Can you post the questions that has been asked in this challenge/ provide link if already posted..that will be helpful🙂 |
What are your thoughts on 'Qunatum Inspire” is a good platform?, Haven't played with it extensively so far but planning to |
I was waiting eagerly for when you'd notice :P <a href="https://www.youtube.com/watch?v=OlKeGxXN2zM&t=1h33m00s">1:33:00</a> |
so i'm currently thinking i'm trying to figure out other ways or ways to talk about entanglement that are a bit more intuitive than um you know the math right and uh it's uh because there's one thing that bothers me about entanglement which is the notion of that you can have like whether it's it's got some sort of strength associated to it right like a lot of people think to talk about like so here you hear people talk about like um the maximum tangle like a bell state and uh sort of all the types of entanglement as well right like not so maybe strong correlations and that's something that's a bit um it's just that like i'm i i think i have trouble thinking about this in and or trying to understand um what this really strength is or what this really means right like uh uh maybe there's a way someone mentioned one of the live streams something about entropy and um maybe a way to think about it is like whether the information of the system is fully on the fully stored you know the system level versus some of it is at the system level some of it is at the um the actual qubits right so but i don't really know so i'm kind of thinking about uh uh trying to approach this from another angle and uh so for example if we think about if i'm thinking about like branching right like i did i did a video the other day where um a video sorry that was a tweet it was a tweet thread actually about um why the three c knots create a swap gate uh and i kind of played with the idea of um how did i call it like relative superpositions or something like that like so sort of the idea that when you have two qubits for example right like one cubit is in in the plus state and the other qubit is in the zero state and you have a control you you you apply control not like controlled uh by the qubit that's on the plus state you can think of these as uh the system branching out like because essentially the control kind of it will have it will create two branches right like one bra one branch where the control is activated um and one branch where the control did not activate and uh this you know these branches don't have to have the same uh same probability it doesn't have to be like uh with a plus state would be the same probability but if it's not in the plastic if it's in in some some sort of like halfway superposition it would have different probabilities but anyway so you have the branches right like in in one branch you'd have the not gate applied to the second qubit in the other branch you will you you will have no no gate applied to the second qubit to the target qubit and basically this is a way to talk about entanglement right because um i mean this is entanglement i get it maybe not all branching creates entanglement but this particular case it does so they're entangled in the sense that uh you're maybe in the sense that like you know that both branches at the end of the day can't really uh can't really um exist or maybe they can exist um i don't really know what's the interpretation of that but you would kind of have these two branches right you can have these two branches and so that would be a way to talk about entanglement because basically kind of like yeah so one thing depends on the other um i don't really know how that differs from like from a classical correlation in the sense it's hard to think about it intuitively like i don't really um i think i don't really grasp this um and so how would you think about non like how do you think about like not non um max like something that's not maximally entangled in this sense is it that you would have branches where i don't know how to think about these like a non-maximum entanglement in terms of branches like it doesn't really make any sense because when you think about a branch it's like you think about a branch right like that's that's that's it maybe it's got to do with okay it's got to do with whether the branches actually happen or not maybe because obviously the second qubit is on the harmony like on the plus state and the first one is in the place state then the control knot like if you apply control knot that's not going to create entanglement because you're going to branch out but the two branches that you create they'll actually interfere and they'll interfere and they live in a state that is the same state like if you didn't apply the control not in the first place and so there's no branches anymore and therefore there's no entanglement so that will be the other extreme now is there anything in between like is there anything in between these case and the other case it's got to do okay so i think it's got to do with interference because in one case we have absolutely no interference and in the case of maximum maximum entanglement and in the case of an entanglement then we have total interference right total as in like it leaves us in the same state now what if we have a little bit of interference so how could these so this little bit of interference would literally mean that part of the information won't just reside in the in the branches okay i don't know if that makes sense um what would be a case right like imagine we have maybe a i don't know like what would be a case like a control knot on uh on a on a gate that's an applause date i don't i can't come up with an example actually i can't i can't easily come up with an example um like in my in my head right like like what would it be a control knot on a state that maybe has got some like some some phases somewhere in there and so when you do a control knot you're moving the face around maybe and and because of these you and dab maybe with a state where it used to be minus but now it's plus or something like that but still i don't understand i i think this still doesn't help me kind of reach the or understand what it means that like not all the information is spread across the system like what it means that's sort of like a non-maximum entanglement that's that's that's passing that's really what's passing me i i i don't know i i don't think i'm gonna i'm not thinking i'm gonna figure it out now it's just it's you know but at least okay so in terms of branching that sort of makes it seems like that should be a thinking path that we could follow so maximum entanglement it's when there's a maximum interference no one time sorry um no interference and maximum and like no entanglements when there's maximum interference that would be probably that's a good way to think about it i think and somewhere in between i have to think about an example where there's a little bit of interference and that will be the case uh when entanglement is not maximum that's a neat way to think about this i think i might be totally wrong i i don't know i really know |
This is great! Your great!, Thank you so much for the kind words! ;) |
i really really don't know whether this is gonna work or not uh we'll go to twitter oh here i am here's my face uh so it seems to be working um what are we gonna do today so i kind of i think i've i need to be kind of the uh the more exploratory kind of stuff you know kind of asking questions and i recently i kind of realized that i've been doing too much stuff that it's been more like you know community oriented and i'm a bit selfish so i really wanna i really wanted to try to understand a couple of things that have been bothering me for a long time so like when i started the channel i started with a couple of videos with d-wave systems and then um i keep keep reading and hearing people saying it's it's it's bs it's you know but like it's not really clear to me and i really wanted to kind of um go go deep with it right and i think the the culminating point for me was uh when i um commented on on a chicago quantum post recently uh let's see where was that i don't know if reddit is a good chat room for quantum computing to be honest was it these oh yeah this is quantum check request and so the the so so the quantum detector said like that's bs um the reason is like i found the tweet funny that we're working to reach thousands of subscribers in 2020. this is kind of our channel um uh what did i say yeah yeah yeah so they basically say um you know you can keep up with the research and the progress in their financial portfolio optimization including we including with quantum annealing computers um so there's kind of two things that i'm not fully um understanding here right so let me just open some notes maybe so i can keep track of these so one is basically what is really oh what what is really quantum any link right i hope you guys can read these so this is something that like i know more or less i remember but i'm not like sure either is this it is this is this way the bs is so this is kind of a question another question would be is portfolio optimization oh what am i doing sorry is portfolio optimization with uh with the with with quantum computer with the quantum computer at all like right um or is it like or is it just d-wave i mean there's a lot of questions that i want to answer and i should probably keep those uh because so there's the medium article which i'll uh okay i'm recording just a tab so i'll just open it up in a second and then there's there um there's an actual paper called portfolio optimization of 40 stocks using a d-wave quantum annealer maybe i should start here with the paper um because i when i ask for these it's like hey um you know where where are they comments where are their comments like did they comment did they where are their comments man i i asked oh yeah here exactly so i i just hate i just hate twitter threads uh it's not really um sometimes they're not linear and that kind of bothers me a lot because i i tend to mis uh i tend to miss basically um some of the replies so um they say you know uh our first portfolio results favor our three custom algorithms simulated an alert genetic algorithm and fat-tailed monte carlo when lucky in our tweet yesterday our quantum answers underperform the 60 asset benchmark i don't know what this means though the underperform doesn't mean there doesn't mean that it's actually working or so maybe i should just you know go ahead and and try to read these first the actual paper i'll put it now in the screen in the tab that i'm uh that i'm like basically okay so yeah that's probably the best thing that i could do with the time that i have today um trying to get an understanding of this but that's my goal and i'm i really want to try to spend some time on these um as i wanna as i said wanna uh really kind of get to understand uh oh sorry you guys didn't see what is there a way to have like notes on the tab or something i should switch to actually recording the full screen maybe i just wanted didn't want to pause process the the the video afterwards and so um maybe anyways i i wrote down the questions what is really quantum annealing um is there where the bs is and and is portfolio optimization uh with quant with a quantum computer like bs or not like um and what it what does it mean for these cpps at all i mean it's like yeah you can do things right like it doesn't mean like if you're not aiming at getting something more just for some reason more efficiently than a classical algorithm you just want to do it because maybe you know it's not about efficiency but it's about like the the actual end result um and i fear that's kind of where the portfolio transition stuff goes it's less about efficiency um it's less about efficiency i mean based on my experience in this whole thing right now it's really really hard to justify that a quantum algorithm or to find sort of quantum advantages really in terms of speed and so when it comes to when it comes to portfolio optimization i guess what you're looking for is you're looking for better models so you're looking for something that works better than the classical models maybe because of the entanglement maybe because of i don't know um so like if this is where the bs is like i'm fine with it right like i i'm i'm all in for uh trying to understand uh or trying to you know play with things that might be just different in nature um but if the ps is is you know that that just doesn't work then let's figure this out okay so let me open the the paper so i hope you can see this portfolio um okay so this is july 6 2020 provide the position of 40 stocks is in d-waves quantum annealer we've got introduction validity of the formulation classical methods and using an annealing kind of computer and hopefully we can reproduce these i really want to go and reproduce these like um introduction because i think they claim that they claim that you know it's everything reproducible so i i definitely want to do these um although i might need access to the wave uh which uh to be honest i don't but like let's see how far how far can we get introduction blah blah blah validity of the formulation classical methods uh brute force genetic algorithm random sampling heuristic approach simulator as a monte carlo using an annealing quantum computer the optimal portfolio developing the cubo to number of assets in the portfolio um a fine transformations of the cube i guess this gets into the mathematics visualization matrix okay that can let's see okay so a discussion explosion uh so what can we see here can we see completion time by method so they are claiming this is faster or at least the the seems like this the completion time by method is is smaller i don't know what c q a c q n s by method r versus quantum annealer computations quantum and classical so the quantum [Music] lower is better okay so the classical is better okay let's start from the beginning maybe let's get let's get to the bases okay uh yeah okay so um the abstract we investigate the use of quantum computers for building a portfolio out of a universe of u.s listed uh liquid equities that contains an optimal set of stocks uh okay so the goal is to build a portfolio so you're picking a portfolio you're picking basically out of a full of assets you're picking a portfolio that it's optimal i guess in terms of maybe some kind of return schema or something i'm not i'm not an expert in in in in um in the stock market so i'm gonna i might have to learn something extra here and there starting from historical market data we look at various problem formulations on the d-wave system so starting from historical market data we look at various problem formulations on the d-wave system here after called d-wave to find the optimal risk versus return portfolio an optimized portfolio based on the markovic's formulation and the sharp sharpie sharp ratio a simplified chicago quantum ratio a cqr is a chicago cornridge and then a new chicago quantum net score ckns approach this first classically and then by our new method on v-wave our results show that practitioners can use d-wave to select attractive portfolios out of 40 us liquid entities okay so the goal i guess if you've got like 40 entities and then you're going to pick a portfolio of some i don't know maybe you know six or five or whatever out of these that it gives you an optimal uh return versus risk kind of um what i would kind of so the it seems like they're simplifying that this is a bit you know this i might be cautious when i see this right a simplified chicago quantum ratio so they're they're simplifying something based on a i don't know what the what this is mark of its formulation uh i'm gonna open another topic because i'm not gonna see this but i'll if i find something useful sorry portfolio theory okay so there's a hall there's a whole it's a whole like portfolio theory and at least name risk and expected return okay so there's basically some mathematical equations here let me show you these guys so this is basically uh okay so we should probably read through these as well later try to understand whether this has any influence in the results but like okay so you're picking they're picking model they're simplifying it it's the and then and then they run it both classically and um good um and quantum uh so the challenge we approach in financial portfolio is to maximize expected returns while minimizing variability of expected returns or uh risk so the defined risk as the variability of the expected returns this is a buy and halt strategy and not a meet or high frequency trading strategy it relies on previous period risk and in our case one year of daily adjusted close data and the underlying variability and relationship of equities we believe investors can improve their chances by selecting the right combination of stocks among the major challenges in financial portfolio is how does an investor balance long-term investment inverse between expected return and volatility and this is why we tackle this question for a variety of methods this problem is particularly well suited for an annealing solution either classically classical simulated thermal annealing or quantum annealing since we wish to consider n equities in which each equity may be included in a portfolio or not so this yields exactly two to the power of n possibilities okay so uh for a potential list of equities as small as 40 this becomes nearly invisible in a work say on a workstation when we approach the entity of the s p 500 uh we very quickly run into solution space which is computationally infinite that is we do not have enough member in the observable universe to run through a bridge for a solution this work is structured as follows okay so there's some exploration with a sharpie sharp ratio where that is the range of covariance of a portfolio with the market over blah blah blah okay so this is then details on the actual model um we'll come back to these in a second from here with developer chicago quantum reacher okay and it's the covariance of the i've asset against the entire market this is a slight improvement over the sharpie ratio in terms of computation as we need not consider nominal assets okay so they so these index simplifies the computation that's what they're claiming uh we can also formulate security matrix form we explore this form okay so they go ahead and do a matrix form of cqr um which i guess may be applied but that's why you took 8 model-ish like applied to a state or or i don't know what they want to do with the matrix we explore this formulations by a variety of classical methods this hybrid causes some different set of mathematical problems in formulating a consistent quadratic form finally we settle on the chicago quantum net score which is given by okay so it seems like they've got some trouble adapting the mathematical formulation to feed into the d-wave stuff and so they adapt it um and then okay we'll dig into this i'm just really going to fly through uh in this first video of validia formulation and its current capacity to use source problems which are from the in terms of analyzing model this our practical challenge is not is to provide from d-wave and an acceptable model on which you may begin its computations consider following image image one voice image one here comparison of cqr and cq ns scores against the sharpie ratio what are you comparing i don't understand that uh our formulation has a propensity toward contoured conservative side in investment terms however it also demonstrates that the pres the present formulations are near the efficient frontier of investment portfolios this from an empirical perspective the solution passes master we develop the method in more detail okay so this is basically what they're saying is the method makes sense but uh we can dig into this later as well then classical methods so it does seem like it it does seem like uh they're trying to use indeed the wave system here is for um uh tractability like so they say for 40 portfolios we can you can you cannot just brute force these because you've got so i guess i'm guessing the idea here would be you go through each possibility you calculate whatever ratio you're using in your model and then you just you know sort those and pick the top five right that is of course intractable with 40 because you have 2 to the power of 40 possibilities so you can't you cannot just force it um and the claim here is that the wave system can can't do these because yeah uh i gotta pause for a second i'll be back in a second you will notice difference okay so i'm back uh you probably didn't know this but i just um yeah uh took some time off uh it's a couple of things to take care of but anyway um better this way maybe so what i was going to say it seems like it seems like the issue here is um or they're trying to use the airport to uh sorry the airport i'm at the airport but they're trying to use the t wave system uh to uh basically be able to actually do something right because the 40 portfolio the proof force is not doable so here's a performance discussion the smaller asset universe are able to do to loop through the all binary solutions and then perforce roughly for the assets um that's not possible this generic algorithm uh okay it gets to a local minimum deeper than our monte carlo method with 950 million samples and does so very quickly our difficulties in tuning the parameters for number of evolutionary steps probability of leds in in size of initial population even with essential random guesses and these parameters are generating which is a low enough energy level so that uh we can determine whether the quantum annealing social media so we can determine whether the quantum annealing solutions are legitimate okay so here they're what they're saying is that with the generic algorithm they can at least get something comparable like they can they can compare with the two web system uh output to at least uh check whether the out the the the the system is not like bsing or something random sampling as mentioned earlier slightly more than a workstation camera without any place algorithm just randomly sample as much as we can um we're able to sample roughly to the power of 29 portfolios of potential to the power 40 this means most of our effort is spent around portfolios of size 40 plus minus square root of 40 percentage wise this doesn't cover much of the entire spectrum uh but we approach point four percent of the meat science portfolio okay i don't know okay heuristic approach but what's the point of using the d-wave system then original test of our problem comes in the form of simulated thermal learning solution using statistics of random mattresses we're able to tune in the parameters of our simulated kneeling solution to deliver very deep local minima additionally this style solution only covers minimizing grease skin portfolio based on the cooling rate of the thermal annealing okay so using an annealing quantum computer the optimal portfolio in our case is one which maximizes the sharp euratia however as presented by the sharpie richards of portfolio is not computable as a qbo so then they have to basically um the main thrust of this research is in fact okay so the aha okay so the challenge is to formulate it in a way that the d-wave system can eat it um blah blah blah developing the qbo we'll get into the details later on i'm just flying over things right now um because the point is like if if they have to do some simplifications we'll see the results and i'm definitely i'm really we're really going to replicate that embedding scaling and hardware considerations [Music] inspector shows how the system encodes and embeds assets into physical key weights an attempt at changing the formulation by manually embedding terms to respect the reordering of assets does not yield substantial improvements um ethane transformations of the qbo qubo and export profilers of different sizes will present a different matrix to doa for each desired size of portfolio add a penalty for exploring portfolios different sciences one obtaining accurate values on example the addition for these follows closely from converting a cubio into a nicer model this transformation blah blah blah okay so this these are more technicalities um ah i think we're gonna sneeze no visualization initial landscape is critical and learning how the wave finds a solution it also is our understanding of how matrix transformations adjust the landscape to improve the probability of something correct as the values matrix c q n is calculated if you place a penalty on the smaller portfolios by adding the sheath subtracting the shift turn places the penalty for larger portfolios um okay results are explaining the waters as follows download one year of daily market data for a specific date a set of n assets in the universes current as of that moment halt data for all experiments calculate covariance of each asset with the market say how much this varies with the market in respect to the market based on log returns calculate covariance terms between assets calculating they're lying in summary values including sharpie ratio and chicago point in score for an all asset portfolio okay so kind of hold all the 40 assets okay so what they do is they okay collect okay so that would be the yeah because that's probably the word i mean that's definitely the worst you can get like just pick out pick everything right and you want to pick a smaller portfolio um that maximizes or minimizes probably the the ratio the right the cubia for each portfolio size portfolio size this i don't get um this part i don't get uh but it's because i didn't get into the keyboard representation part but run a classical progress scale with me in our case a generic algorithm to see one best portfolio and its values okay so run basically one of the classic alternatives so then we can compare it with the d-wave results execute like this an appropriate range of portfolio sizes usually result to see the generic encryption compare values classical so they use the d-wave to seat the generic algorithm okay these ii so they are not using the d-wave system to choose the portfolio to choose the answers but to inform the generic algorithm and genetic algorithm i'll have to go into details i guess the following figures 34 give some idea of how well the quantum computer performs using the ckns against the sharp eurasia we see that in the sample that have purchased the efficient frontier in a few cases highest return for that level of risk most points chief one will also miss the baby shouting uh we see classical random sampling on average for all portfolios which is where most of our portfolios where run this shows that d-wave is not picking randomly or average solutions but good ones life classical expected to understand the deviation i don't know if you can hear this as a baby in the background discussion and collagen positive semi-definite considerations practitioners of numpy will know well that numpy is prone to running errors in particular we find that numpy computes covariance matrices with slightly negative eigenvalues what [Music] okay where's the codes so let's find the code next steps references thank you where's the code maybe in the medium article so [Music] please see our archive article uh here on youtube you can play this here there is our maybe if there's no code i just should try to do this myself so kind of based on the paper go back to the paper um what about it what if i just search for these code say anything github maybe researchgate [Music] maybe in the webpage oh that hurt so portfolio contact team blockoff and security use cases more car logging block offerings during use cases it's okay maybe maybe there's something no uh do you wave for your portfolio and it's just one red hole quantum markov is a dui sample code could this be there could this be that who's this user so data averages covariance prices say anything in the code that kobe it's embedding composite okay could that be the code who's this person i don't know if this is the course i don't know whether this is these perforated wave sorry i've been checking i was checking other tabs i i apologize for this i started the video like that and i can't i can't change the um i guess i guess we maybe can just start working on these to be honest so is there enough detail i guess this basically so here there's the there's the formulation right so they they say we can reformulate cqrs in matrix form explore these relationships around classical methods do you wave requires a linear quadratic form we attempt to rectify this by exploring the following which i don't understand and i mean i really don't understand but i don't understand one thing so it so so what what are they calculating with the what are they calculating with uh with a d-wave computing computer so rw is a weighted portfolio and alpha is a real number and most experiments we choose an equal weighting uh brute force genetic algorithm our general consumption gets a lot coming in deeper than carla method random sampling heuristic approach one second yeah i running out of time but i tried this that days don't understand where they are calculating with the which means the sharky rage however shutter is not computable so trying to throw sensors in fact how to formulate a qbo which when presented to d wave pretty similar to the classical sharpie so they're calculating the actual ratio or the numerator can be expressed as a simple dot product where we expect the return and the relative weight of the ice acid respectively the denominator okay so the d wave computer is supposed to give a weighted like an answer of like the weight of each of the portfolios that minimizes that ratio and still they use these as a starting point for the generic algorithm later maybe keyword two number of assets in the portfolio this is what i don't really get we're dealing with a single asset portfolio we only consider linear terms in the keyboard and transfer matrix i think i have to so do you wave i think i have to go through this first maybe try to get something there up and running that's what i'll that's what i'll do in the next uh in the next video stanford quadratic and constraint binary optimization is from the science restaurant falls there the key problem is defined isn't an upper diagonal matrix it's going to express more consensus okay all right terms could write a fishing variable but what's the what's the way that do you wave define an objective function objective value q1 i guess q0 q1 being the qubits and then so i'm guessing what they're doing is kind of define that rate as the function and then as a function of like different portfolios and then try to kind of get the um okay so each qubit will be an asset probably i guess that's the point and then you want to have the 40 assets represented and you want to get an answer of uh what's the probability associated with each asset i guess that's what i guess that's what you want to do but i don't get like one thing that i'm missing here is how does these then fit into the um genetic algorithm genetic algorithm it gets to a local minimum deeper than our monte carlo and that's so very quickly our difficulties including the parameters three number of evolutionary stats for ability of altitude and science and potential population it was essentially random guesses at the parameters which is learning to learn developing the keyboard i'm betting scanning hard considerations um because i don't get that step like i i don't get that like uh around a classical so execute the d-wave using a proper range of portfolio sizes and use the d-wave to see the generic algorithm this is what's not clear like i get maybe the d-wave is used in here to work work around the brute force or the the the the fact that you can't just brute force the thing but that doesn't still give you an optimal yeah i don't know i i it feels like there's some pieces missing here so i need to find i need to find more about these in terms of what are the other like is there any reference code anywhere that i can to take a look at otherwise like i i can what i will do is i'll try to implement the dwa stuff they talk about in here uh and then but i really don't know what the genetic algorithm is that they're using and how does these uh so we further give results of the cqns by method once one will notice that the waveforms are in fact depending better is awesome with the current methods but unperformation i got with i'm using dwf as a seat guarantees that you know from better okay so that's the reason as we disallow anything worse than the c to propagate through generations of solutions uh okay yeah so it's basically okay so you create an initial population based on what d-wave tells you and then you you tune the actual uh you you tune the actual uh genetic algorithm not to go worse than these um and so you know you can only get better okay yeah i i don't know i don't even know where whether you can call this bs or not ps or like where is the problem in here i might be like to be fair it's like i don't know where um i really don't know what is there to evaluate at all it's like it feels like they're using the d-wave system just to get a better first gas and and it it's it's maybe less about efficiency but it's about the quality of the gas right if it's about the quality i don't i don't think that's bs but like if it's about efficiency yeah that's another that's another level it's a discussion at another level i guess but i don't know let's try to implement this okay and then we'll make we'll get conclusions at the end of everything so we'll see yep gotta go bye-bye |
please take a look on quantum money and semi quantum money i can share you some papers, Happy to do so if you share some papers ;) |
Hi, I am Jeff from Chicago Quantum. |
um and i basically just kind of wanted to quickly hop in uh you know i just want to quickly quickly just hop into the into a stream to kind of maybe do do a bit of a out loud kind of testing um i don't know i just you know also even if anyone's joining and you're interested into knowing a bit more about the details of this idea the entanglement hypergraph idea then i'll be happy to answer questions as well if any if not you can watch this offline as well uh condo activate ehs and then we'll just go to my simulator notebook for the prototype and open this up and i'll just quickly actually share this stuff in twitter see if anyone's interested in joining oh give me a second i gotta go open the door cool i'm back in there um see that's working that's working i think so yeah um okay perfect so what i was thinking did this open at all oh yeah i did open another tab uh there we go so open this up and then i'll just quickly drop a tweet that says um so we are feature complete stream running at twitch.tv uncertain systems quantum computing why is it so slow computing computing okay perfect so um cool now um first let me try to let me just try to take a step back and try to understand i'll just i'll probably be talking to myself for a while but if uh if you've got any questions let me know if not i can watch this later uh and ask questions offline as well um so what is the whole you know what is this whole idea for right and actually in the quantum daily if you go p if you go you know get the list of articles that i recently published there um there's a bit of an introduction to the concept and i'm thinking i'll i'll definitely add something to um figure on certain dot systems uncertain certain dot systems there you go that takes you to my um to my page in strange works and i'm going to put something in here as well soon about these a bit of a more maybe sort of a version of these uh a version of these as well so no i don't what did i do yeah okay i think that's the right okay so this is these are the posts and that was the first one that i did i probably probably should just kind of compile this in a big list or in a big post and then put it somewhere but anyway so i i what i primarily what i wanted to do today is i want to kind of do a bit of a first fly through and fly over sort of testing scenarios or testing cases that i'd like just wanted you know to use the tool for and at the same time have a bit of a random thinking around whether you know kind of want to take these in terms of next steps right because this whole idea was born out of the need for a so two things right so one thing was a a need to kind of maybe understand entanglement better and and see you know see visualize entanglement in a more clear way and then as a side no as a sort of a sort of a byproduct of these i thought you know this might lead into some kind of more efficient um representation for a simulator right um and i'm now like i finished the prototype and i'm happy because it just confirms that it it does what i expected this thing to do there's a couple of quarters to to polish obviously but it does what it's what this is supposed to do uh i should run everything um yeah but basically i kind of thinking about where where to take this next basically right um and so let me maybe first it's just going to be an exercise at least for me just kind of walk through uh have a bit of a really explained idea to myself if if you've got any questions please drop them in chat um i'm more than happy to um to answer questions if need be cool see let me just so let's just start from scratch right where did this all this whole thing maybe i'll just use paint and my um my digital paint and my digital pen here to express and explain that a bit better cool so so and again it's just also i'm doing this as an exercise for myself to um give me a second because i just peel coffee that's not good that's definitely not good so get some paper for this cool all right okay so cool there we go if you have got any questions just drop them in the in the chat as we go on with the explanation perfect cool so um let's see it's not working yeah that's working uh cool so the the reason or the idea why these kind of um sort of bubble tab in my head as an idea is you know i've always had trouble when you have a state vector right the states like the formalism tells you you know that you basically have let's say a um a two qubit system right and you know you start in the zero zero state so you you represent this with that with a cat like these um and basically you evolve this state by applying you know the gates right like the hardware gate the x gate and so you know you you may think regardless of this is cubit zero let's assume this could be um one and say just keep it keep it one and this is cubic zero right so if you apply if you have a circuit like these that applies like an x an x gate and this is cubic zero and this is q b one right and a harmonic gate here so what what you'd usually do is you'd evolve these into first like the zero one state right that's this x here being applied into cubic zero and then the harder mark would basically um evolve your system into the 0 1 state plus 1 1 state right so this is now your whole system and and so another thing you can do is you have for example another type of system right which you can use you know control operations on and so you have a qubit that you apply a hanamark harmonic gate to and then you apply a control not gate and that's the typical bell state which basically you know takes you from the zero zero state to the zero zero plus uh one one state and i'm omitting here all the complex um you know coefficients and and what not the amplitudes in the phases just for simplicity um but basically you have these so you have this set up right and it's like you know intuitively especially when you're starting right like i guess when you're getting this thing you're getting better these things you're better at reading these state vectors because this is basically state vector so if this is a and b this is your wave function if this is a b in terms of coefficients this is a b in terms of coefficients so um you know the state vector in this case right you have a two cubic system so you have basically four potential elements right so for the zero zero element it's it's zero because it's not in the wave function for the zero one is a for the one zero is zero and for the one one is b right so that's kind of the way a a really naive simulator would work is by keeping track of the four possible states and evolving them so you can see that exponentially grows with the uh with the amount of qubits which is why simulators can just can't just go over a certain amount of qubits um and there are already of course smarter ways to you know represent that in a more compact way so you can have more efficient simulations um but whatever that's that's the the way you you usually would look at it right but even if you take a look at these and these right like there's no quick intuition or intuitive way that tells you you know um that things are not whether things are entangled or not like you can't easily like yeah if if you're getting if as i said if you as you get good with these you probably are able to recognize you know pardons easily but usually the recipe to figure out whether it's entangled or not is okay can i express can i express this wave function as a product of the two qubits so can i factor these out right um in this case if you've got this here right you can actually factor this out because you can basically take the one because the the one is shared in here so you can just say well it's like me having the zero and and one state and then having this multiplied by the one state on um you know this is this would be um so this would be q b zero right and this is kind of cubit one so so you factor this out and then you know this here i'm so dirty this is basically the plus state um so you know essentially you kind of have these so so you know it's not entangled now here you can because you cannot factor anything out so it isn't tangled but it's you know it's like as this grows as as you know the system grows that becomes obviously cumbersome because you have to figure this out right like what do you whether you can um whether you can do this or not and really i haven't found any tools out there that help you do these you can also represent systems using a density matrix but i think i'm not going to get into that because i don't know enough about it in terms of how much it actually shows you about entanglement directly but anyway the whole idea behind um the this this ehs or entanglement hypergraph project is to work with a bit of a different um work with a bit of a different representation in mind right let's say look what if we what if we start with a system already as a in a representation that is already sort of just the factored version of the system not not the full-blown expanded version of it with all the possible measurement outcomes but just the single qubit representations just just keep that right because that scales well so if you have 100 qubits and you need 100 data points right okay they're complex numbers but whatever you have like just just that and of course you you can't then easily see for example what's the probability of measuring a certain outcome out of all these because you need to compute that right um whereas here you see directly it's like okay what's the what's the probability of measuring the the one zero the one's your element well it's b squared right the bond rule um or something like this uh yeah it's b squared um so and this is how the idea kind of uh was born right in the sense of you know you have to qubit system right uh q0 and we keep track of just these the state of q0 which is in the zero state and we keep um this is by the way the wrong way to say and to do things it's just i i'm gonna use the zeros the number zero to represent the zero state that is in reality a um a complex number so in reality a qubit is sort of a two it's it's a two element array right one for the zero component and one for the one component so in this case the zero would be basically a one plus you know zero i in here and a zero plus zero so so you're just saying you you you're saying that this is in the state zero so that's the technicality here but i'm just using the shortcut here to say it's it's the zero state um and so you you keep the qubit like these and then you you have the other qubit also in this state and so so far that's it right you keep these two two two numbers in your system and so now you have to figure out a way to evolve this system um and and in a way that it does not explode in terms of size because that's what you want to avoid so and again this is sort of thinking thinking through the um efficiency from the efficiency perspective of this whole idea right um so you know it if you apply single qubit gates it's easy right because you know that there's no nothing is happening there's no kind of correlations being built there's no entanglement being built you can't you can't build entanglement by just applying single qubit operations so you know if if we had back let's say the circuit um the the bell states are good right so so it's it's a hot market in here and then it's a control knot so the first operation is easy to track right because it's just telling me please apply a hard mark to keep it zero so our system would transition into into these right so it's still we have not increased the amount of information we store it's all the time too it's all the time yeah okay four complex numbers right like it's all the time um two times the amount of qubits right so for each qubit we have two complex numbers so remember this is in reality um two complex numbers like a 1 over 1 over the square root of 2 and 1 over the square root of 2 for both components so since it's two times it's not exponential right it's two times the amount of q beats um and and and now but now the problem you know now so the first challenge appears which is okay how do i represent a uh a control node operation right because this is an operation that can create entanglement and this is where one has to you know this is where basically you can you can do it the dirty way or you can do it the sneaky way uh the the early by dirty i mean proof force right so you'll you basically say okay cool so at that point i know that i have to do that so i know that this is represented by a um you know mathematically a four by four matrix so all i need to do is you know temporally i i do the product of those so i temporarily kind of give up on my save on my space savings and and actually build the um you know the four element uh the four element state vector and then just multiply by by the whole matrix and then as a result i get another four element vector which i can then kind of replicate back to um to my node structure by saying you know i'm going to have basically my structure is going to evolve now into a hypergraph where um i'm just realizing that there's so much that goes into these that it's it's it's complicated to explain even the mental process that's gone through this in terms of why a hypograph and and why would you do it this way so um what i'm saying is here you have two nodes which are independent right um i will try to explain this backwards the reason the the the way that you have to think about entanglement is that you have a set of possible outcomes which are just not compatible in terms of you you cannot have a mixture of those um of those outcomes together right so if you have the state if you have the state 0 0 plus 1 1 that's what the state vector is telling you that you either will measure these or you will measure these you will not measure zero one or one zero right so the way that i thought of representing these in this in this structure is by basically having a kind of two two copies of the system so you have a you have four nodes right so you have one node representing zero and zero so this is cubic zero and cubic one right so this is zero zero and one one and so you have a system that actually has all these these data points in here these are these are related with hyper edge the reason is a hyper edge and not just an edges because as the system grows and you have three qubits then you know a cro an edge that connects two nodes is not enough you need to kind of group the three nodes so so we keep it more we keep general we keep it general by just talking about hyper edges um and i'm not so sure even if this is the optimal structure um to power this but it's it's definitely a useful one so so this is you know this is sort of a way that you represent entanglement and this is how you're going to know that your system is entangled it's because you cannot get rid of these edges you cannot like factor things out right like if your system doesn't have any hyper edge it means it's not a it's not entangled or if it has you know hyper edges but then you know there's a third qubit somewhere that does not belong to any of these then it means that there's no entanglement between q2 and any of these right but q0 and q1 are entangled at the same time so um so that's that's kind of the idea of the of of this it's really this is just the core concept right so you have nodes and then whenever you come across a controlled operation or a two cubed operation it doesn't have to be a controlled operation they're also non-co they're also two qubit operations which are not which are not controlled operations and then you'll potentially have to build hyperedges right but once you have these hyper edges you have a really simple structure that you can run a factoring algorithm on right so let's imagine that i have a system i'm not even sure it's going to be correct but i have a system in this state so i have two hyper edges here right that's my that's my system so i can because i know that i i know i'm drawing this on purpose i know that these ones are q0 and this is a q1 just as a convention um i can think of these i can think of a strategy to factor these elements out right kind of calculate whether i can factor things out and and merge these hyper edges and you know because there's exactly one element that is different right i can factor out the rest for me like i can easily say look there's this you know this two notes which are the same for the same qubit in the two different edges so this should actually be connected this should all be one big edge right um and by future of connecting these then these two states which were kind of mixed in nature right it's like it could be either zero or one but it's not that it it is the plus state because it's entangled now now really kind of become the plastic because i'm merging i'm merging the hyper edge in here so um so you would end up you know the algorithm basically what actually what these does right like what most of the stuff in here is doing for you is is doing this factorization um automatically and we'll take a look at these in a second um but then you would end up with just a system that has one hyperedge that is like that and if you have just one hyper edge is the same as having no hyper edges um from from you know from that perspective and so you know that this is not entangled so the one way this concept could be used is um is to you know you you just load a state vector run the factorization algorithm and then you get you know at the end you get you know whether it's entangled or not so you have an easy way to to see these now i could have used like why i why i'm complicating my life with hyper edges and whatnot i could have used just some traditional factorization algorithm or any other data structure i just i think hype edges are hypergraphs are really visual in here and i think they're really helpful to um you know to kind of see that also apart from these the point is i needed a structure that i can evolve easily programmatically because the use case here is remember it's not only just to say oh give me a state vector and i'll turn it into an entanglement hypergraph and then i can factor things out there that that misses the point because if you i need to if i need to get and parse a state vector i'm already on a losing end in terms of performance because i it means that i have to read like a potentially exponentially exponentially big data structure which is you know it's not i want to be able to have a system that can evolve in an efficient way so what this does is you know as long as there's no control operations it evolves real efficiently as whenever there's a controlled operation then it starts um it starts to kind of uh grow it's sort of the downside of things here that if you're not careful you're not gaining anything like the the more you know the more entanglement you have the more the hyper h number grows and actually grows exponentially to the amount of controlled cubiets because if you if you [Music] let's say if you have the toefl if you have you know two cubes in the harder mark like if you have two cubits um in the plastic and then you apply a toefl gate right the way this will map into the system is and actually because this is an end condition you will really have to have four hyper edges right the hybrid zero zero zero zero one zero one zero zero and then one one one so this is the only place where we apply the the actual target operation because the two controls are in one so and these all get wrapped into hyper edges ah thanks for the thanks for the nice comments this is these all um uh you know end up in hyper as hyper edges and and as you can see now we've had basically two control qubits and so we have like four edges four hyper edges so our our structure actually does grow exponentially and that's not good now i'm currently working on you know from a prototype perspective that's fine it's it's something that like okay so i can i can simulate low entanglement circuits efficiently like 1000 cubits 5000 cubits is no problem but as long as like if you have growers algorithm that creates an entanglement like it creates a controlled operation across the entire um across the entire uh you know set of cubiets like then it just explodes because it has like a huge control zed in the middle of the division operator and that just explode right but there i believe there's a more efficient way to um evolve these but this is kind of going to be pos prototype right so because if you pay attention there's some sort of symmetry here as in like to be honest i only need to keep track of you know kind of this branch potentially i don't need these as long as i i store some information in here that tells me that hey you know there's all the branches in here maybe i can keep the branches in like these right and then um and then keep two nodes in here somehow right that tells me look you know somehow this is a branch that it's compressed um and as long as i don't you know clash with another operation that is going to then modify something from some of these uncompressed branches which then i can probably carve out from here there's going to be a smarter way to do these but i think this there's something you can improve there but from front of that perspective that's all kind of fine um i just you know it's nice because you see the entanglement like you see you see what are the values that are entangled and you can easily say if i measure this qb you know one to be one um it means these hyper edges don't exist anymore and so my system is now you know only made of these two hyper edges so i think it makes analysis um more interesting and it's visual in terms of you know the type of entanglement that you have and it's a great tool to explain um face kickback which i'll do now in a second okay so i'm not making anyone look dumb come on people that's not it's just i've been thinking too much about this stuff um if you spend enough time thinking about things like these and you end up just looking like smart but not that really so face kickback is a concept that a lot of people struggle with um because it's not um it's not a it's not it's not obvious right let's let's open quirk and and see it in action so what face kickback is is you know the effect that happens when you um you have a system like that right where two qubits are in the plot state um and now you apply a control um sorry one is in the minus state it's it's better so this is inviting state and now you apply a control x and you see what happened so actually this state hasn't changed but the control state has changed like it used to be a plus and it's a minus after the controlled operation which is super counterintuitive because you're like dude i just i just did a control like how it goes against classical intuition where one you one would think of this okay this is a if it's one then apply this operation but if it's not one then don't apply this and yeah sure that creates a superposition and what not but how the hell is that supposed to turn my state from a plus to a minus right because that actually has an important effect which is if i now how to write the thing then i'm not i'm not back at zero anymore i'm back at one i'm actually at one so that's this is a concept that i myself struggled a lot to understand um and and these these type of entire entanglement hypograph helps you helps in visualizing these um in a neat way which again i'm not claiming you cannot visualize any any other hat like you can probably take just the wave function within in notation and do the same but it's just a bit more mathy and cumbersome and it feels like the hypergraph version of it is a bit more a bit more clear so let's let's let's actually replicate these right so we have a we have qubit zero and qubit one right and we start in the zero state and in the zero state then we apply a not gate here so that turns into i'm gonna i'm gonna split the system like this so now we're in the zero to one state so far so good no entanglement and again we're very being more efficient because we have only two qubit data points we're keeping track instead of four right now come to harmar gates and so we turn this into the plus state and this into the minus state i can also write these things for simplicity um as a zero plus one zero minus one i mean not simplicity this is a simpler way of seeing things but this is going to help us see the phase kickback effect so this is my current system and i'm omitting here um again all the you know complex compo complex um numbers that are associated with these things um but they are there in the simulator so so this is what happened now an intuitive way to think about um a control knot with this type of structure is that you don't have to actually do a math multiplication you can actually think of this in a really dumb basic classical way and say look the control is now going to split my hyper graph into two hyper edges because here i have a zero plus one but i i'm trying to force the thing to to apply a condition only when it's one but it's not not only one and zero and one at the same time so a rule for a control knot would be now i split this thing in two right and then i modify this state accordingly so so this is gonna give me two hyper edges one hyper edge is gonna have a zero node and it's gonna have a zero minus one because i'm going to leave this state untouched and another hyper edge and i'm marking it now with the stripe because it's just easier because it's two nodes right but usually i would wrap it up in a circle or something so now you have like a one here and and and but now you apply an x gate right next gate remember what it does is it turns a zero into a one and a one into a zero so it really will turn these into this state which if you're a bit familiar with quantum staff you know that these states are physical physically equivalent right because the zero zero minus one and the minus zero plus one they all have its relative phase what counts is the relative phase of the one component towards the zero component and in both cases it's like 180 degrees um so they are all equivalent but mathematically we've ended up with something that is mathematically not equivalent right so we know from an experimental perspective these things should be equal and so and here's when here's when the so the face kickback becomes visible right because now the question is i have this i have this system so how can i how do i factor how do i want to factor these out i want to factor things out to see if this system isn't tangled or entangled or not and okay i can cheat to be because i know i know it's not entangled because um i mean for such a small system i can take it i can check it with quirk and i can see that it's not entangled right there the both values are independent um oops but i i pay attention that like it bothers me the fact that there's a global fight you can think of this as you know this having a global phase right it has a global phase of 180 degrees which for the sake of manipulating the structure you want to get rid of because you want to make sure you can compare things at the physical level and say is this state physically physically equivalent to these other states so i can check whether i can factor them out so one of the things that you know that um the factorization algorithm does is it first runs through all the branches all the hyperedges and removes the global faces but you cannot just remove the global phase you have to put the global face somewhere else and and what what this algorithm does is um it puts the global phase at the hyper edge level the reason is that the hyper edges are at the end of the day translated to specific components in your wave function and these components will have a you know its own complex amplitude and so this is something you can do i i have a couple of videos that go a bit more in detail on how i figured these out um but it's probably too much in detail for now but the idea is that this this system turns into the system like you know this this high price stays the same but then this one gets corrected so these two so so this is you know the it's basically turned into the convention of the zero component is going to always have you know no phase and but now you have a um like 180 degrees phase right which is like a minus one uh from uh from from a sort of mathematical perspective in terms of the because it's a complex amplitude so it's um it's like 180 degrees in the complex plane would be a minus one um and that now sort of belongs to the hybrids the reason this can belong to the high price is because in theory i don't care what element has these all i care is that when i when i wanna if i wanna ever translate that into the state vector i'm gonna have to do the product of these so i'm gonna have to do like the minus times one right um and i don't care where these minus one sign is because it's going to end up at the front anyway so it can it you know it can just stay at the hyper h level and now i can make a comparison because now i can say good are these elements equal can i factor things out well yes you can so these elements in this element now they are equal they are both a minus so i can factor them out so you know you could think of these being as sort of as an in-between step which is not you know doesn't make much sense but it's just to illustrate these you're connecting this element but now you need to connect these two right um and in order to connect them in a way that is consistent with the mathematical formalism behind state vectors and what not if there's any complex amplitude associated to an edge it needs to be taken down to that element right because now you're going to make a pure state out of a mixed state so these minus one now comes here and guess what you end up with zero minus one and zero minus one system totally unentangled and this is the minus state and this is the minus state and voila that is actually phase kickback so it's it kind of you know we've moved from a plus state to a minus state and sorry for the medicine here just by virtue of of running the factoring algorithm right so that's the it's what um it's what this system is you know supposed to make you know make you make it easy for you to understand how you know how these kind of things happen so i think this i think these ideas gonna have or it definitely has some kind of educational component to it that at least programmatically it lets you manipulate states in a way that you can see things like this happen uh while developing that i i also kind of learned a couple of interesting tricks around um things you can do to collapse you know to create superpositions and stuff like that but i'll leave this for another stream um let's play a little bit because i'm i'm taking a look at the time and i really i think i have max like 50 20 minutes more for these um again if you have any questions you know uh feel free to reach out like i've got the you know if you're not if you don't know i've got a youtube channel um where i'm archiving all these videos you can reach out via twitch as well the chat or also um i've recently purchased the uncertain dot systems domain which i'm really happy um and you can just reach me at uh shoot me an email at daniel uncertain dot systems um in case you wanna in case you wanna just put in the chat in case you wanna reach out um and wanna play with this this is available in github in github by the way um so it's still really you know it's still really it's a prototype right i i think it's a nice idea sort of both ways as i said from a um from a simulation perspective because i i believe there's potential to have you know more efficient simulations but also from an educational perspective and just as a tool to manipulate um a system in a way that you can kind of maybe you know design an algorithm or try to figure out you know how to implement a certain trick i definitely want to move away from just talking qubits and also talking like you know um actual numbers for example if you um so in the in the quantum daily if you go into quantum daily um i have a series of a series of articles that explain that in a bit of a more structured way i have to link them all still um but they all contain links to actual archived videos on you know me building that stuff uh but you have for example you know sort of what we what we've just let us just explain now um uh you know sort of the initial state and here i have a bit of a nice innovation for that and uh and kind of you know i take you how i take you through the process of how to implement a control knot and you know and things like these right so i think it's uh and i'm definitely gonna make a more polished version of these and make it public out there but let's play with these now i have it i have it basically feature complete i have the prototype feature complete in a sense that you can apply one qubit operations as long as you have the matrix so for the one qubit operations i do refer to i just i do rely on the actual matrix multiplication because i think it's just it's just convenient i might change that because ideally if you have a if you think about the fact that you have a hyper graph maybe you can think about the formalism actually being more around um transformations and how you can transform the hypergraph rather than just multiplying things around but because it's just um a qubit isolated operation these you know for for for operations like the xk8 the z-gate and the uh how to mark 8 i just rely on these and then i have two methods that implement um so i have a method that implements two cubic gates and it's also using the to be honest this is a bit of a cheating it's not cheating but it goes you know it follows the non-optimal approach that i described before so it actually uses the metrics as well but i have it there just so i can also run non-traditional style gates right for the for the controls for the controlled operations i have a specific operation that it's i think a little bit more efficient probably not because i'm expanding it anyway but um this is a place where i can definitely make it more efficient where you provide the controls like any number of controls the target and then what what is the gate you want to apply to the target and it's going to apply the um by default like the you know um all ones type of end controls controlled operation um so the way this works is like these let me just think we're done with paint now the way this works is like the following so so you load the whole thing and again this is just a prototype right so these first two cells contain the the bulk of things so and i'm really not happy with the names and and whatnot right but you have um let me start from scratch actually so what you do is you create a system by creating a hyper graph right so system equals type graph and you give it the amount of cubic you want um i will initialize this by default to zero if you if you if you if you um yeah i'll call this is the mass it's gonna be easier if you do these what these also prototype does is it draws the hyper graph it draws it in a um a way that at least guarantees that it's readable because uh you can do this in a rubber band style where you kind of wrap things around circles but then i realized as the amount of cubes grow it just becomes unreadable and i am definitely moving into a more friendly representation of these but this is just a working uh working example so you see that you have the cubic zero it's in the and this is again this is the you know real underlying data structure for the qubit it's the cubit state so it's it's got it's 100 on the zero um and it's 100 of the zero state as well here so both cubes are in the zero state and they are all belonging to one hyper edge called system the reason is the the visualizing visualization library that i was using um that i've been using doesn't allow for some reason to have no hyper edges since i had kind of make up the system one which basically means it's they're not entangled so this is you know you can have three cubiets and you have three and as i say you can have three thousand cubits right that that that shouldn't be a problem it takes a while because it just needs to render these so i think the so you know it's it's in it's more inefficient on the visualization side here than on the actual computation side and i hope i didn't just break things now um but let's see oh god i should have put the lower number it's definitely the painting that's can i just delete that what if i delete the cell delete the cell all your problems will go hyper graph say 10 draw oh god i think the process probably kept running so i think i might just have to restart the kernel and run everything see if that works yeah there you go so it's 10 it's 10 10 cubits again it's not perfect the ordering is just messed up it should go from zero to you know nine or whatnot and it takes too much just to paint this picture so i'll just i'll just keep this small anyway for the for the sake of of these stream and i wanted to do some testing but i'll probably do another stream tomorrow while really focus on actually hands-on testing these um but the idea is that you can evolve the system by just saying hey apply oh actually i think um i got a yeah did i actually integrate it here a um yeah there's this function that was basically sort of a result of a pull request by someone from the discord server um thanks a lot for these where you know it kind of allows me to apply to not specify this but i'm i'm sorry i'm just going to keep using the other ones for now so i uh whatever i can just do it like that probably that's work that's going to probably work anyway so i think if i i think that's the way it works right so yeah uh you know what i'll just i'll just oh no it's okay i know i know it's just comma separated so i can just say q0 and then yeah so you can see that for qubit zero i don't know if you can see this it's changed and now it's in the one state right it before there was a one here i don't know if it's too small hopefully it's not too small um and now it's a one state so this is the x gate i can also you know apply harmar gate instead and and it basically yeah you see now it flipped the ordering now the q0 is at the top before it goes into the bottom so it's a bit confusing but that's why i keep the labels in here visible um the visualization is just crappy but you know it evolves the state like that and what's interesting is when you now apply ah what's wrong with my typing today i apply a um i can apply a control gate like this or i can apply it uh with the um controlled operation right so if i do this now what you can see is that we've created these two um do i still have it on paint no that's already the other example anyway so you you oh let's write that example so you can see that actually works and at least in terms of the um face kickback so you can see that now you know things are like that and if i so you have the two branches this is one branch and as you can see it also has this complex number here because it's something that you know we factored out from these two um and you have this the one one and the zero zero um hyper edges right and what you can do here is also factor qubits right for now this is works really naively where you have to give it all the key like if they are entangled that's the problem you just have to give a list of all the qubits um i'll i'll improve that it's just a prototype but what this does is it tries to factor things out in in which case it just won't be able to so you end up with the same system right um if you know if you if if we had a and i and actually i'm in theory can even give it the steps so if the factorization is a multi-step factorization because it does h-by-h comparison um you will be able to go and walk through each of the steps so you can see things like like the face kickback happening right um but let's let's let's do the example so let's do the example that we had that we just did now in pain so so first uh so to keep it qb1 we apply an x-gate to keep it zero we apply a harmon gate we apply another harmonic gate to q1 and then we apply a um what do we have a control x gate so we end up as you can see you end up with four different hyper edges because it's not factorized yet right um again i know that's highly inefficient that's not the point right like it's exploding exponentially but i'm it's just a prototype so i'm gonna be working on sort of transformation rules that allow you to run controlled operations in a more efficient way so we don't expand to the full full set of you know correlations but now i can try to factor things out because i know i run it in query and i know that this is not entangled so i should be able to factor things out and end up with a minus minus state and i hope that doesn't leave me now in a stupid so that should this should work let's see so it does all the factoring stuff and boom you end up with a system that's now it's just it's unentangled right so you have you have no hyper edge just a system one and i don't know if you can see the little minus here so this is the minus state and this is the minus state for both qubits so it basically actually works so that's that's what we end up with it works at least for this case right um yeah again you can you can run you can also you can also to do apply the [Music] um apply the control operation is controls controls target and gate so i can also the equivalent would be that i would say you know these are the controls this is the target and it and it's the next gate so this is another way because it's going to be you know otherwise you cannot implement something like a tough legate right um so you know this should actually lead to the same outcome possibly yeah cool so this is another way and this is what's going to allow me to hook up a bit of hook sort of put in a bit of efficiency in terms of how this is run but kind of that's that's really the roth idea right and it's feature completely since you can also measure qubits so so now let's say we have a minus minus system right in quirk you can go and say look i wanna i wanna measure these right what this does is that actually assumes the measurement is deferred so it puts this qubit into a classical state of it's either zero or one right um and it still gives you the state vector assuming that you haven't measured yet but that's the idea that's what a measurement does is it destroys the coherence it destroys the the quantumness of the state and it's now either zero or one right um and you can also do pause selection which is basically what it does is it measures and then and it then just keeps the zeros and this is also supported in here so you can also just say you know even after factoring before factoring whatever but let's do the after factoring so i can also say s measure and i think i have to give it a a list of qubits and i say just measure q with zero so what you what you see now is that actually we went from a non-entangled system to something that looks like an entangled system and it's really cool to see this because it actually kind of tells you that entanglement and measurement do have a lot of similarities and you can see that now the qubit 0 is has this little m here which means this qubit has been measured what this means is the system you cannot mix them up again i cannot apply hadamard to these qubits because they are now collapsed right so they're not quantum anymore um but you can still you still have the whole structure and so you can still reconstruct the state vector out of these because you have all the numbers all the probabilities and whatnot and you can just multiply things and get the state vector as if you hadn't measured yet um but that's the idea and so uh you know basically you know you can see you cannot merge them right because they are measured and you can all but you can also you can also do is pause select and i forgot what that looked like as it uh posted like zed so possible like would just measure and select stuff in the um zero state by default so i can i can replace that with a pause select and what it will do all it should do is yeah it just keeps one of the branches right it keeps one of the branches and i i just spotted the back that should definitely not be an edge it should be the system edge so that's definitely a backside of the fix but it dropped the other branch right um [Music] so basically you kind of see that um yeah that's it's feature completely from that perspective [Music] yeah basically that's it really i'm bad at explaining stuff uh i'm really bad at explaining stuff so i hope that was somehow understandable uh i was just kind of quite happy to i haven't gotten because i don't have much time to work on this tab you know with the day job and and a bunch of other obligations it's always complicated but at least the prototype now i'm convinced that this is somehow doable and it is to some extent useful um so yeah um i'll do another stream tomorrow probably or in two days i'll see it depends on my time um where i'll really get to actually do some testing and the testing will really just be me kind of running through different circuits and comparing the state vectors at the end and then you know seeing if there's any flaw i i know for sure the factorization doesn't fully there's something i'm not it's not fully you know it's not really well rounded up it works but it's not like as you can see like it did now it didn't now work as i expected it where it would just leave this in the system edge it just kept an edge which makes no sense uh because this one component here makes no i mean it makes sense mathematically but it's just not how i liked it i'd like to have it just as a system um yeah and basically you know uh so this is this is this is it right so it's it's a bit of an educational tool so i think the the factorization stuff here i think i can also you can also add the steps as i said and if it's a you know it kind of walks you through the step it merges branch by branch and so you can see what what a thing is happening and and how is phase kickback happening um when when you're factoring things out um so there's you know an educational component of it that i could explore there's an efficiency component of it that i could explore with it um i don't know we'll see my plans would be to turn that into somewhat like a um a more robust and actual an actual library that you can work with um maybe integrated with quirk i don't know like maybe you know kind of be able to have a display here where you see these branches somehow um in a hypograph way nicely laid out so you can see the correlations um but i don't really know it was an exercise at least from my from my side to kind of you know learn more about how systems behave um without thinking too much about the state vector formalism um and at least it opened up you know my eyes in terms of how you can interpret controlled operations as in like h creation or edge splitting um transformations of the hypograph which i believe kind of can lead to a bit of an efficiency gain in terms of simulation but i don't know at some point it's like at some point things just will explode right if you need to modify something that you've got compressed then you're going to uncompress this um and that's when things go uh go south but the same with mps which is you know the matrix product state model which is also a compact way of evolving a system has the same flaw right like if entanglement grows then your your tensorflow items flow your tensor network will also you know not be as efficient as you'd usually expect it to be but anyway cool i hope that was interesting i i really wanted to do this as an exercise for myself as well to kind of you know take a step back now that i was sort of feature complete and think through okay what is in here and how can this be useful it was useful for me um in a bit of a uh you know um selfish way to learn a bit more about stuff and how you implement things like this and yeah i don't know if to take it you know down the educational route i then you want to have which is probably what i'll do i think sort of an educational tool slash maybe a sort of a bit of a debugging tool or a bit of a because think about these if if you think about grover's you know growers shorts algorithm it could be if you if you move away from qubits and you end up you work at the register level so your nodes actually have you know decimal numbers for example instead of just key beads um you could kind of play with the toy version of shore and see how the algorithm really exploits entanglement for these uh to achieve its outcome right so it it's it's got a bit of an educational slash you know development tool kind of or you know prototyping tool maybe prototyping tool type of uh vipe that i'd like to maybe capture in a way that it's useful some for someone actually trying to build a quantum algorithm which is break and break an actual algorithm down in a way that you don't have to do all the math like um you know factoring things out and stuff like this this allows you to do it in a programmatic way which is uh in yeah from that perspective would be even it would be even nicer here but the problem with quirk is it it's quirk really stays at the cubit at the quick really stays at the cube level it doesn't go at the register level um in a way i mean it does to be honest it does because you can have the input gates and all that stuff i don't know i need i need to reevaluate that a little bit but yeah i think that's uh and i'm also i also worked on a cyto escape export which is basically exporting this whole thing into a big json thing it dumps it into a jameson jason string so you can then because i'm playing with you know js based visualizations of hyper graphs which are nicer we'll see if that goes into a tool in and of itself something like quirk you know um i'd like it to have integrated to quirk i think because quirk is awesome just basically the flexibility you have here is just awesome so if i would just have a an entangled hypergraph display it's probably what makes a lot of sense it wouldn't be for efficiency perspective but it would be it would definitely be for a an educational or sort of debugging perspective and you would kind of be able to have a display here and actually even step through the factorization yeah i think that's i think that's what's that's that's what makes more sense that's what probably that's what's worth investing the time in i and even if even if craig does not like kind of maybe i like it as a an additional to the tool i'll at least have it for myself and maybe offer um sort of a quick version that actually has this gate available through through the sign but that's probably yeah that's probably the best way to go about these and yeah because with these you can like with with quick you can build cool um you can also build as i said right like in play with sure at the register level i like the fact that you can do things like these and actually yeah that's something you can't usually do with so this you know with all the simulators and it's just so quick and intuitive to work with so you can have the modular exponentiation part in here blah blah blah like you can build something quick maybe you know what yeah i think maybe there's a way to have registered displays as well well at the end of the day it's these right but you have here the elements it doesn't matter i think it doesn't matter it really doesn't matter because here you already have the it says decimal zero decimal one the symbol two so it's good enough that's definitely good enough yeah that'll that'll probably be the best the best next step i think so to put this into into quirk um and uh yeah so i need to have a nice visualization that i'll actually maps with this one uh with a style at least here and um because yeah you can i'm not so much interested in the efficiency component of these so anyway we'll see it's just a personal interest perspective i'm less interested about efficient simulations because i know eventually quantum computer will just beat a beat in the simulation or at least that's my feeling um it's more about the educational component of it and i think no matter what what algorithm you want to build there's always a small version of it you can build and to test stuff out so i think that's not it's not a big deal we're like oh yeah i need these you know algorithm whatever to prove that stuff like i think there's always a way you can just build a small version of it maybe not but who knows anyway i hope you enjoyed the stream and see you next time bye bye |
This is the clearest intuition for phase kickback I've ever seen. Plenty of vector arithmetic, but THIS is an intuition. Amazing work :) |
Thanks!! |
Thank you!, I'm happy you liked this! Looking forward to making this into an actual usable helper tool |
basically um what's going on yeah so these these hyper edges are basically um one has two notes and another one has three notes right that that's usually how you deal with these they they don't necessarily know about each other so so you kind of programmatically have to think about okay so i have a hyperactive has three nodes but are any of these other nodes maybe maybe that's maybe that's um maybe that's a way to do it right that you're like are any of the other nodes it's like consider related by another um hybrid because that would impose a constraint and i think that is true for the case where they're nested in the case where they're not nested because if they are nested then it means that um then i don't know what it means hmm because you know the the thing here right is that if i know that this is one it could this could still be one correct no this must be zero because i'm in the greenhouse branch so here i was documenting the rules oh god things are just going crazy slow anyway so so be patient with me i'm changing the machine soon but it's just a disaster disaster so the rule the rule number one would be that um how i would formulate it so sort of if i know i am in a particular hyper edge um then i know the rest of the notes in the hyper edge are true right and i i know this is just a funny it's it's not it's not formal at all but um not knows but elements right that is that is important because i don't know how to write like how to write the case that you know if you have if you have this nested case right it's it's like if i'm like i cannot i cannot just say that oh i know this is one so i can tell for sure that this is one that is not true even though i move thin i bridge but this is because this hyperedge is embedded into a broader hyperedge right so that's that is only true for like top level hyperedges in a way yeah that is kind of like top level hyperedge then i know the rest of the elements in the hyperedge are true like um how do how how would i i don't know how how would i formally describe these things right are true because the hyper edge that's the thing is the thing is the hyper edge by which is the future of just the the way it's defined these high pressures are just you know kind of happen to be independent and they all relate the notes not the hype edges [Music] is there anything special about this stuff nestor hyper edges nester hypergraphs hypercards maybe there's a better structure for me you know what i mean maybe it's like a like a tree right you know what i mean maybe maybe subgraph one sub craft two like maybe maybe it's it's rather a t-strike tree structure what i should use in this case to be honest i don't think so because there's also a lot of age cases run like that might look like that and i kind of i think that it's gonna be not doable with the tree it's just that we we have to we we have to this is a something that we have to yeah make sure that we can calculate efficiently so that we can say okay so so for this type of edge like am i you know are all this contained into another hyperaction so that's kind of what matters this kind of bubbling up yeah you know or kind of whenever i'm answering questions when you ask the model to answer questions right and say hey i know this cube is one so what can you tell me about the other key bits and then i'm like well this is inside this hyper edge but i can only tell that's true right if it's a top level if it's inside another high price and i need to know what is in the other high bridge and then i need to decide whether that's something that i can infer or not this seemed to be the three cases i have i have simply like hyperedges that are alone i have high bridges that kind of cut through and i have hyper edges that are totally nested um then this of course another case maybe of hyper edges cutting through two levels of nestedness like that would be incoherent if i was just kind of saying now what if what if you know what if these two are connected like that that makes sense so probably not make sense because uh it's inconsistent right because if i i know this is one end of these are two ones but then this hyperedge is something that it's just invalid so okay anyway let's go let's let's let's let's go where i wanted to go which is basically take a look at yeah so opening vs code made no sense and i'm gonna i think just i'll keep it there doesn't bother i probably about this but whatever um so let's take a look at these and let's take a look at um famous entangled states let's see if we'll get something interesting you know we have the g the the yeah the ghz these are these these are not interesting can be written as zero zero plus one one tensor product plus zero minus one one tens product minus hmm how would i even go about writing something like these yeah well that's actually telling that if i know this is plus then i know that this is zero and one if i know this is minus i know this is zero minus one one yeah okay so what this is just telling but this is yeah but this is something that we can then kind of later on say look so i i know the g so let's let's let's start from what we know so the the g h z is definitely like these it's just like that it's no no secret in here of course if i now ask the system what if what if the top qubit is the plus state but then i can transform these um typographs so i can this is another question um yeah yeah this is another question so this is this is uh but it's it's an answerable question so that's that's easy what are all the famous exact quantum entanglements so what are all the things that we be interesting to see mixed and tangled states this is really generic i want something quite computingish um bell states oh so okay what a spin squeezed states um this is something else i don't know known stayed okay how how were the other ones uh so for for three cubics there was the gh set and there was another class of entanglement three keywords and tangled and the double u states let's let's go through these ones so you know bell state is just a um just just the upper part here right so what about the w state so the w state is okay so those are all the ones that are um yeah that they have these these three okay that's an interesting cool circuit that that okay so this probably will take a while let's see if that helps me clarify some stuff so we have a state where so what we know here one way to specify this is that once we know that one is a a one right then we know that the other ones are zeros so this is one way to and and and that is always one one so how how will we do this so so we have zero zero and zero one one one and i guess the thing is yeah i'm not so sure if that's a good blueprint to kind of start always with like mixing up all the states right but we'll see because because because because because maybe if i would just follow the construction procedure which i can try to do okay this is a there's a y rotation [Music] the angle is these okay so cool let's close the rest so if i this is pretty this could be pretty messy right because if i know that this is a one then i know that these are these two are zero right so one way that i could do this is i kind of could could make the circles and then yeah could make kind of hyper edges you know one two and then three and then pair it with the other ones so it's it's a is there a cleaner way to do that hmm another way would be that i know that effects for example the top cube is zero right it's kind of making making it as a conditional if i know the top qubit is zero then i know that either y so then i know that um that either you have zero one or one zero right so it's documented zero you know what one could do something like that it's cool because i see this quite some so if so that i know that this is the way is that correct does this make sense because because you you're basically saying i know that if it's first of all i i know that if i'm in one of these the other one is not true because it's the same qubit and automatically i know that this is a zero right whereas here but i'd still have to say but i still have to say that if do i need another do i need another hyper edge here this is a pretty nasty this is pretty nasty or complicated one to come out just just out of the blue maybe you know what maybe maybe this will actually help me construct it better so if i open quirk because then i would still have to say that uh you know i can still kind of draw this right so i know that if it's one then this is zero zero that's that's just the here and if it's zero then it's it's wider one of these so and that's but that's pretty messy in comparison to the g8 set right so what if i do ah so a rotation of what is two what is this whatever um so what if i just do i have to specify degrees i think i'm just gonna say degrees right so it's a 109 or 47 degrees let's give it a test first okay so oh in radiance okay okay closer it's one 191 so i'll just say 191 no 191 cool um one what am i doing one ninety one there you go and now i'm doing a um control hard mark control not ctrl naught and x gate so control hallmark let's see if this works control not and control not and not gate yeah there you go there you go that's the state that we have so [Music] what's the state here so okay so it's like okay so this basically takes us into like a one third of the three zero of one third of zero and one and and two thirds of one that's what this rotation does without any like the the there's no relative phases that's good so this means because i'm trying to i know i said i wouldn't do that but it feels like it's a more natural way maybe maybe you know following the rules that i have see if i can get to a result that i like that it's better than this mess and then maybe eliminate some new things so i have zero zero and zero and i have one one no no what am i doing so we're starting we're starting with all we're starting with all zeros so we're starting with we're starting with all zeros um but then we do a rotation which basically what the rotation would do is it would just kind of rotate the state right so i would and i would really have a kind of like a um one-third zero plus two-thirds of of one and then zero and zero awesome then i apply a control auto mark so the control harmonic would tell you is would tell me it's like split the things apart right to split these two apart so then now i have like enough to keep this like a third of a zero and then two thirds of a one and now the rules would tell me apply harmonic to the second qubit the second qubit is the har mark is just the rule b take a zero into a plus state take a one into minus state so that basically stays zero and that stays as a zero plus one um that stays zero and here we have our first hyper edge and that is the first high branch this means at this point in time i know that if this is a plus state this is a one right so in this i can i can i can see that here if i do an access control for the plus state here i can tell that this is yeah so that this is what what no i guess the first the first weird thing is here what why not why is this like that 80 and 20. if this is the plastate right ah okay that's confusing oh no why i thought that's the point right if if if this is the harder mark and okay but i get why it's because yeah yeah because that control doesn't so that control basically it there's a portion of the zero that should be also counted is that correct you know that is that is half that is half plus and a half minus right so it's actually yeah it's actually a half so a half of these is you know um 1 6 right and so if i say 1 6 plus 2 3 then we get yeah kind of roughly 80 percent so i think that kind of makes sense ish yeah this is this is the way it's represented and again if i if this is this being a zero doesn't tell me anything about any of these so that seems okay the next step is a control knot down here now this is an important rule okay so now i gotta split this farther but i'm staying within this edge right so i'm staying so this is still like a third a third of a third of a zero it's still a um two-thirds of a one and now i get a zero plus a one this is what we're kind of doing as a mixture because it's an electron match right and now but now it's a control knot so yeah okay but basically what that means is that um yeah that kind of stays the way it stays isn't it uh that stays the way it stays it stays so i can i can i can mix it up i do this but then this turns into one into a zero and so the question is what does it tell me now this is an important question like after these control knots if i do a control display here yeah you see that is cool okay that is an important rule because here i know it's a plus state but after doing that it stays mixed up that's an important rule so i don't rejoin things crazy i don't rejoin stuff so it stays like that the control just mixes it up i know that in some cases i shouldn't be doing these but yeah that's another that's another edge case to look for right um that control has just mixed things up why why so the the so you see you know now now this would be this is what the control knot has done no of course ah stupid ah of course it does that's not a special rule man that's not a special rule because it's a control knot so it's it's it's gotten these zero and it's got a zero okay give me guys give me a second i'll be right back it's all back there sorry for that um cool so where were we we were here with this step here the control x let's see if we get that right now um well for one we'll need a one here right no whoops there we go cool thought i had lost i just wanted to go so we need a one here and now okay so now now now it's not it's getting interesting so so now we have well this is only going to be the case right if it is zero it can't be one it actually will be zero this we know for sure it's a bit of a bell state right in here but then uh but then the truth is now and now it's going to be that's it's getting complicated right which is like does it include the zero or it doesn't i think it does i think it definitely does because and again right that doesn't mean it is just it's supposed to be a mechanism to help us answer questions zero zero one one i think we'll end up with a really similar structure to the one here it's just i i think i have a i think i have a big issue and i try to do this from scratch with like trying to kind of cover all the cases with hyper edges and i don't have to that's the point because i mean the next thing that happens is a control knob between the first and two and the two qubits in this case oh and by the way what i was trying to do before is like that's obvious because a plus right if i'm controlling on a plus state here it's a full match here but it's an lcom match here so i actually i need to answer the question i have to separate these into two and then take that would still into a plus into a minus yeah exactly so there is a portion of zeros involved exactly so now there is another c naught gate here which basically tells me that you know like well there's a full match here and then the only thing that i have to do because they're full matches here as well let's just change these so that is an easy change and note that now i'm basically ending up with what i kind of had here right so this will be a one in black place this would be a one and this would be a zero right so this is kind of what i had here and uh and then yeah and actually that's pretty cool you know like it's uh now an x gate and so this means bam i ha i actually so that's not needed oh wait a second this is wrong i mean hmm that's another technicality but that's that's we're going to solve this now which i mean is like why i i shouldn't have two zeros here i wouldn't avoid that but okay that is now a uh zero and that is a beautiful zero and a one exactly now i want ideally so you know that is the trick here right the technicality that i have here is that because i've had to separate these two but i already had a zero here you know what i mean like that zero is redundant in a way i already have a zero so that's that that's the the thing is when i when i do the splittings here i have to think okay so zero one but i already have a zero um and the trick will be here that i somewhat have to make sure that that the amplitudes are okay like i think that's the that's the thing i think that is just becomes irrelevant right and um i wanted to do another test wait a second which is like is this mixture so so one thing that deep that i did before in my previous model is that i was keeping the weights in the hyper edges right because i was interpre it was the interpretation was different so but now i keep them here what i wanted to check if it's it's something that it's accurate in terms of what one would um one would see here yeah it is accurate in a way that we we can know the proportions but we but it's it's incoherent right so this means this this 33 percent chances will get a zero and 66 chance we get a one and like you know you see that here as well um but it's uh yeah that's cool so that makes sense so the these amplitudes so these you know probabilities kind of stay there um the question is like does this keep being 50 50 because that is also an important technicality after all these like what is the state of this giving yeah there you go there's more zeros than once and i think that's crucial that is crucial because this is this is okay so this is what's happening here right so when i have to do the split um [Music] when i have to split right so that originally was so that originally it was a zero then we turned it into [Music] a zero zero plus one so it's like fifty percent and fifty percent right but then but from these 50 you know this kind of house go here so it's it's actually uh 66 two thirds and one third now that's literally this like yeah two zeros and one one so actually actually it's actually here you can have like a two-thirds and one third should it be the other way around though shouldn't it really be the other way around why does why doesn't these change after the control now that's interesting i thought it should change shouldn't it well no you still have two thirds and you just you just change the size but you still have two yeah hmm so wait so the amplitude stays that is something weird actually because here it does change right after the not gate but then why doesn't it change why doesn't quirk change it after these uh uh well that makes sense right because we know there is zero this is before the naught then this is going to be zero do we know that hmm there is a lot of things which confuse me right now so do we know that if this is one then this must be zero like how interesting because okay so this is yeah this is definitely next level for next stream i gotta go now but this is definitely next level because here it's it's just awkward be right because i'm like if it's one i must be able and that's kind of what this edge here was the purpose for right if it's one i must be able to say that this is zero whereas if i keep doing it like that just because i merge these two things which kind of tells me maybe i just shouldn't merge the things because essentially if this is zero then i then i know this is a mixture all right so if it's a one i know it's not a big shirt but then what is it like it cannot be a mixture like i mean the answer was here right it just got rid of it the answer was here okay so so yeah so but then then you wouldn't have amplitudes in this case you just kind of have complicated complicated this thing with the applet is going to be complicated as well let's see okay but that's that's then that's the example that we're working on now because that's what's gonna that's that's stretching the model the w state cool this this part makes sense i know it's zero it's kind of like yeah that was even intuitively what i had built before right so it's just a matter of how do i answer the other questions like what if it's a one right how do i how do i tell that it's certainly a zero i know it's not a mixture so i know it's either well maybe so the thing is i know it's not a mixture i know it's not 1 0 nor zero one so it's going to be either zero zero or one one so that is maybe the only thing that uh i'd be missing to clarify here because that would still be i'd still be able to answer the question like what is then the value this qubit i know that it's none of these so it could still be zero zero zero one one one so why not one one one one so why not why not one one one one that is the like how would i would i be able to answer this question from this model my my first answer is that it doesn't but let's see cool see you on hopefully thursday bye bye |
okay so for this episode we got something really special so the guest today is Craig kidney basically the creator of query the tool that you see here right Craig also runs algorithmic assertions which is a block where you have the actual quirk link here and then it's a this is it's been one of the biggest inspirations for me especially because the way that Craig writes it really feels like you can kind of follow his thought process and this is all what you know quantum intuition is for me as well right so it's kind of about sharing my thought process as I learn and as I try to solve things and I get stuck at cetera right and and and it's kind of at the end of the day also disappear it with the turn the qubits office series which is to kind of share someone else's way of thinking about a particular problem so this time I asked Craig because you know a regular pasta will probably do easy for him I asked Craig to solve a little bit of a more difficult task which is fix this implementation of course on prism so I basically built a Kerberos algorithm that it's Blacky right so it has has a problem or a couple problems in it and then the challenge is you know figure out what happens what's going wrong and fixing you know with with a couple of set of rules that you'll see now in in the video as I switch to to his reply and spin yeah I'm not gonna do a step by step I'm just going to leave you guys with the video because I think it's just a mind gold of course my stuff you know knowledge and I think just a bonus in which endows the way in which correct things about it it's just that's just crazy yeah and I'll probably do a follow-up where I really kind of break down the way you approach the problem because I think there's a lot to learn about it and even fixed some of the stuff that was not supposed to be fixed but you'll see that okay so I'm gonna leave you now with a solution from prank and I hope that you try it and I hope you enjoyed I mean really at that point and kind of crossing a thousand subscribers was also a bit of a bit of a milestone for me and I'm kind of you know feeling excited about the message and kind of about the spirit of the channel and I really want to continue continue these and the next couple isn't going to be releasing a couple of other things around puzzles in ever I'm building up intuition adjusting that and I hope you guys are gonna like so yeah I'm not gonna steal too much to more of your time he is yes Craig enjoy all right so recently this person who makes a youtube channel about learning quantum computing sent me a puzzle to solve and apparently it's about fixing a broken grover search that's supposed to recognize inputs that have exactly two bits set so I'm gonna go look at this puzzle and see if I can solve it alright so this this puzzle is encoded into the tool that I made called quark and I'm quite happy to see that almost all of the tool boxes are available for use since otherwise I'd be pretty constrained okay so what do we have here we have this one box which is probably the Oracle for counting the number of two's and then we have this double a box so I think the first thing to do is to try to figure out what these boxes are actually doing and the easiest way to do that is to get a look at their matrices by using something called the state channel duality where we set up a bunch of entangled qubits operate on the entangled qubits and then because of a nice coincidence in how a court displays states we get basically a visualization of the matrix so I look at this matrix and I immediately see that it looks like most of the things are about the same size and there's kind of this diagonal that's different so I think that this is this is what the grover diffusion operation looks like so a diffusion operation will typically look something for reference like like this yeah so you can see that those look basically identical and if I repeat this and that you can see that they're exactly undoing each other and this is a self inverse operation so that's what this a a is it's it's exactly this game the other thing to try to figure out is this count gate this one is a little trickier maybe because it looks like it's going to produce output that's not being uncomputable anyways let's let's try the same thing with this but I don't quite have enough room to work with here all right let's get this out of the way temporarily and get that out of the way and move this down too and apparently the output is going on to those two so this particular pattern that was back here where you had a one bit set and you're applying a controlled Z is completely equivalent to just having the Z as replacing where these controls the Z like the one is completely irrelevant to that anyway it's so apparently I'm supposed to do that and then I'm guessing that this is okay it's not self inverse no right okay I need to so currently the trouble that I'm having is the fact that I have these two extra bits that I don't want to have to care about that are right in the middle of my state display so I'm gonna move them out of the way like like this and then shrink the display a little bit and we can see that that's still not quite right I wanna put these in the wrong order okay so what we can see now is that we have this diagonal so whatever this is doing it's only applying phasing gates like it this is the right shape we're just seeing that it's somehow become entangled with these things and we can see the nature of that by conditioning on this bottom give it so I'm gonna condition one way and condition the other way so clearly what's going wrong is that this is not actually a self inverse operation the way that I thought it was so what can we try instead of that let's let's go back at the initial state by holding undo and let's look a little bit more closely at how this count reacts to different inputs so if I have one of these that's on I can see that that's on and if I have two of them that's on I see that that's counting correctly and three of them that's also good and if all four are on then that's also looking fine it seems like oh I think all I have to do is I have two uncomputable all right so if there's a little bit of a hitch in the video I'm using a trial version of the screen recording software that only lets me do five minutes at a time anyways so it looks like depending on the number of things that are set here it is giving the correct output like I can see that this is often this is on when there's exactly two here so that that part is good and all I need to do is to do the second part of it where it gets rid of this intermediate state in order to to basically uncomputable that we've made so all I have to do is have to implement this count in Reverse and that's actually quite simple I just uncountably just repeat these steps in order to perform the diffusion except looking at this I can see that it's it's still not quite right and probably the easiest way to check that I'm actually on computing this way that I think that I am is to see if I can feed in states that are changing an incorrect way so what I'm going to do is I'm going to apply this this cycling effect here which is cycling through all the inputs and this is supposed to do something and this is supposed to do the exactly the opposite and then so at this point we should be back to normal count and actually it does look like this is working although it could be that there's some hidden phase presence so it could be that there's some phase kickback occurring and that's probably why this this doesn't look right and the way that we have to see that is by doing that state channel duality thing again so let's get rid of this for a second and let's get our state channel duality bit back and set up entanglement between the two things I still have enough room I'll just I'll just bring this stuff back later when I need it it's up some more room okay although actually looking at this all of the little arrows are pointing to the right and I'm betting if I uncomputable to be okay oh I'm just good at so many people oh oh so I do actually have the inverse here all right let's back up again so there's something else that's broken is it the diffusion oh right so earlier I showed you the example of what the diffusion operation was doing which was this operation here this one right here so the thing about this is that this inverts the state where all the qubits are in the minus state this initial state is set up to prepare all the qubits into the plus State and the state that you're inverting has to match the initial state so what I did here is I just apply these X's in order to get the qubits into the minus state it's maybe a little bit clearer if I just directly show that it's like that so you can see here that the states that are being amplified the ones that have a little extra green bits here are the ones that have exactly two ones inside of the little display that's showing here which is a little hard for me to show with my mouse because it goes away when I when I move towards it and if I instead use plus it doesn't work anyway so that fixes it and then we just have to get a rate a few times so we're going to apply our Oracle again because there's so many solutions oh I because there's so many solutions compared to the non solutions we're actually pretty far into the regime where Grover is going to step too far so we actually need to lower the amount that it's that it's rotating so we can do that like this and we can see what might actually you even less than one step let's see if if this goes to exactly zero as as the angle changes and it stops at about 1 okay does this one go to exactly zero then as the angle changes does it go too far hmm okay this might be tricky okay so this this is a little bit tricky oh I know what the issue is the issue is that this diffusion operation also has to be at an angle yeah that's that's exactly what it is okay let's let's let's see if we can find a way to get it kind of just right so let's try number 0.1 that's 2% and then if I change it to 0.2 and the leftover is 3% so it looks like that got a little worse so maybe I should go down instead or I should just try and complete the other numbers hmm seems like any amount that I rotate is actually making it worse so it could be that we overshot with the first step and I should actually go backwards so I'm not actually allowed to remove this gate but I can I can make it go backwards from from here instead of removing it no no that that also doesn't seem to to do it and it's getting kind of annoying for me to constantly change these angles manually so I'm gonna switch to these parametrized versions which can be controlled by an input that's elsewhere in the circuit and then count through it like this and just see if okay it looks like it just gradually gets worse in Psych it starts it starts off at about 14% and then this is actually the chance of getting the right answer is actually going down over time so I'm clearly over rotating which makes me think that I actually need to apply this over here and I need to break the rule that I'm not allowed to change this a little bit I guess I'll just cancel it out and then do my own thing this second part okay let's see if we pass through zero yes we do pass through zero and when are we passing through zero so it's roughly around round half of the way down so about here yeah and then we can kind of die out that looks really good all right there we go so this angle is roughly a half maybe it or maybe that's 1/4 plus 1/8 so this would be this would be 1/2 plus 1/8 and then I do the same thing here yeah okay there we go it's it's actually fixed now so the things that I fixed were first of all the input state didn't agree with the diffusion step that was being applied and secondly because there are so many solutions it was over rotating compared to what it should have been so I decreased the angle of rotation although I might have kind of cheated by having to cancel this out and thirdly it wasn't uncomputable shear which I which I um computed by by doing this and I guess that solves it |
Ok , this completely changes Grover's Algo , great intuition...., I still need to give more time to Grover's Algo , this intuition , really questions my understanding of Grover's ....<br>Thanks Daniel , great puzzle . |
WoW! That was clever🤓👐 |
quantum Fourier transform so qfg this is what we so this is where we left last time and what I am trying to do today is I'm gonna try to explore a bit more the actual problem because I think in the last video we just jumped into building the solution that was proposed here without really understanding what was I haven't in person I was I didn't have a good understanding of the problem so I want to just back up a little bit and then just just to remind you this is an example where we we're preparing a an input that we know it's gonna give us 0 0 1 as an answer and this is the actual so this is actually step number one step number two step number 3 issues like this is the actual way you build the QFT but I have the suspicion that as in most of the cases in quantum quantum algorithms is that those they exploit the structure of your initial state so the reason these steps work is because we we know that this has been prepared in a way that you know we can exploit certain certain properties of the state might be positive a negative uncertainty might be that the phases or the things are set up in a certain way I don't know the but I think that's and that's what so I want understand I'm not super I don't really like the way this is presented here I don't really like the fact that this is so that the example given is just they just prepare something which obviously has some rotations here which they are you know they're thought in inside right because those are exactly the same rotations were making here so I understand up for the sake of the example it's it's good but it's so basically I I've done two things I've opened up this morning basically I've done a bit of googling around and I also start taking a look at this block which is from which is written by Craig Guinea which is basically this block was shared I was not aware of the existence of this one of my subscribers of the channel shared that with me and I think that it's really uh I haven't really gone through all that at all just read a little bit the beginning and but I saw that there is a topic about how do you how do we prepare the states I think that it's definitely worth taking a look at because also I again ended up back a davits Cobb chick block where basically explains a bit the same thing and at least the problem is a bit clearer because here he explains it in a way that so we've got a function we've got a wave right and so I wait a second this was not here I think this is this was this was where did I find that I was reading something that's not what I'm looking for kana discrete Fourier transform let's just discrete Fourier transform I just found something this morning which was really eye-opening in terms of understanding the problem maybe this one yeah III uh yeah yeah this is this is this is it I mean because basically I understand I understood thanks to that what DFT really is right so the idea is imagine you've got and I was really not far away from that but I couldn't really grasp what was the de or like the Y was a discrete whatever so you have it makes you have that wave right so what you want to do at the end of the days you wouldn't know how like what are they what are the components that that make up to these so this is definitely not a sort of it maybe we can call it a pure way for it's it's sort of like it has has its own weird shape so it's it's obviously the result of composing one or more two or more waves so you want to find what are the think what are the the periods I think of those functions that make up for this one but the way you do that is first you sample so first and that's why it's called the screen I guess so first you take you you take your because that's sort of an like a continuous function and first what you do is you you based on on on a period right T you sample with equal equally spaced I think and that's that's kind of also what what we're gonna use in the corner I have the impression this is one of the reasons why the quantum algorithm uses sang the way these but let's let's go step by step so this is we basically sampled those points right and that's our input vector so that's our input so this is the input of the problem so you've got that sample you don't have the full thing you've got that sample and out of that's and out of the that sample you wanna you want to basically have a I don't know what you want to have but basically what you wanna have put a papa papa you wanna have I think here David explains it well so you wanna have is frequency filters blah blah blah so as the frequencies is what you want to have okay so we wanna have basically the frequencies of the of the waves that make up for that input exactly so before we what is the fridges form so you've got a vector and it maps into a new vector so this is our input that I you just you know checked in here right so this is this is your sampling of that and then we want to turn it into this and that's seems almost one-to-one to what the IBM guys you know to understand assume input signal consists of a true signal with frequency f10 and amplitude tool so it's described by this equation and a noise okay so here's an example of okay a true of a true signal that's exactly what I meant and in the sense that okay so this is explaining imagine that this is your input here right so this is your this is your input so imagine we know a priority that's made up of those two things off of this and then a little bit of noise and you know this plus these kind of make this signal right so and and why this is useful I guess one of the applications is if you're able to to get this and these out of out of just analyzing that and you can cancel the noise and do all this stuff like that right so that's kind of I guess how a lot of the stuff works now with noise cancelling or sort of basically signal processing whatever so um I guess what you want to have okay so that's the output you want to have is it correct after the FT so you wanna have two frequencies as a correct one frequency stand amplitude - I know the amplitudes okay look at this amplitude - an amplitude point three and this is what you have here so you've got point three you've got two so that's what you wanna get out I'm and those are the frequencies okay so this is the so here here you've got so here we've got the amplitude and here we've got we've got the frequencies so you've got frequency 10 and frequency 50 exactly I understand all that all those dots so that's what we want and that makes this approaching the problem much more understandable so we basically want to we want underst we want to get and how this is gonna be useful for Shor's algorithm I don't know and and but we'll see right and I think this is probably useful for other algorithms as well so we want to basically get that so we wanna we have we have this input signal and we want to understand which frequencies this is made now I wanted to for this video I wanna say I wanted to understand how can we represent that on how can we represent the inputs and the outputs like what's the concept and the quantum version of it because I think that's gonna help me so my idea is this is gonna help me understand the actual algorithm so this video we're gonna focus on understanding how the inputs are encoded and then that hopefully will make it easy to understand the algorithm so and I'm gonna I'm gonna I think here so here basically I'm gonna skip the first part where they were basically it is so they explain basically Shor's factoring algorithm sort of the overview I haven't really gone through this and a bit of background on the Fourier transform and then I guess the signals and signal processing and stuff like that um but I think this is where it's interesting probably so if you have a computer possible 32 states there is the thing that's the president computer for custom computers I'm quite a computer can rotated states through particular weighted combinations of the classical States called superposition is exactly okay so you can write down possible states of qubits this we know all this stuff so yeah so you have five cubits and you can basically write some things like like like this here right or like this here or like this here so a lot of fun combinations in this post when I say periodic state I mean a superposition where the weights of the classical states go like 0 0 0 0 not 0 0 0 0 0 not 0 ok so it seems like the because I'm guessing the and I'm making a big guess when he talks about periodic states it's this is about encoding the input because at the end of the day here and also here that's what it that's what it happens right so this is our input is and this is the period right one period two periods through third three periods four periods so we want to encode those things okay okay I understand so the the way that that he is suggesting to him to encode that is by saying that so we're gonna have a super position where the amplitudes are going to be the ones encoding the actual samples right so let's say this is 0 25 this is like zero 75 right so but they're gonna be equally spaced by by states where they'll have zero probability okay zero zero so four zeros and then one amplitude four zeros and one amplitude and so forth in other words the classical states that have nonzero weight should be evenly spaced and I'm I'm I bet you that this evenly spaced is gonna probably play a good trick and to understanding how the algorithm works um it's just my intuition is telling me that but um and they should all have their nonzero weight okay the same ah so that song wait a second I'm not completely right I thought that this was the value you were encoding but that would not make sense okay for example for example okay so an example so here we've got an example so if you've got this state 0 this state 5 or here has 0 5 10 15 20 25 and 30 is a periodic state it gives one divided by the square root of seven way to the States zero of blah blah blah and all the other ones are aren't given any weight so you see there's a difference of five in between in decimal okay so is a product state with period five okay okay yeah a more compact way to write the state is with summation notation at this point a certain subset of readers are probably thinking something something along the lines superposition that I bet the quantum computer is just secretly in one of those states okay let me just remove this the ask producers are real so we the specific quantum of pressure we care about in this boss is the QFT what the gift he does it's like it takes the weights of the states pretends they are the samples making up an audio file but they're all equal and your samples won't be equal figures at what the frequencies in that audio are then uses the strength then uses the strength of each frequency so maybe that's just the first step and then we're going to tweet the amplitudes to actually make them become the samples but that would be weird because the amplitudes I always sum up to one that sort of the probabilities figures aren't what the frequencies in the already are the strengths of each frequency as the new way it's defining the state of the computer a good predictor for whether you even slightly understood that last sentence is whether or not you mind just got blown hard it's all the time being blown hard and if you start with the classical state and apply giftee the resulting frequency spectrum looks completely flat the classical state all the action is in the faces not in the magnitudes by contrast the frequency spectrum of a periodic signal is definitely not flat like the spectrograms from earlier the frequency spectrum of blah blah blah if the current computer was really in just one of the classical states how could properly about the spacing between the possible states been getting into the output exactly so you see the spacing is crucial here the frequency spectrum tells a very clear story about what's really going on concrete example the green rectangle on the left is a chance display it's showing for each classical state the probability of the measuring this preposition at that point would return that state okay so that's exactly what we've done since it's separated by but do three four five six is a bit more nice probably not the same example um definitely not because we use five cubits here's more for the purpose of this post all that matters is that it can be done okay so we're that's that I'm happy that he's ignoring the QFT for the moment we're caring about the inputs and the outputs first so we understand how the algorithm is built the rectangle on the ride is a chance display of showing the probabilities of getting various outcomes when measuring the output state it has ten evenly spaced Peaks why ten because the number of frequency Peaks is behaving just like it did with really the input state spirit is ten so the frequency space output has ten Peaks just to check that if the opera's the number of Peaks equal to the period of the input okay before we continue I want to address why we're bothering with frequency space at all if we were rather if we are rather the period of the signal after the period of the signal why not just get it by looking directly at being put signal keep in mind that the green displays there is a non available in real life yeah sure so why don't we just sample the initial signals several times and watch for patterns why can't we figure out the period by sampling the input signal gee there sure are a lot of multiples of five in here the problem with noticing that a certain multiple keeps happening again and again is that as you'll see later in the problem we care about the signal where something from is going to have a random offset if we sample the number 213 that could be 50 times 4 or 13 times 2 or 13 or 10 times 21 with an offset of 3 okay let's just go see Peaks are resilient little imperfections preparing a prairie quantum state I've explained that that's probably the interesting part I've explained that if we had a priori quantum state with unknown period we could sample from the peaks in frequency space in order to learn something of the period but how do we prepare the periodic state in the first place first in any first in an easy case if the period we want is a power of two let's say 2 to the power 3 then preparing a periodic steady simple started with an N cubed quantum register initialize to 0 do nothing to the first 3 cubed and hit the rest of the qubits with a harem are gay each qubit with a harem our gate will transition from 0 to 2 this to this superposition putting the overall corner edges into the state damn that is a product state it's an cubed but can I pick the NI one if I pick if I'm just gonna start a new one I liked it I would like to stick to the example of three because maximization is easier so if I do this so there's an if I do that yeah okay yeah of course because you're leaving dot zero untouched here exactly so those are and now we have here so you see this like it's this one in between each right or in this case there's three in between each okay I'm sure entirely house is gonna so an alternative way to create a quantum state that has period aid is to hit every cubic with the harm or gate and the register we want to prepare into a register of size three then measure the order register and try it again if the result is in zero pretty crazy preparing all the periodic states instead of doing a three-beat edition like model age we can do a different model with some other number okay for example is it the progressive product quantum state with period 7 that applies a Fourier transform to show that there are seven peaks in the output state ok I'm gonna skip that for the moment I think we can dive into this later on in another video on how to prepare those states but I want to go a little bit farther without getting into the depth here we've got a simple example where we prepare let's do it like this so we prepare that with with period 1 ok ok ok those are other ways to prepare all the types of bring order periodic States figuring out periods from frequency samples but I think that's already the that's already the other part of the key of T that I wanted to avoid getting into before understanding you know or wait a second we're still in qft right history if you had periods from frequency samples here's a bit of a puzzle for you good it shows circular prepares the periodic stage using model addition like the last sections what I'm going to hide which modulus I'm using your job is if you're out the secret modulus three peaks so the R is 3 but remember when running an actual computation as I'm six preparing States with an on periods turning appeared into an extra square ruin put it all together okay but I didn't want to go all the way through the entire Shor's algorithm I wanted to understand why so the problem is that IFIF it feels I think this article that could be an interesting view on orthography but it kind of goes all the way through I really wanted to understand how PFT why QFT works and how does it work really um so what I get from here is that the periodic to predict states in computers so this is the input but still why why does this work if the amplitudes are all the same right what happens if what happens if we now do the actual this one so I'm gonna I'm gonna actually I'm gonna do it on this one so I'm just gonna I'm just gonna remove this and and do that so this is the and what if I replace this with that that's not what I was expecting that's not what I was expecting and that's why probably using different different learning material to learn one thing it's probably not a good idea always um good good good but I'm not giving up on this um so I kind of understand a little bit that concept I don't know if that's exactly how is this supposed to be done QFT how is that your sample I mean how can you how is that your encoded sample like I I don't get that right especially if we come back here so accent a quantum state and maps into the quantum state blah blah blah cubed Fourier transform so what does it kind of look like for a larger end yeah that's just a place the thing and I don't like that explanation because it's just trying to understand a little bit more how to prepare the state at all maybe that's just me just misleading so what I'm trying to understand is how do you prepare the initial state as your sample because I assume and I understand the sample is like in this case for example that right so how do you prepare that like doesn't make sense to me to me that those things are encoded in the amplitude because you would need to make sure that the all the amplitudes fulfill the rule of that if you square them and add them they are one so what makes sense to me is that those things are encoded in the and the faces right so you're basically so what it would make sense which is again maybe what was happening here okay I see this I was bashing on I was bashing as bashing on how this example was build but really that those things here I think so let me sorry let me just do this here was it like was it like a thing this was this was pie I think this was pie and a half and I think this was so the okay so the sample is encoded in in the that what is u1 give me a second what do you see one really gate as it is it a rotation on the z axis or one you want it's a cool to the okay it's an R sad okay equivalent to the rotation in that axis here so it's literally like you could use an RZ it's the same thing so it's so you're encoding that into the into the so say face okay but I'm curious if so if if that would be a valid thing to do I guess so if you have a bigger period I guess so I'm not so sure okay okay so but this is the this is the way you do it so you include your samples in this case is 0 3 5 4 - 0 3 5 4 here's a big here's kind of complex and that's actually interesting how do you okay so that was a bit of a nice journey I think we've really I mean we learned one thing which is I think that that actually makes sense it's good that's a good step um but I'm gonna I'm gonna do another video I do want to keep this video is too long so basically we're still figuring out how to encode that exactly is and why does this make any sense like why we're like how do we map between how do we map between like here we've got real numbers and then we are getting into imaginary imaginary numbers and then we'll see how that how the actual qft works and and and turns that into into the outcome that we want which are the frequencies studded off of you know of words that function made off cool perfect |
so much more time but I'm gonna do I wanted to definitely do a follow-up on on the cue cat paper q cat paper that the pharmacist design thing so I basically if I pull out my notes from last time II understand Vicki so I like I I think I did on the video at the end sort of a bit of a the XX icing coupling gate and in the meantime some of the author's from off the paper actually reached out and said they are open for you know helping me answer questions and and stuff like that which I really appreciate I want to go through some of the stuff solo first because I think that's what helps me learn and I think what's gonna help people watching is well enjoy the experience more but I'll definitely I'll definitely get in touch with the the group I think that's I think there's a lot to do there I'm okay I want to start taking a sneak peek at vqd and what is these and how is it different from pqe so and probably there was a probably reference on the paper I quantum assisted design or something if I just search for that there's actually a quantum aided design that's the paper that's the paper so if I take a look at the cute I search for vqd it's nothing new here Oh in a second that's not a paper that's for the that's the one for the optics hardware i won the one for it's probably this one here you go so the original find deflation algorithm fifty-two of excited states let's take a look at this so is this gonna be any how findable in our key in archives excited states that seems to be the paper did I click on it I click on it someone this is week here we go nice make my face smaller let's just apply through the paper see if oh before that let me pause for a second because I want to just make sure I'm freaking out these days that my sound is on you know because it happened that I did a one-hour session without any sound and that cannot happen again so I'm kind of having a little widget app here that tells me what that my microphone is working I'm gonna pause this for a second you should well one note is don't go back in here that makes my face lighter as well it's like almost like having a an actual lamp in here so what is about the calculation of excited state energies of electronic structure Hamiltonians has many important applications and is the calculation of optical spectra and in reaction rates while low depth quantum algorithms such as the very you know quantum eigenvalues solver vqe have been used to determine ground state energies methods for calculating excited state currently involved the implementation of high depth controlled unit or ease or a large number of additional samples here we show ok so there's I I don't know your show nabob is la he will show how how overlap estimation can be used to deflate eigenstates once they are found enabling the calculation of excited state images and their degeneracies we propose inclusion that requires the same number of key pizzas vqe and at most twice the circuit depth our method is robust to control errors is companies compatible with our mitigation strategies that can be implemented on near-term quantum computers still use the deflate eigenstates okay so there's the the algorithm in here and then this some mathematical proof and detailing x' error accumulations discussion sampling const destructive destructive swap test and that's the that's the overlap test the harm are overlap test am i mk2 so there's a better way to do that than what I'm done what I'm using for the vql is thing it's a modular multiplication okay once for microsimulation I can show me the references I want to take a look at the algorithm itself try to understand the diagram first maybe it's not all the way up so what have we caught in here so our prepares these from the usual state zero expectation estimation of animal is p1 p2 and PN and overlap estimation then classical adder calculates the overlap of severe the calculating the overlap of something lambda 0 and lambda K lambda 1 lambda K okay so there is a lambda K that's prepared I know what those lambdas are and then classic a lot of calculates the actual expectation value of the Hamiltonian and then there's an optimizer that updates lambda K which I guess is a set of angles okay to minimize and there's a cost function that is the energy for the energy for these set of angles plus the overlap okay plus how much overlap so there seems to be two components to these let me see if I can get my I'll pause for a second click on here whenever I'm having some itches again with that with a pen but that's that's what it that's that's what it seems to do or so why I don't know probably have to read the text but it says it's some expectation estimation and in some overlap but this expectation is this going to be the classic this is this just what vicki with us or or what's the what's the essential part in here or what's the it would stick let's see i'm gonna skip danger for now maybe it's gonna then should help though but for the special master because there's some I can fella prawns are equally small size of Jeannie girl's PageRank algorithm alone has had a significant impact on modern society and its course solves an eigenvalue problem associated with the stochastic matrix describing the world wide web another important example is principal communal assist which is once per application in bioinformatics neuroscience image processing etc the time-independent schrodinger equation provides yet another example of fundamental eigenvalue problem its numerical solution enables properties of atoms molecules materials to be predicted with far-reaching applications in materials designed by Discovery Center a great transition of excited state energies of molecules is required to predict charge and energy transfer processes in photovoltaic materials or thundersense some chemical reactions suggest another however classical methods such as density functional board okay chronic computers have the potential to solve these these and other prongs significantly faster than a non-american interation of car migrants solver introduced in reference 30 is the first algorithm designed to find the lowest eigen value of a Hamiltonian on a near-term non fault or and chronic computer EQ is based on the variational principle and utilizes the fact that quantum computers can store quantum states using an extremely fewer resources than required classically vqe uses parameterize point circus to prepare trial wave functions and compute their energy in a classical computer to to find the parameters minimizing these energy the low circuit depth of the QE has like 200 K since it's obstruction modification has been suggested to enable Vikki to find excited state energies exam for example a folded spectrum method which requires finding the expectation of the square handle tone Ian with chronically more turns or symmetric symmetry based methods which are non isometric again oh I know about that such situations have been more recently superseded by two proposals a method that minimizes the phone annoy Minh entropy and the quantum subspace expansion method requires the large number of high depth control unit Ares and the quantum subspace expansion method currents all our traditional okay we can dive into those examples maybe later as the Curiosity as well our algorithm extends the qe2 sister that's what I thought it kind of extends the key to systematically find excited states that almost no extra cost we achieve this by adding overlap terms into the optimization function okay so it's as simple as adding overlap terms in networks in order to exploit the fact that the hermitian matrices it made a complete set of orthogonal eigenvectors exploding further the fact that Vicki retains the classical parameters of on that states that enable their their repression low depth quantum circuits can then be rarely used to calculate these overlap terms let's try understand that so in theory the real primer is lambda for the ends at state are classically optimized with respect to the expectation value of this Hamiltonian which is decomposed by my I mean I spent 20 years and so it's not a big deal okay we know this we know these are method extensity to calculate the case let's stay by instead optimizing the parameters for the answer state such that the cost function is relieved so that works just you just add the overlap but what's the overlap here what is this calculating you're minimizing the overlap so you want something orthogonal to what what is lambda I okay so the case excited state buying centers in the parameters for the states such as the cost function is minimized this can be seen as minimizing the energy subject to the constraint that that lambda K is orthogonal to the states this is orthogonal tool and what are these the all the other ones so lambda 0 would be and here we show how choosing sufficiently large means minimum of everything is going to be the energy of the case state provided enhances so that's this mean that you first find the the lowest energy and then you find an orthogonal one so you're you're kind of finding the next one and then you do this over and over again that would be cool that is a genius idea this is a well the first term can be used in key the second term is the sum of overlaps of the answered state with states 0 to K minus 1 and can be computed efficiently in front of you using one of the methods given in section 4 so computing that occurs knowledge of lambda 0 to lambda K minus 1 and so and interactive procedures required to calculate the K eigen value that's all I said yeah okay first first time this here is calculated ok we're done with Vicki T we're done it's nice I like it that's them smart come on it's said of something kind of fencing it's like just this it's read more okay so you calculate the lowest energy and then you see QE by minimizing then lambda 1 is calculus jeans then has it's so simple the concept I mean I guess it's it's probably tricky but I mean how to come up with that answer minimizing because the idea is that because they are the eigenvalues they're all all orthogonal to each other right for K 1 then k 2 and how do you find so how do you consider the overlap how do you do that so how do you consider how do you accumulate how do you make sure that it's orthogonal to all of those because the first iteration seems easy to me but how do you do that for for example because if you wanna you wanna find for K 3 or K 2 you want something orthogonal to both k0 and k1 right schematic for rational expression is in figure 1 showcased a progression circuit some period and use the results it's gonna be ok so we know this almost figure why subscribe I'm sorry that's the console optimize updates you can minimize how do you calculate these how do you make sure that when K is bigger than 1 you're the overlap now that I know the overlap test makes makes so this destructive test makes it easier to understand the whole concept here so but where how do I make sure that it's orthogonal to all of the previous ones that's that's the point here the circus is reportedly to compute each of the expectation values and overlap terms for I lower than K the overlap terms activities in circumcision for let's go to section for a low death method formation I mean I know over I know is that trial stay Patricia soon for the universe of the preparation circuit for the eyes proves the computed state your obvious there's a low depth method for our overlap estimation proposed in reference in the Russians twelve can be seen by writing the overlap as [Music] so you're using this wait a second so you're using the you're just making sure that it's orthogonal to the previous one you found how this is going to you that you can find another one because you're always minimizing or want because R so R is the circuit that is the various your enzymes right all of the signs to me with precision by the fraction of all zero bit strings when measuring this state the computational basis listen these are true eigenstates was possibly non distinct energies Smith requires knowing the inverse of the preparation circuit for each previously computed stayed well these inverses of known in theory by inverting gate in the given position of the original version so the device errors may mean that the information is inaccurate in practice it will be fine to the optimum parameters originally found to prepare the e the eyes state using Big D then it's inverse can be found by fixing these and varying the trial state parameters so that the overlap is maximized system you can able to retain the robustness to control error Sanskrit rest okay this I don't get much why do you need the inverse as you're not minimizing for humanizes the expectation value of the of the overlap are in the universe that the Europe doesn't require that you okay well it does if you it does if you don't want to have double the amount of qubits right because you would literally have to have double amount of qubits to do one to do that destructive side because the same number of qubits in Iran I think you and around twice a circuit in the fenders really describe an alternative method again that seems good to have Vicki but twice an hour of qubits because you're doing the destructive swab test okay okay so this is another so what was described in here as I just keep all this stuff okay so what's described in here not in here this is another method okay that the overlap equals these and so this is what's in reference 12 in reference laceration reference 12 it's a nature okay what not gonna by isn't gonna pay for this [Music] well so but I'm happy with the altar of the alternate method method I just wanna understand intuitively what's what's what's going on but I think I think I've got it already so vqd requires it's okay so the surface is mm on a qubits if a larger gate that is available then yeah okay then you can actually do phase estimation can be used to reduce the total runtime of overlap estimation so you have different techniques in here to calculate these so can never create alpha cheaply what is alpha Q P is this like a what is alpha q is they say this is a thing actually what a reference for D accelerated rational quantum eigen solver April 2019 so it's really recent that's so cool oh I had it open already it's alpha v QE I don't mean alpha v QE can evolve I even keep in alpha keeping me okay these I probably might want to take a look at that corner phase estimation on steroids of simulation we similarly Vicki Dion that's hydrogen are one and the same bases for a range of inter nuclear separations and compared to exact diagonalization astronomic to using fry so this is the example they run an example the discussion on air accumulation in general we cannot assume perfect state preparation suppose okay but I think I got the basic idea I will deep dive into this problem later on I just want to make sure that I kind of go back to the other paper to the original one about these simulation and that stuff okay but I get that I get the gist of it so it's the idea is you use vqe to compute the first the lowest eigenstate then you kind of add a change your cost function to consider kind of minimizing the overlap between that state and instead of the next step right I give a white and and the next so you're the the next eigen state you're looking for yeah because that means that it's if you're minimizing your overlap it means that it's orthogonal and you do that iterative lien so you're getting all the you can get up to you know kind of basically get all the different eigen sites yeah I get it I think it's a I'll dive into these probably later on but it's choice of effective Hamiltonian because there's a lot of new ones I guess to the Hamilton you choose and or how do you do all the star sampling cost and then the destructive swap test and these we know about these but I also want to dive into all these part because maybe I could use that actually for for my other project what is these they it's the bitwise product I guess that's what it means is it because I'm you should use maybe that's how do i this is the be twice product headline message for mental simulation bonds her air accumulation our mediation signature constrains okay we'll dive into these into the error correction partly that air mitigation part lender I just wanted to know where is these post-processing explained probably somewhere here Seeger for the softest neighbors overlap of two states to determine precision pretty measures means after applying a circuit upon register in the state Wyler's also tests acting on 2n cubed states required and and cylinder control so upstate leading to ins notion that the same outcome description can be obtained more efficiently without and ancillary using parallel they bail basis measurements so that actually has a name it's a bail basis measurement oh yeah it makes sense because it's a bail based measurement so it's telling you whether they're both zeros or not actually that makes sense the bail basis measurement interesting so in classical logic the so called SRK subtests turns your foreo cards just to two n qubits and that's okay but it they don't explain it that's the post-processing is just that and then one and so if this is one it's not - right so you're really only gonna get I don't know well maybe there's another way to find that what what is the post processing in here but I'll deal with this in the other video anyway I get the gist I think the idea is I think I know enough to just go back to to the original paper here the actual simulation stuff so I can take that off the least so far so that's done what else is done this is done and this is for later and so I think I think there's no advantage into using an msk dress it controls it gate but I won't understand people get more why's that enough those are different types of I mean the MS gate is native to anion computer that's also wanna in the mail that I got from the authors they mentioned that and kind of makes sense I but I don't know if I'm happy with these explanation so but but I think my next would be to understand the design of the answers is it be better yeah good it was better than I expected to be honest I thought it we have I would have a harder time could be some happy I hope you enjoyed the video |
these I think my mine is just about to be blown so I what I'm trying to look here is a second take on the bench time algorithm and ice trying to figure out what's going on like why we have all this how to March and and those are these entanglements are those not entanglements and and actually that's been super helpful to play with that so let me just walk you back through the way and I'm gonna keep the measurements right I'm gonna remove this so I'm basically trying to recreate recreate this circuit and and I was like trying to send you know with my taking a look at the state vector on that side at screen see you know what happened right so that's what we that's a starting point so and and then when we say we have identity here just it's clear those are operations make nothing but at least it's clear it's sort of submit it's it's sort of one-to-one to what we see in Papa Papa in here right especially because here I have to prepare that that qubit to be one so I have to apply an X gate so and and so when when I'm applying here the harm or it's right so we basically end up with is because so because of the one here right we end up with basically two clusters of possible states and you see this because of the color and the reason is because when you play harm our one they basically you end up with the face neck the the sort of - right right so the face would be negative in which like the probabilities are not affected if I take a look at the measurement probabilities is just equal probabilities but you can see that the there is a minus and in those coefficients here right and that's because I we've applied a hot amount on qubit that's one that's it's not zero and so now what we're gonna do is we're gonna create an entanglement but not not within the same dimension right but but and tangling between those two clusters so when I do this right I'm basically basically shifting that I'm not so sure if you can kind of see the entanglement here or not um but I'm basically so what's what's happening is we're shifting where the face is and let me just I have to edit that and this needs to be here good so now what we have if you if you see what's happened here the colors is that the the two clusters now are separated by what by basically so cos 0 1 1 0 0 0 1 1 what is the difference other classes now here so so basically okay so and that's and that's the entanglement right so basically you look at these so what you've done is that um [Music] that the the hairs where the cubed 0 and cubed 1 always agree like are the same 0 0 so this one here and this one here they are then 4 in different clusters right 0 1 and 0 1 1 0 1 0 1 1 1 1 does this make sense I think that's what's really happening here because now what we'll see is that as I add the next Hartman's that we're gonna start losing we're gonna start losing some of those states so now we've lost everything that has zero as a possible value for the qubit zero and this is because this is because that entanglement that we've created here right so yeah because basically because basically they just cancel each other that's what's happening right because if you've got the two different clusters I think that's what's happening BAM when I when I now do that I'm losing when I now do that I'm losing the ones where this is supposed to be zero exactly because when I do that then I'm done right so then then we end up with that the question is why so I think this has to do with the fact that there's there is a one here because if we didn't have a one and we had an identity here then we wouldn't have with the state zero zero zero so you're recovering that that um so but essentially that is a that is that is yeah there's something happening here with the entanglement between those because let's go back to this so we've got here that we've got here two clusters right and the way this the way this is the way this is clustered now is because if I where to now just do again how to Mart's right I wouldn't have with just one zero zero which is what I have here but why because they cancel each other but now the trick is what am i doing with this is when I when I do something like that what's the change is that we I'm moving those minuses right so the colors the colors are moved and now the colors are what is the cluster now so so now the cluster is the glasses is everywhere where okay this is everywhere I see because those so originally those two things disagree right because it says you're in a one so now basically the cluster is because I've I've done things so the way this is entangled now is that for all the pairs where the cubed 2 & 1 disagree then we've got a minus so we have got the yellow color that's what we that's what we have done right so and if I now add another one but not this way but this way we're saying is now do the same for the cubed 2 & 1 so basically what this has this is doing is for all the 4 so it's it's sort of entangled all the states where it's it's flagged okay it's flagged all the states where where there is an it's flagged all the states where there's an even number of ones mm-hmm right because it's it's either 0 1 or 0 1 or 1 0 or or just like it's an or search like 1 1 1 1 right exactly and so as I add other modes now to go back to where we where then there's sort of some cancelling out that's happening here so if I do that we're gonna lose all of those we're where we had a zero I still don't know exactly why this is being canceled down I guess is because there are zero zero zero they canceled because this this and then and then mmm this gets cancelled yeah with this I think that's I think that's kind of what's happening so they they getting cancelled right um and now you've and now we've got and now we've got left those where this one in sorry this way we have we've got zeros and this and this will get cancelled exactly and and now here is the final trick here is how we end up just with this one as a final probability is simply based on the fact dad it's based on the fact to do it to do yes when we so in this case when we have a harem ours we can only keep one of those states because we we're kind of pack we're kind of kind of moving the certainty again into our measurement bases right because we've gone into CERN into uncertainty so if we've come into like a superposition and I we brought him back that's that's what we're actually doing but because we have because we we've got them flagged right the the one that we're losing it's the Wanda so the one that we're keeping is the one that's flagged that's for sure not so sure exactly exactly where why this is this way because if I would if I would now say that is that is not an X but like an identity so ever now end up with total different possibilities and now nothing's have locked so the fun part is when this is an X because because then then things get flagged right so because what we would do if we want to implement that not for one not for the secret being one one but the secret being zero zero my assumption is I just add X X here right and then and then and then that's it then you've got it right because then when you do that you end up with yeah zero zero right so the trick here is the trick here is it has to do with this one being entangled that's that's what allows for the cancellation kind of for the conservation states that happen but it's interesting to see that that's the type of entanglement you're doing right sis it's your entangling your entangling between between two dimensions or time and tangling with the sign so to say with a minus sign with a face and and then by going back into the certain that I mentioned where you've got the certainty or bring about the certainty and on to the measurement does that dimension you're along the way losing the information you don't want to you're virtually losing it right it's still there but you're losing it and so when you measure you're gonna get what you want so pretty cool but I'm trying to still figure out what's the effect on and why so the reason this is a one it's important because that's what's that's what's allowing for the type of entanglement exactly that's what's allowing for the type of entanglement and at the end of the day it's the same you're dealing with a Doge algorithm just with two qubits but that's what's a lot that's what's allowing for this type of entanglement and I guess there's different ways to look at it but that's that's really that's really that's really it so why though we're getting hmm well because because we're recovering originally what we had here right yep attend those are those are not it's that's definitely interesting I think I'm gonna even make another video on that but that's that's been that's an awesome tool to to explore that i think i think that's i think that that we're onto something here cool |
live it says you are live so and i mean i might not be hearable right but uh maybe i am i don't know i'll i'll share i'll share the link of your twitter you can get started set up the set up the process let me see if we both are here yep you are here you are you are here everyone can hear you i'm hearable that's cool okay cool so obvious that's awesome man obvious works that's awesome we should have yeah on this yeah obvious works okay so first thing in the stream people don't use xsplit um or at least if you're using this pay uh pay with pay for it probably cool so uh it means it yeah it means are you are you uh okay so are you ready yeah i am let's wait and see if some other people show up okay i can start maybe because they can uh go back and check it so uh the first thing yeah yeah yeah yeah yeah let's go ahead let's go the first thing what i'm trying what i think i should do is i should use the amplitude display to basically divide the whole circuit because it's qft as you said square qft and what you have done is i liked what you then did is you actually implemented the qft from its basic but you replace the whole circuit with mystery boxes so yeah that's nice and so the first thing what i'm doing is what i'm doing is i'm just dividing the whole circuit into sections so that i can see what happens after each section and yeah let's let me put i can put it here to make it more convenient for me so here it's quite uh you can say evident while you're just looking at it that uh the qft should work like this because yeah the hadamard gate is perfect this uh there should be s here or yeah uh yeah seo and the control not down there so it should have this particular display because everything is zero so that it makes sense that it would have a phase zero and uh you can say uh for two cured for one qubit uh and qfd is nothing but a hadamard gate and then you move on uh applying phases whether by i don't remember the formula but it's something like e to the power 2 pi something something so it goes from s t s z s t and goes down right there so this output makes sense it should this is the output this is the you can say the correct output for two qubits uh four keyboard doesn't matter actually if i even if i do this it doesn't matter so this is correct correct and this also makes sense because uh because it's a queer qft and every qubit is zero so the phase should be zero and uh you can because as if let me just to prove my point if i put it in plus yeah if i put it in a first in a superposition then i should have this kind of pattern you know i mean this is quite evident pattern so yeah that makes sense everything is fine till here but as we move here what we should see here is uh we should have a pattern or let me just put a qft here simple qft so if this is a qft and this is zero so what pattern i should be expecting is i should be expecting every qubit or you can say to have phase zero um yeah i think everything is to be phased helping phase zero and it should be a superposition because yeah that makes sense so this is the output i should have should be having here but everything is fine until here and it everything get uh you can say jumbled or is destroyed here so this is the particular gate which is wrong and let me try correcting it because to me it seems that you are just implementing the qft from its basic so i think let me just implement it this might be wrong i don't actually remember the circuit actually but i remember that it was um you can say the amplitude of z gate so this was the wrong this was the wrong one i can i could say that i don't exactly remember uh what you put there mm-hmm how does it go oh just uh um the box that you removed uh was in between the two hadamards just so you oh yeah okay okay let me just can i control zeddy i can control certain work so yeah this is the box here so i think yeah i shouldn't put it here that makes sense because the gate the output of what i'm expecting should be here so its its correct supported here so if i am removing this rotation gate and if i put i think i get it if i put it uh put a i don't move this let me just move this for a while and yes if i put a t t t versus t and if i put uh a z to get a formula z gate and change it to minus one by eight i think and then i put a header mark here yeah see this works yeah yeah yeah yeah yeah so i think yeah that that actually uh gives the correct output for the qft for this uh four zeroes am i right uh diagonal because this is not exactly uh turning the qubits off so i can be sure that i am right but this is the output we should be expecting for q4 cubic so can you hear me daniel am i audible and that'll hello we are still live here we are still live hello so uh i just got a text from daniel uh that there seems to be an issue with his machine but i think yeah this is this should be correct output let me just check that if i'm right or wrong because he suggested that it's a qft so if i open a quark and let me just put a qft for qubit qft so yeah yeah this is the output we should be getting for four or zero qubits and this is the output i am getting so yeah i think i solved the puzzle and what my intuition here was uh what i did i tried putting an amplitude gate after each section and divided the whole circuit into each sections and then just follow the basic rule let me i don't remember the uh the exact formula that how the uh you can say the gate goes i remember that is something like two is something of to the power of two so i use that logic and everything makes sense until this point but after it was you know after this r3 box yeah it wasn't at three boxes rotation three box that things got jumbled and mixed up so it was pretty certain that this was the wrong gate and uh then i just you can say i have implemented qft so i sort of remembered that for four qubits i think this should be the correct way of doing it because it just goes from one by two one by four and one by eight so it makes sense that this works out and uh we get the correct answer so i think that's it i think that's it for i don't think i can add anything more to this uh this was the whole puzzle and uh i think that's it and uh actually let me just because uh due to evolution daniel is cannot make it to the stream anymore so let me just uh talk about myself my name is abhijan mishra and i run a blog by the name of quiftex so if it's not an issue let me just open the blog here so here you go this is my this is my blog and i post things about quantum computing so what my or you can say motive behind this whole um blog thing is i'm trying to build a community around quantum computing i am pretty or you can say new in this field myself i have to spend about you can say six months or ten months in this field i'm learning and what i want is i want to meet people who who have same interests like me and i want to learn and grow with them so yeah that's everything there is to would be about me is to know about me and i would like to thank uh uncertain sister you can say daniel columba and quantum intuition channel for allowing me and giving me a platform to be here and to show myself and to you can say yeah i think that's it i don't have anything else to say so thank you and thank you for joining in and ciao we are live cool because i i haven't stopped really you know the string on my end so um this should be uh still going on so um cool so the the re what you should always do when you try some of these things right especially because this is quantum computing you have only i mean even in classical computing you have only tried for one value right like you have said okay so it works um it works for the input of zero zero zero right and and kind of you got it right in the sense that you fix one of the boxes that had a mistake but um but that's really not everything that there is to the puzzle right right so for example what happens when you uh you know try with i don't know try it with all the classical inputs right like what happens when you like that's the kft still or two for the input of three uh and stuff like that and kind of compare it maybe with the you know you can use another register to compare with uh to compare with the qf and then you will kind of see uh so and if this is definitely yeah there's definitely some issues so you see that doesn't work so so that's that's kind of that that's that's the that's my point right like it's always um like you gotta you gotta kind of uh to make sure that a box works you're going to try all the inputs right what's beautiful about quantum computing is that usually if it works for classical like if it works for non-superposition types of inputs it should work out of the box for superposition inputs right like because of the linear combination uh nature of things um yeah yeah so yeah okay let's think about it again let me see okay this makes sense uh okay what can i do is i can implement the qft again on another tab yeah sure yeah you can do that yeah that driver makes things took so you can that's that's a good that's a that's a good smart idea what you're doing right because you're basically saying i know how the implementation should work so you know let me let me kind of do back where i'll just compare it you know where it is for this doesn't work but what particular blog am i actually missing it up so that's the only way i think i could solve it i don't know this could be there could be better ways of course yeah no that's good i mean it's it's uh it's the it's the idea right so so i think i think this is right and this is right or am i missing something oh yeah i'm missing something you're missing one last part yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah yeah i'm understanding okay so this this at the bottom this is the exactly so this is uh this is the right click you have to it is right let me just check it for one one and one and one so yeah that is correct that is correct that is exactly what i should be expecting so this is the correct way of qft and it works for the two output and let me just try it let me just plan on super positioning from triangle position because then i should get a particular value uh yeah it works okay this 50 is worth it so this is a correct implementation and i'm sure not because i'm pretty sure about two inputs because i have mostly user into for like on non-superpositions and all zero so that's completely opposite because uh if you put all zero there should be a superposition it's right what i'm saying is uh now i'm pretty sure that this implementation is correct which i just did here because uh for superposition i'm getting zero and if i put all 0 i get a superposition plus plus plus plus plus so that it works this is the correct implementation i'm sure about it let me just put an amplitude after every step exactly check after every step and then see whether the states look the same and and then maybe that should allow you to that should allow you to then pinpoint what the issue is right because you can then correct let me just uh do it again no problem yep so one one one one oh can you just go go on and check the stream is stream is okay or not because it would be too crazy again if i had to do this everything again so yeah it's okay yeah you just won't check that no no it's okay it's okay it's okay don't worry and if something doesn't work if does it something doesn't work out we'll do sort of a joint video offline if you want explaining everything because yeah it's definitely it's definitely been a mess it's still the line i think we should do it anyway i think we should do another video together sort of an offline one just record the call yeah yeah what we can do is uh uh what i'm what i'm doing is i i like still queue and i've been doing some those few projects on cube we can work on in future i think that that would make sense or any article we will see we'll sort it out in future that that makes sense yeah yeah however yeah so it's still the third rotation i guess yeah it's uh wait a second it's second and third rotation or it is it could be you know it could be second rotation only because that would affect the third rotation as well so it makes sense that if this is wrong let me yeah that would make sense yeah because i it [Music] okay yeah exactly let me just put it out there you got it ryan there you go oh yes let's put it down [Music] yeah that's it man that's it yeah i i know i know but just just making sure oh come on yeah that yeah it works yeah yeah don't worry it's good practice yeah yeah it works so you see so basically that you know the kind of takeaway here right is is i mean it's kind of the trick here right like you make sure that it works for all the inputs right because uh the fact that the box works for one input doesn't mean that it works for all the other inputs and and kind of that's that's crucial um and so the the puzzle was designed to work you know for some of those errors to for some of those errors to pop up like only with certain inputs um and so yeah that's basically it uh yeah i mean thanks thanks i i've have you have you talked a little bit about qf about uh cubetiques and your your your block and stuff like that i'll do it again basically i did but let's just say that i deleted the previous video because okay so uh my name is aviyan nisha and i read a blog by the name of quivtix and it's just q f t i c s dot com so it's nothing special it's just a normal blog i try to share things as i learn them so what what's my basic intuition around this cryptic thing is i want to build a community because you see i'm no pro in this field i've just had you can say 10 or 6 months in this field myself so i'm still learning but what i want is i want to learn with people people with similar interests so quick allow me to share my views get people's view so that's it so if you want you can go and check out cryptics and thanks daniel for um having me on the stream and this is a good stream oh man this is a memorable stream turn the qubits off i didn't turn the qubit off i fixed qft yeah technically i'm different exactly you fixed you fixed the qft exactly that was uh and this puzzle is going to be available as well and some variations of these pencils are going to be available in the webpage as well as part of part of the first levels of the level 10 which are kind of you know designed to be around fixing algorithms rather than turning the qubits off i mean the name is you know just because of the series is how it's called but yeah cool man um thanks a lot and then i hope that we we can collaborate in the future again and uh you know i i'm thinking really to do another video with you and uh where we kind of explain uh that puzzle uh a bit more in depth and other ways that you could approach this as well because i have other ideas in mind so i think that's uh that's cool as well so yeah i'm going to end up the street yeah i'm going to end up the stream now um um thanks for holding and thanks for going through all the the hiccups with the stream but it's going to stay it's going to stay in the cringe history of the channel but i think that's the beauty of it right like it's not um it's not always you know uh beautiful and easy right so yeah um yeah i think that's it yes i'm gonna stream the in the stream and now and uh and have a good have a have a good time and stay tuned for more stuff and more videos and definitely i encourage everyone to visit your your blog yeah cool man thank you bye bye everyone bye ending this |
I learn more about Quirk with each video. I'm still impressed anyone was willing to try this live. |
Link To The Blog : <a href="https://www.qftics.com/">https://www.qftics.com/</a> |
Great work guys!! Really impressed, just saw the team on Qftics is putting out great content, Yeah , its all about the team.. |
um good i think it's working now so what i'll be doing today just a bit of a shorter video uh i just wanted to actually do an example with paint on the equation that i built up yesterday and just to make sure that this works well yeah wait a second i think i got again the same problem with the mouse yeah okay cool that works so we had basically i think we had um uh what did we have i forgot what we had what we were calling something i think we're saying c a c b x y based on what i'm seeing here and then this is being transformed this is being transformed into [Music] e d right and uh what sorry no full screen needed for that yeah and and and and we're trying to figure out um what's these here right and so the idea is that it uh it's supposed to follow these equations i mean it's just like if you you work it out from here right you you basically you know if if we we call these you know the result then you basically get that you know e times d times the result should be the same uh like x times a times c plus b times y times c right and so uh we know that e equals c and we know that d equals there's something i think i did i don't know that's correct and two times the the inverse of e i don't think that is uh i think i called that differently all the time where did that go okay so what i was calling e is the result um for the state being equal for the state being non-equal no the other way around okay i call these d and d e okay but then i said d equals c [Music] which is true rather than a thing if d equals c then c cancels out doesn't it i think it's simpler than that because if d equals c basically if d equals c then basically um it means they cancel out because you can factor these out so r is really just x times a plus b times it's good to reduce stuff you know divided by e and we're saying that e i mean that's the same as as doing e e to the minus 1 times x a plus b y x a plus b y which i can easily calculate so okay cool so i think actually this makes more sense so wait a second so e is for the states with it's not equal uh and i think that's wrong i think this should be chronic products because you won the thing is if you this is supposed to be more generic so if if here we have more more qubits right like a like you know there's just a you know one a two a two three whatever like it goes it goes farther right there oh no wait a second that won't happen because if they're not equal that's when it should be they should be all equal but one it's a thing that's actually wrong if they are not equal then we shouldn't be calculating these things here should we be calculating these things regardless no because okay so if the nodes are equal then it's c they're not equal then yeah that's okay no that that is correct the thing is let's be real we don't need these because the assumption is there can just be one element that is not equal so actually i can i can i can just simplify this way more than i thought oh that's good it's good to revisit this stuff after a good night's sleep yeah yeah yeah we don't need all that you're you're right we don't need all that because we don't need to do all the multiplications and stuff but for c because they might be equal so if they are equal it's an awkward way of doing that but yeah if they are equal um i need to i need to here i actually need to do it so so here i do need to need to say if it's not it's not and if this is not then do this else c equals i think that is i think it's it's nb cron and i think i need to do c with these i think it should be the chronic product because the idea is this thing grows up right it grows it grows up in terms of it's a bigger and bigger state yeah but i don't need c that's the point i don't need c right because c is this c is okay then it's e that makes things slice life easier okay so i need c i don't i don't need because c is is there so i'm just saying okay let me just think about this for a second okay so x times a times c plus y times b times c right [Music] again those are those are states and those are complex numbers so those are actually matrices of like column vectors of complex numbers so if i add these things up right right that's kind of that would be the wave function describing these must be equal to r times e times d e being the sum of these and d being sort of the the merging of these but then d equals c and so c just factors out of the equation okay cool so actually it's even easier so c is not needed i just need to capture that z z is not needed x and y are the amplitude so these are the complex numbers right [Music] cool and so what i needed to do is i wanted to make an example so let's make an example let's see if i can uh make an example down here okay so let's say that we have the state let's say we have the state right uh let's make a simple one right just just the first a simple one right zero zero and this is like square root of two square root of two and so this is merged into zero zero plus one and then what's this z right so an according to the function or according to the equation that we've just built is that you know well i mean the the the so okay so um so the part that's equal c we don't care so we care about the part that's different so a so a is basically an array that's like 1 plus 0 i right and 0 plus 0 i so this is a b is 0 plus 0 i one plus zero i this is one right and then x and y are equal which is basically um 0.727 plus 0.707 i right just because i mean this is the you can check this out but this is the regular stuff uh it's one of the square of two it's like a complex number this is x y right so what we're doing now is okay an e [Music] so what is e right so e is the okay so is this thing here so it's the blast date which means that e actually is is 0.707 plus 0.707 i 2.7 uh now what am i doing yeah i think that's correct seven zero seven i is that correct i think that's correct let me just double check so let me just double check right no it's plus zero oh sorry the the the no no no i'm i'm mixing the things up it's it's because so this these parts are all zero you're right so these parts are all zero so these parts are all zero sorry the same here yeah yeah they're all zero because um yeah there's no imaginary component it's just that they're both one here because it's in it's an array and this is just a number right so good to correct that good to double check that plus your i plus your i good and now so now we're have to in theory do is uh so the transpose of these right then x times so basically x times a is going to just turn is just going to turn into like this is just going to turn into zero point seven seven zero all right plus so it's just like basically these right so 0.7 blah yeah so basically yeah so if i transpose that the multiplication of this transpose or complex conjugate uh and these will just be one this means the amplitude this means that is one which kind of makes sense because at the end of the day you've already put those things down here and so you're just merging them and you want that to be correct so i think i think that seems to work for these yeah i don't know let's see okay so let's let's we just i just need to make the conjugate transpose of that and then so it's it's t and it's just and t e and it's the normal product so and and then x times a y times b so i should print e print um a let's print a and b print x print y print e then print te and then print that okay cool so maybe i should just a so if i if i do this so this is going to be the result this is going to be t this is going to be e this is going to be y this is going to be x and this is going to be b um yeah i forgot the commas everywhere that is awesome okay so a is is these psd is correct x and y are these okay so i think t is correct but then what's what does not work well is the these only integer slices with another integer although errors are valid index error come on is it really is this really the issue uh simplified recursive oh come on you want i mean oh again it's a is it amp it's amps exactly no and also i need to change the print statement and i'm so dark you can't even see me but i mean who cares amps oh yeah baby look at this look at these oh wow okay so what is the problem now this seems to have worked so this is the result and uh can i reshape array of size four into shape to one okay why do i have an array of size four so this works once so now key 0 q1 i mean okay what are we doing so this this seems to work correctly right so we're remember the case right so let's remember the case it's we've got two cubits um um so that's the that's the that's what the amplitude should look like print result um okay i always think if i am being believed being zero we can merge these da da da da da simplify recursive and so it's basically now we've got some issue with a reshape and then this happens because of e okay so because of e this initial is a noun why would these have size 4 y that makes absolutely no sense i mean i can i can i can replace this by by that so it doesn't break which is okay because i think that we would have to do that anyway um but still hierarchy key error q0 okay because q zero is not anymore because something's wrong here something's wrong q zero should always be should always be there so we do for all the cubes in m1 right m is we're given here so go through okay so what does m in every iteration look like print m um print m print amps okay that's weird it's just it's just um okay one and it already breaks so this is uh it happens here a what is the problem um for n in me1 oh oh i get it it's not no it's here it's um it's these it's the length of these right it's the length of these that's why it's complaining because here it's not defined okay so complains q1 missed it somewhere else notes oh yeah they're definitely here cool okay so now it does now that's tough okay so now it does it has the two edges it does two iterations interesting and now it's the problem counter shape side and shape two one but why if i'm what if i'm using the length of nodes how can that shape to one i really don't get it i really don't get it notes n that's weird so i normalized so can i can i print notes end and then print okay oh why is it a matrix of two i know sorry it's two no no it's two it's it's it's okay it's correct length two length two because it's already in that shape i don't get it is it already in that shape why is it already in that shape oh because it's already reshaped oh huh okay so i reshape it once already no that's weird okay um print notes and i don't i don't have to reshape it because if i reshape these two and i do these then that's that already has the right shape i get it so this is just i got it no it's in that's the point everybody has the right shape i think that's the idea either these or i just you know reshape that okay oh still breaks kiara kiki oh what what q0 what why come on i'm so close check which cubes are entangled yeah q0 y is not there q0 q0 q1 h okay so let's um okay i think i think m is the problem maybe yeah i think is the variable name is it create edges is it increased edges or is it these where's these yeah it's in create edges okay so let's print m and so happens now it seems like the simplified recursive has finished so i'm see i'm not putting anything else in there it's all correct oh yeah this i'm praying still so i don't want to print this cool so i'm printing m what happens now what is m so i've got h so this is the merged edge q0 q1 then another edge q0 q1 then another hq 0 q1 then another q0 q1 so what is the problem i have h and cubid and so this is complaining that oh i think i gotta okay i think i know what the problem is i gotta do probably gonna do like these and then and then make the assignment actually okay yeah now the next hair has not recount um if this will work i'm gonna be super happy create edges okay count so what am i counting here um check with which qubits are entangled so for keeping t if the qubit is oh what am i doing here so i'm checking the keys i'm checking the first okay so i'm checking this so i'm checking the keys checking the first key now what a second cell check which cubits aren't tangled because i want to get which ones are not entangled and i i want to check for each edge if that qubit has got the same node value so what am i comparing here for cubing tea securing t i want to count how many different values were there right okay i get it uh ah yeah so this is this is counting how many oh how many elements of these how many elements like the first one are in there exactly yeah i think it's be like i can i can it's fine i can um i can i can do like cube prep is basically tq tq keys zero so i get the first value and then i say for e in ntq for engine tq right equal true so all equal to then so if so if t q e is different than q prep like if i queue first it's fine so i find anything that's different than the queue first then oh that's stupid the item so all equal is true if they are the same and so if no that's not what i wanna because i don't wanna oh yeah and if all equal and and this is still equal then all equal but if suddenly article is false then this will stay stay false and then at the end of the loop i just say if not if all equal then it's not a time build exactly so what i'm saying is i'm getting the first value i'm setting the all equal true and then i'm iterating and as if i find something that invalidates these then it's gonna just keep all equal falls forever and uh okay what's the problem now the keys are just not subscriptable oh come on what am i doing what am i doing where am i create edges oh come on quite a bit why i get lost so much um so q first is um come on i'm just tired of i don't know enough if q first is none then we set it right q first equals t q e else all that stuff i i'm just i don't want to waste time looking for the right ways to build these loops um i don't care so if the key first is not that we initialize it and then we just check the rest okay what's wrong now um okay key error okay okay cool cool cool so there's something in here create edges yeah okay of course this edge is not here right now create edges needs the new amplitudes does it have them no amps and so what it should do is it should not check on edges but it should check in amps to get the amplitude but it's just already what's in there yeah perfect so um cool now create edges so how do i return these wait a second so so simplify recursive returns one can i just return a tuple i think that works in python return return m and amps and so this means that here and simplify i get m simplified and amps simplified right um delete edges and then here say amps simplified they are simp and then amps simplified is what goes in here as amps and then that should work no key error oh come on create edges great edges what am i what am i doing here for e in m hyper edge edge this is edge okay no it's it's it can be e i just have to i just have to use euid for these eoid for these i think that should do it h26 doesn't exist oh i need to give it the i need to give it the uid aeh26 is a new one that is the problem the thing is can i give can i give because those are going to be unique so can i give a hyper edge basically um uid so what i'll do is if uid is none then then get this one else selfie id equals your id and we're going to assume that we know what we're doing and so when i create the hyper edge here in the create address actually gonna i'm actually gonna give it a e and i think we should change this and not overwrite e because that's really stupid so enjoy the hey it's your idea oh look at these it doesn't break that is awesome whether it's correct or not it's another thing but it doesn't break what i think that's not correct i think the the the amplitudes in here are just wrong but let's take a look at uh it's four notes left which i think makes sense it's just um why is the is the delete edge is not working well or what's the what's the model without because there's an empty edge in here in this system edge something's off again remember the expected result is that we have two branches right because we have a hormone harmart and then we have a c z and a c x so uh so basically we have a zero zero then we have like a plus plus then we have like a zero [Music] plus and a one minus and then we have a zero zero one one um zero one and then one zero but this one has a minus one and if we merge these two then we have a zero and a plus and if we merge these two we have a uh one and a minus right so we should really have these two branches which kind of makes sense but they should still be somewhat entangled and it doesn't seem like they are but let's take a look at at least if we print if we don't draw it but we print self notes self edges myself system and i think i'm going to leave it here even if even okay so i'm stupid okay so for n in system nodes print n for e in system edges print e thank you genius state and then amplitude print print i think i should make that a function print notes now print edges and then print nodes so system print print raw okay so print print raw call it and then it's just a matter of [Music] raw itself it's just this not system now it's actually self why is it so slow i don't know okay so nodes is like yeah that's that's awesome that's actually correct look at these so one is a minus but it has the minus and the zero component oh wait a second why are these things like that that's the reason why i get this here okay so it seems like i'm [Music] so i think what i should do is it should be back at uh that is nonsense wait a second that is total nonsense um okay what i should do is the following yeah that that's the problem so so this should go here so it just happens normally then then abc happened like these things just just happen as you know oh what the heck uh what have i done exactly and i need to reshape that as well so i reshape that to have you know these but like notes over here i do actually reshape like these um but then i don't the actual assignment to notes which is what then goes to the result doesn't get reshaped because if if this would get reshaped then we'd be in trouble so we need to just reshape this just for the just for the the stuff in here that doesn't work and do one okay if i have i made another mistake with every shape there that's it um just reshape numpy change the actual shape of the array instead of just returning it actually gives the okay now but a stays so why is what what's wrong here notes and this is note sheet man i mean so some notes notes notes notes notes notes these are all just um oh man [Music] why is these why is this printing a column why is this why why does the damn last operation print a two by two metrics because i don't get it why so print these print that and then print these so yeah okay that's that's just all chill man i mean it's all good it's uh and then the amplitudes of oh i think the problem i think the problem is with the amplitudes is it i think these amplitudes need to be reshaped yeah that's the problem is here okay that the result of this is not what we want that should not that should be a number not a matrix okay so but what this means is the issue is okay so the issues with the function now the issues with the calculation that's the issue this is the issue here that this is not a so what if i just give it e what if i just give it e okay that's better but we still need to reshape it so i don't know why maybe that's wrong but um okay i thought we had to transpose that okay but the e is already oh because e is already like that it's like alright reshape i don't know if it's the right thing to do ah i get it okay so it should be t e but it should be that you shouldn't reshape that what you should do is you should do the transpose so now should be correct yeah that's correct that's correct i think yeah but that's not good that's that's something oh like that's not good that should be a number not an array right so that's the so i think there's something so if i print transpose i should get a raw vector okay and i don't i should so but then it was so wait a second so it was correct i need to reshape it and i need to transpose it i need to reshape it because i need to reshape it because i want this to be vertical but then i need to transpose it because i want this view transpose that's correct that's that's okay it's just one element something's wrong with these so the transpose is correct now um should i just do like this but now it's complaining that one size two one okay i gotta i gotta i gotta probably fix that the next time it's uh so there's something there's something off with and i got it to work but there's something off something off with the way these these works here yeah what's off is that i get the the uh e is just notes which is not transposed right but it needs to be it's not it's not it's it's a row and i want a column so i want to do the reshape right oh i gotta reshape it here as well that's the that's the issue so i reshape it here as well no i don't have to reshape it here oh damn it damn it i don't have to reshape it here no because i'm just assigning it i need to reshape it for e because i e is e is that like that's what e is e is this thing here i need to be this to be a column vector so i can then take the transpose of the column vector oh man yeah the problem is the problem is here the the problem is this doesn't have the right shape so x is a number so this should be column there should be columns so there should be columns this row times column right if i print these column so transpose times column so it should be correct so i do transpose time column times column okay that's not correct why it's not correct what is this multiplication doing should i just do maybe i should do the dot product i should use dot but i thought that's the same so i thought is mp dot oh yeah there you go so what is the multiplication doing so i need to do the dot i need to do the dot and i need to get the element okay yeah baby awesome cool and now i can print these and i think we got it so there are four notes okay no but it's not correct right because the the this is still somewhat lost like the edges are not these cubes are not attached to the right edges let's take a look at these so when we create the edges delete now delete edge apply simplify recursive create edges add note so if q is not in untangled self add okay so this is not working i think it thinks they are all not entangled maybe that's the reason so all equal is true and then we take a look at all the edges right if if q first is none thank you first equals this value else all equal equals all equal and um tqe being the same as q first so this means they're all equal right because they're they're still equal like you know like the same so if all equal then append cue to not entangled and if i print not entangled yeah they're both there which is wrong why what is what is wrong with this it's kim t right i keep it in in in the t matrix then for edge this is the right value here t so for e in m for q in m e oh i think there's a mistake in here oh uh yeah i'm over i'm overriding b um how do i how do i do that in the other if uh uh if i think i do this somewhere else i think i do this in the simplify if it's in the keys yeah exactly create edges so i'm basically overwriting these accidentally so that is why so if q not in um t dot keys right then do that else else just do the assign uh no or just just do the assignment regardless what's wrong an array um [Music] all equal oh okay these are the nodes i see so um because these are complex numbers so i should i think i should compare the imaginary in the real oh i have self i have self equal right state equal okay i have this state equal so i can use that in uh in the create edges so state equal exactly there you go oh yes there you go there you go that worked it worked it worked it worked it worked it worked it worked can i see this yeah okay all right let's see so one is yeah i worked it worked it worked so the zero is a plus the zeros are plus and the one is a minus yeah cool gotta clean that up but basically the simplification seems to work voila and uh even the numbers right like the the just like that's the the points 0.7 that we wanted and all that stuff so awesome |
Cool |
just yeah this is so damn loud this computer but so let's let's see um to be honest i've been so i've been now out for about a week or so uh you know cygnus involved not kobe doesn't matter whatever um it's just difficult to take care of everyone and keep doing and keep going grinding with all this but i've kind of come to a realization which is that we're in february basically and i haven't just i have i barely have done one exercise from the from the list here so i'm i'm almost like i'm almost like going to just rethink the whole thing and i don't know whether i would just not spend the whole year with one very single super focused goal of just going through all the exercises i think i think that's what i'll do so i think i'll i'll try to be a bit more realistic and say well i'm not going to be like i might work on the whole entanglement simulator on the side a bit maybe do a couple of videos now and then but i'll definitely leave that aside i'll definitely leave that aside and i'll definitely just not approach you know qft and super determinism as of yet um but i'm rather i think gonna focus on on the exercises because like that's like a 300 pages book right and i've barely moved from like you know i've just done one exercise from the wave functions and i'd like to go even more in detail like to not just shy away from the math because that is important uh not to do so and uh yeah so this is it so i want to take a look at i want to try to i wanted to do a quick session today just a very quick one uh probably half an hour or so and and try to um [Music] maybe do a bit of a bird's-eye view on on these um and kind of try to have a bit of a realistic understanding of like is it going to be doable or not but i think it i think it should be why not right uh i i've spent a fairly big amount of time focused just on problem one but i think it's good because and it helps me and i'll come back to these in a second because i i wanted to i've had some more thoughts about these um today but i was thinking one of the things that we that i could do is i could maybe change the approach a little bit and um and actually go and say look we do one exercise the first exercise of this chop the first right so we go like uh breath first and then we go um deep in each of the topics [Music] and i could just go i think i'm just gonna go and read the existence so so wave functions is about wave function so that's page one and so there's about like you know wave functions and whatnot and then you're calculating things like probabilities of finding things in places and whatnot and it's all like that that's actually helpful i think i think i have i think i have a hint on why this is like that and it's got to do with the the fact that uh this is awfully similar to the dot product [Music] dot product the dot product formula with cosine or something like that i think i saw this in a video that i was excited from a federal product look at this exactly because if i and i remember well from the other product i have two terms like that right so that kind of makes sense i have two terms like that and i could just replace them with these and then maybe there's some uh stuff happening there that makes the other term disappear and whatnot but that would explain these yeah but you can come back to the other video for that later uh i guess and um so that will be definitely that's a wrote unblocky and unblocking thing but then we have problem 1.2 um so we have no we're talking about normalization we're talking about orthogonality we're talking about expectation values that is definitely helpful what else do we have problem number two three we have hamiltonian uh in real values i'm a real real valid hamiltonian and so the question is what like okay show stuff about the eigenvalues and what not the periodicity of the system um [Music] 1.4 is look at these superposition of plane waves what the hell is like there's an integral okay so this is going to be interesting proof that these wave functions are normalized in orthogonal for free particle computers so i think it it it pretty much is like these writes expectation values it's about orthogonalities it's about like you know we have a wave function stickers have a real wave function up to multiplicative constant um okay show about you know the reason about momentums okay so so we're doing a bit of you know position and momentum analysis and stuff like that 1.6 propagator okay so that's cool because i think this is then path integral right quantum mechanics propagator i think that's path integral which would actually be pretty cool we'll get into that awesome propagator what the heck is that problem seven uh prop calculating propagators proving things like that there's some heavy math involved in here but that's good oh god um normalized yeah i cannot use that completely normal simultaneously expanded alternative expansion it's a lot of manipulation like i feel like it's a lot of manipulation but let's see okay so now we go to the free particles so what do we do in free particles compute normalization factors probability density i mean it's kind of a bit of the same but just working with a free particle model isn't it problem 2.2 spherical waves should satisfy the stronger equation for free particle of mass m except at the origin give a physical interpretation of this non-conservation of probability okay so some some showing your question things okay so this is just it just goes to be more in depth you know with the actual free particle mode which is it which is good because i think it's gonna help me strengthen that uh that perspective oh god baby this looks dense this looks dense position probability density evolved wave function cool so it's basically the same that way for that but it's focused on the free particle what is simple potentials show that the angle of reflection equals the angle of incidence the snails snails low with a snail slaw smells low oh that's now slow the thing here okay cool i didn't know that's how it's called okay so here we here's a b so this is this problem is more like like the whole thing interacting what is it simple potentials okay maybe i mean that might be interesting because it seems there's something to do here with the environment interactions for example problem three consider particle incident on infinite planets it's all about like it crashes with something where there's a potential some potential energy or something and then infinite square well okay and then it probably culminates with the infinite well a square well model cool but it's going to be good to put some of the stuff in practice so this is wave this is still more wave stuff this is about like you know what happens when there's potential changes or whatever the harmonic oscillator i guess that is going to be standard stuff about the harmonic oscillator i just want to spend enough time on all of these problems so that's why i definitely think that i'm gonna change the whole roadmap and i'm just gonna focus the entire year and getting the exercises done and getting acquainted with the math and you know losing fear like what the hell is this oh my god like i i don't know what i'm gonna be able to digest things like this and that's the solution right um angular momentum what is this about consider an electron bound in a hydrogen atom under the influence of a homogeneous magnetic field ignore the electron spin the hamiltonian of the system is boom the eigen states bom and then the uh [Music] for each of the following states calculate the probability of finding the system at some later time is the angular momentum eigenstate is it an angular momentum another observable or yeah okay so this is uh yeah so this is cool next next we will do quantum behavior so a qantas has only two energy eigenstates [Music] party measurements is this when we are getting into the discrete stuff or not a pair of particles moving in one dimension is in a state characterized by the wave function like these discuss the behavior of and the limit what would the solution look like so the normalization constant and the in the limit this is sharply localized amplitude describing the situation where the two particles are at the distance keeping complicated this is damn scary man this is super scary quantum behavior consider a netron interferometer composed of three crystal slabs a beam of neutrons is split in the first slab reflected and redirected at the second and finally superposed at the third and final slab a phase shifter is placed along the route of one branch giving a phase difference to the neutrons with which it interacts a spin flipper that can flip the spin of a neutron is placed along the route of the other branch by placing the detector in one of the final beams there's an interference pattern the spin flip device is based on the operation of a static magnetic field and a time independent magnetic field perpendicular to it what must be the relation between the natural magnetic dipole moment and the rest of the parameters in order to have maximum speed probability ah god that's complicated okay so this is i i see this is a bit more generic like you have like more of you know a set of kind of like specific setups a bit of an experiment kind of thing and then you have to reason about like what's the behavior gonna be and you know okay so that's definitely going to be interesting general motion a particle of mass m is bound in the spherical potential well find the energy eigen functions minimum value of the potential [Music] general motion the particle from this bond is bound in the central potential the particle is an eigenstate energy momentum consider this operator expectation value what does that category even mean that section actually a particle of mass mu is bound to central potential is everything bound to central potentials an isotropic harmonic oscillator is in an eigen state of energy and angular momentum with energy eigenvalue boom viral theory theorem quantum mechanics so that's the thing the virule is the expectation kinetic energy of the quantum system to the potential uh okay so that relates the kinetic energy and the potential energy cool it's like realistically how many hours can it take me for like a full problem right because in a lot of these problems i'll just have to actually research a lot of stuff right because that's the point but i like that i like i like just kind of jumping you know um uh jumping to to um kind of like more the deep water right so okay many particle systems maybe this is gonna be [Music] uh there's gonna be some entanglement stuff in here cool so consider pair of three identical particles for simplicity suppose they are moving around the dimension they got the spin variables each part is described in terms of real wave function well localized show that if the two particles are feminine then there is an effective repulsion between them compare with the case of two identical bosons solution the given okay crazy man it's crazy but at the same time i gotta go through this i wanna go through these because it's gonna force me to learn even just like things like okay so what are then the fermions and the bosons things in here like how did this relate to the how does this relate to the problem uh is there any entanglement okay no there is actually i thought there was no that would be consider system often and perform a measurement i kind of find the simultaneous measurements gives an outcome that can always be predicted from the first mentioned measurement show that this property entanglement is not shared by states that are tensor products okay so these are the more fundamental this is a more fundamental exercise right about entanglement and stuff like that cool so there's gonna be something well that is inch that is what is this section section six what is the chapter six why is this going so slow man quantum behavior okay so there's going to be some entanglement stuff in here general motion and many particle systems so yeah that's going to be about many particles and then approximation methods you could see a product from that especially the first time the column potential only in the original very close screen the difference becomes negligible consider the difference as a perturbation and calculate the first order correction to the energy of the ground state what the is even that man approximation methods a particle of massive moves in one dimension subject to harmonic oscillator potential the particle oscillation is perturbed by an additional weak an harmonic force find the correct ground state and calculate the expectation value of the position operator in that state corrected ground states oh it's so i don't stay perturbation theory okay so in quantum mechanics perturbation theory is a set of approximation schemes actually related to mathematical perturbation for describing complicated quantum systems in terms of a simpler one okay cool an additional perturbing hamiltonian representing a weak disturbance to the system if the disturbance is not too large the various physical quantities associated with the perturbed system can be expressed as corrections to those of the simple system these corrections being small compared to the size of the quantities themselves can be calculated using approximate methods such as a sympathetic series okay that's complicated stuff man but yeah i think i hope i'm not gonna regret that i hope i'm not gonna regret that decision but i uh and scattering so what's contouring it's coloring no what is this is this the oh god that's the that's contours integral contour integrals right i have no idea what is that how do you even do that i i've i've saw these contour integration oh god along paths in the complex plane so basically that's it yeah i think i think that's what i think that's what i will do i think i just have to take it easy and go one by one and as i said i think the strategy that i'm gonna follow is gonna be i'll go through every single first problem of everything of every you know first uh session and then uh i'll go as deep as i can try to make sense of the solutions making sure that my solutions match their solutions and that you know um actually we can make this work and gain understanding and get comfortable with the whole mathematical stuff as well and i'll definitely i've already learned a lot by approaching the first exercise which is almost i mean it's soft but i just wanted to make sure that my my intuition and my actual you know the formal part of it is correct as well right so you know where does this come from and whatnot and stuff like these and then um the whole thing with the current density is still a bit unclear where the formula where is this formula taken from and and what but now i understand and know what the current density is um yeah i'll i'll leave it i'll leave it here for today um i still have a couple of the things that need to be taken care of but we'll stick to it we'll definitely stick to it and i'm gonna try so that that's the roadmap uh for those who don't remember uh but as i said i think i'll just i'll push everything to 2023 and i'll literally just keep these as like 20 or these two points is like 20 22. so i'm gonna because i just yeah i'm i'm gonna i'm gonna i'm gonna spend all the free time i can uh all the time that i have for these streams that i that i can't uh in this year just focus on on solving all the exercises in this book and let's see what happens wish me luck bye |
fire machine go live cool let me just share that in tweeter real quick and we get set up with hypercar stuff i just have 45 minutes literally 45 minutes today but we'll make the most out of it um where do i go stream manager there you go these i don't know why it's under the edit button but or tweet that boom awesome um what do we do today so first of all i already i fixed right after the stream um first of all why isn't the why isn't the environment activated terminal new terminal i thought that was happening automatically there you go automatically cool um so jupiter notebook i gotta learn how to type without looking at the keyboard that's a looks like a good skill to have which i don't but um what we're going to be doing today is um further experimenting with the um splitting and the expanding here because i've already now i realized one of the consequences of getting rid of linear algebra is the fact that now we actually have to deal with a single qubit cases specifically which i had before this means you know splitting actual single qubit nodes into edges it's something that that the entire library does not currently support um so this is going to work for later for sure uh notebooks this is where we are with the testing um there you go so what i did last time is it it worked oops i hope that terminal didn't up with this now um it's still working right even if i close the terminal run all run all yeah awesome perfect so it works right so what this is telling you is this one qubit right cubic zero and after so after the so when you run the hormone rule it just puts the qubit state into super position because that's what the rule tells you to do right it matches a zero cat matches a zero cat and he puts a plus cat um but when we apply the x rule then we it does what it what i expected it to do which is um it basically splits the note into two notes and one in the zero state one in the one state and it applies yeah well it applies the rule right because that's what it's done it's basically you know there's no like there's no difference in terms of where the rule is applying right um like the zero turns into one the one turns into zero um so that's why you have still a zero and one uh i wonder if i would see anything probably i would not because it probably will be sensitive to print representative to actually print the uh the amplitudes just to keep track of the problem is the way i did it here is not so comfortable to that so per edge we do all the qubit labels and at the end we print okay and a dna print basically amplitude so amplitude um is it amplitude i think it's amplitude so i print and here i don't because it's the system edge yeah actually actually i never thought about what the implications are having a system h that doesn't have an amplitude associated to it that that's something to keep in mind as well at some point um awesome perfect what are we doing so yeah let me re-re-reload this stuff kernel run all so what i want to do is i want to i i run the x rule first run the harmon rule afterwards yeah so that that kind of makes sense right because the edge that has the minus is the edge that's in the zero state which is you know here if i if i if i do a pre-roll here which now i can do easily because there's no something like these right so the the minus is in the one component and here the minus is in the zero component if i were to factor these things out now and actually get the thing that i want which is back the the that would still be the plus it would still be the plus state but i'd have the minus on the zero component but i can't merge them because my factor my factoring method does not account for edges which one with one cubic that's the problem that's that's the next thing that i that i need to work on but first we're trying to see whether we what this works at all right um so cool that's good um what else should i test so what am i testing at all i mean i'm testing that the rules work and that it's actually doing the expansion as it should right so i'm trying it with one qubit case and it seems like it does now the contraction or sort of the factoring it's it's the next step i first want to make sure the expansion works across all the use cases so it seems to work for one cubit right this means we get um so so i get to keep it in the plus state and then what the x rule does is it splits it splits it because it splits it because it's not in the z basis and then it applies it runs a rules through the system which turn the one where the minus was into zero and blah blah blah so that all works well if my system is in the set basis like things stay like it's just one edge is the system edge right yeah that's all cool there's no edges um i forgot the coffee it's fine um yeah so now two edges um two notes okay now two key bits so we have we've got two qubits let's try two cubits let's see what we can get out of these we've got two qubits q0 q1 both in the zero state right if i apply the x rule to q0 that one changes if i apply the x rule to q0 again and into q1 like you know chill right so that one changes um [Music] what if i apply this swap rule which i've never tested which takes zero one and turns it into one zero one zero into zero one and the other cases leaves them alone and that shouldn't split anything because it's everything in the z in the zap basis right like remember what the expand so what the keyword expansion should do is it just should expand the qubit but the actual expansion oh you can actually oh you can actually jump to the function i didn't know the it's a good it's the good you know good features of an ide actually let me just get some light into here awesome um so what this does is you know it's pleats only split edge split edge split edge but a second actually does do this splitting just there go split edge yeah so if if it's in the set basis then there's no splitting we like that we like that all right right i hope it's clear what i'm doing if not it'll become clear soon as i'll do a bit of a wrap up on this but basically if i put one qubit into the one state let's say like this and now i do the swap rule and i map it like these didn't work that's good stuff to fix split a split a b drop shift type ball has no length what oh the parentheses are wrong cool colonel but i know because it's printing split a split b i know that it's it's it's trying to split things so but actually i can get rid of these to be honest they're not really needed but let's see it worked it worked the software will work neat right because before the swap yeah it worked that's awesome it worked now if so it works if i don't have to split stuff that's cool what if i put one qubit in the plus state and another qubit in the minus state that can be interesting or one qubit in the one state let's put this way let's let's do one qubit in the honor mark state right so uh one cubed in this one cubed in the one cubed in the so we have one qubit in the plus state and one in the zero state and again i'm still the printing here is is still not working well in terms of i want everything to be aligned properly but that's excuse you that's q1 and now i would expect the swap rule to do what to split this cube into two edges so you're gonna have two edges in the system one with zero zero and another one with one zero and then ah and then run the swap roll we just flips things around let's see okay let's see it didn't break that's good and what do we have we have so we have two edges we have two edges which is good um and the system edge is empty um we so cubit zero right is in the zero state here yeah and it's it's it's actually it worked right because if i if i will merge this now the plus state is in the key it's in qubit one you see because it's one and zero and here's zero zero so if if my if my factoring would work which in this case should to be honest can i can i try can i try factor how did this work because it doesn't work for one cubit but it but it should work for factor qubits so i just give it a cubits okay so if i just say so if i now just say you know kind of printing in between steps system factor qubits and i give it the qubits which is basically these guys right and then i print the system i've got factor cubits like yeah it worked it worked it worked you see you see boom it worked awesome so now the plus that is in q1 the zero state is a q0 i mean even if i just make sure things get state preserved and whatnot like if i run an x rule here so i get a minus state yeah then i get a minus state here as well awesome let's try to make that just real quick a bit a bit nicer which is in the print in the print row whenever i print the state [Music] whatever i print the state in here right so what i should do is state equals these right um or not like i should just say system simplified state right so simplified state i should do this simplified state which is a function that will basically simplify the state so what i will just do is you know if self state equal [Music] if if you know state is equal to uh to zerocad you know um return basically a um so what we're going to return we're going to return just the you know can i do that is that going to work i'm a dirty programmer so i don't care about else ifs so you know plus and a minus state so at least i get a you know for the simple cases i get a nicer representation if it's not then just just return a string state hmm right because that's yeah so that should at least make it a bit nicer ah oh stupid self self kernel run all and here i definitely don't wanna i don't say if this changes this this is being yeah there we go what ah self just basics you know just the basics is what i'm missing sometimes programming basics okay but it kind of works so let's try with three qubits so if it's not you know if it's not in the pl if it's not in the cool yeah but now i don't need so i need a better way to add the amplitudes in here um definitely so this is something that you know kind of doesn't need doesn't need to be that big um i need to definitely have to find a better way to do that but um cool the amplitude what i need to do is the other way it's the other things i just needed i just need to get these and these and basically you know kind of um if length of p helper is uh 0 is bigger than 0 then p helper append what the yeah so i append the amplitude in here that's probably a better way to do this and uh that's the only place where it makes sense to do this awesome that's just cosmetic stuff but um okay let's do more tests let's see if that runs nicely and let's do more tests right let's test with three qubits but i don't have a three key but i can build a three qubit gate easy because it's just rules man it's just rules ah it's still why does it still write it so crappy print raw oh i haven't saved that's awesome right so the rules for the the rule for um for the toefl it's actually pretty simple look at this minus zero and with zero and a minus and here we have the amplitudes of these edges and then this edge is just there's no system edge so what i should do here is to be honest um [Music] the same right but just cosmetic but it's just amplitude um you know no real amplitude would just say system like freaky freaky freestyle three qubits so not entangled um but what i should do here to be honest is you know um it's not just the length what i need to do at all is system empty equals true and so you know if i get one of these then system empty is false and then just have an overarching if here you know if system if not system empty then we do the rest yeah that's better so i don't print unnecessarily that stuff you know cool it's running you know that's how easy it is did i save yeah this is running or one yeah there you go why take so long cool that's it's cleaner yeah yeah it's cleaner so they are not not entangled not not non-entangled let's say not entangled it's probably better but anyway um so what about we write uh you know it's a bit of a i think it's a it's a it's a good trade-off that you can build rules like that so you don't have to you know kind of build mattresses it's much more efficient to just say you know you you build these rules like that so we know that one one one goes into you know one one zero and one one zero goes into the one one one right so you have the ccx rule and um if we have a system with three qubits i'm going to keep that one there so insert cell below and i'm going to do a system with three qubits and i'm gonna print the system so that's all good and now we'll do is they're not entangled and so what i wanted to do is i wanted to put the um i wanted to put the first two systems the first two qubits give it to your qb1 and the plus state so we got a printing step in here and i like that and um [Music] and now and i'll run the ccx to be honest if that works i don't know um ccx rule which we will map it like let's this what happens okay look at this well that actually looks like it works right it's cool man because yeah it actually it seems like it it works and if i if i factor the qubits and print the system so it's it's you know sort of optimized or more or the most compressed in terms of that's essentially let's see what quirk science work quirk quirk quick quick quick so we have a plus and a plan state and then we do a toughly gate and uh huh yeah okay it's fully entangled right but it's like does this make sense so what i'm saying is one edge like if this if q0 and q2 are zero then the one in the middle is plus this is this is this if i do zero and zero and i take a look at the blocks view here we have a plus that makes sense that's that's correct this is one one it's one that's true one zero is zero so if it's one zero then the velocity here is a zero obviously yeah and i guess that's not of course that's not a unique way to represent that um i'll chill but it works so yeah it seems that it works i i only have to um work out how the factoring of one qubit can like whether that works at all or not because this is the two key example this is the three qubit example right so this is two qubit tests this is three qubit tests and uh if i insert a cell above we can do again sort of the one qubit test right where i put a cubit so we have just one just wanna keep it right we print the system and we have a um i'll do these things here so i'll actually copy the whole step in here right so that gives us a minus state and now we want to do a um a next gate right so so the x gate will be just like that and that leaves the system with the amplitude in here right because we keep always jumping simple there and and now the idea would be factor key bits and see what happens which i guess it might not it work what it actually worked it is the mine is what doesn't actually work like it'll be cool to print the global phase to be honest it'll be cool to print the you know the that it actually has a global phase to it but it's not super urgent but i what that worked i definitely gotta revisit the these methods because i'm not sure that you know how can i make this this is just the space in here it's just these right so i should just like uh you know just like that and if i reset so it just looks nicer because i'm going to be mostly working with with rather simple numbers instead of just complicated things i know that the the you know as soon as i get something more complicated in here yeah it looks better so what about i just um what about i just kind of kind of you know do something like this stupid cosmetic stuff but and probably i want to do the same at the end actually you know to kind of close the oh i got the right okay cool so if i just i don't have to have this these separations printed in here right yeah okay but uh you know what they say hacks are the best that's so dirty that's so dirty that's not how you print stuff anyway um good so actually i can get repeated these things probably as well and these things here oh fun not this what okay yeah that's nice a bit too much for the one cubit case but that's okay um okay so yeah looks like basically the tests they are it's working better than i expected i just feel like this maybe something a little off in terms of well i can always print the state vector right state vector to state vectors i can always say print raw raw here um just below what do we do it just uh yeah yeah yeah so just here kind of like print self to state vector um and uh see if that looks nice but yeah so this is a um not about way to test stuff awesome how is the n a looking like if i add a uh okay so now i have the state vector and i can check as well you know the things are correct what if i add like a fourth qubit in here how will that pop up okay so it stays it stays here like factoring does not take it out and factoring should take it out that's that's that's a thing you know what i mean like it should just be um because this qubit is not entangled so that's definitely yeah that's maybe the next thing to do is to dive into the uh dive into the factoring it's not entangled not entangled zero minus a exactly and so but here like you know q zero one and two are entangled but q3 is not entangled so i definitely want to take that zero out into another row that's like n a and not applicable not applicable not applicable and then zero right um cool yeah but i mean you know that looks pretty nice i mean i don't know that's not it really to just take out the linear algebra is that all that it takes just what have i done let me just maybe you know commit that new terminal so okay commit basically you know um testing rules and no linear algebra okay push that's cool what have i what have i done though basically i've built the so we've got the simplified state concept we've got the print row we've got the ease match replace match rewrite concept right so that seems like it's working right so what the rewrite is doing is what it's doing is first of all it it sets all the tracking oh maybe let's try to visualize the tracking yeah yeah yeah so what about in the print row here so what about um the stuff that something changes maybe i'll mark it with a star something like this so if if the note is replaced right i guess here i can do that it's like if yeah okay so that shouldn't be the state that should just be or i can do this yeah no i can do this here say replaced it's just decent then if um if this is replaced and replaced equals like a little star and this is always going to be replaced right and hopefully the replays is true and that's not okay cool that's not turned off that makes sense it's turned off at the beginning so uh notes edges and then what else is this happening print row so through all the edges and then we go through these here as well so system empty but also you know if you replaced and basically replaced so what this should do is you know for every every time that we run and print something we should see what has changed hopefully oh some indenting line 683 683 some indenting where is the indenting problem yeah well that doesn't that's not supposed to be indented oh wait a second it is supposed to be indented why not replace this like this and you know if replaced and replaced like that and here it is supposed to be in that dead that's correct why it's not kernel run all oh that worked okay yeah okay so this is what changed then the two things changed that makes sense and here yeah we're seeing what changes right so this changes so key one is the only one that changes and here we see that the operation changed these three elements only right um i'm not so sure about like whether you should flag the no that's probably not that's just the stuff that it's been changed by the operation i don't know how useful that is but yeah um i guess i could just also leave it like you know um just with a spade just with a space here and then kind of just add like you know a bit of space all over the place anyway you know kind of here and here whatever yeah but that's just been yeah that's just cosmetic stuff that's all i guess all cool well we now need what i now what i'm doing is i'm going to drop it here because i think that's i didn't expect that i managed to get that tested up to three qubits but it seems to work at least for these rules um there's two things next i want to just make sure that i can round thing you know round everything up a little bit more and um you know basically all the stuff that is around factoring and the namings and whatnot and and then kind of bring back make sure the measurement works as well do it you know general testing and also merge there's a bit of a there's a couple of things here with the creation of the edges that i'm not super sure they work the way i want this to work for example as i said right we want we want to make sure that the zero is kind of out and it's moved out from here correctly and i think that doesn't happen because of this part of the code that's commented here um that's kind of doing the last mile um factoring so to say yeah basically all these and so when this is done there's two major milestones ahead one is merch stuff from jordan jordan is helping me with the project a bit on the um you know things like visualization and stuff so i've seen a pull request where he uh you know separating the visualization stuff and it's making some of things better in general so i'll i should merge that as well and then the next staff and next thing to work on is for sure um once all this is you know well done and worked then we can start tackling you know bigger bigger dimensional systems like not just qubits but kind of have registers actually and have you know two three four five numbers in here instead of just one right because this will allow us to implement rules like you know module exponentiation and things like these with this kind of system which is a more intuitive way of defining the operations because i can mathematically then you know do something like ideally what i'd like to do with is like for example you know the mod the model experimentation rule there will just be a rule that says you know um for example like match like match x right but that's a symbol so this means kind of match every edge and assign that value to x and then um or you know something like these right and then it's like you're saying match x and i don't care about y and then um and then sort of keep x and do you know um or max match x and y yeah and then do you know leave x and then do replace the y by you know something like like these right so ideally like you know and and i know this is obviously not working now but i know you know there's a lot of new ones in here and i know that you might end up building something that is not unitarian would not but need to work out these in general but like this would allow you to then build choice algorithm prototype shortcut with them really quickly and remember the goal of these is to have a tool that can be used programmatically to explore algorithms and to kind of build you know build things and and see how the evolution is you know how the different registers look like how the entanglement looks like um and analyze kind of what's happening uh in there right so that's kind of my my ideal goal is that you can do stuff like this and then you know that is the quirks equivalent of doing you know these and then taking taking these right and defining defining r and we're not like as default value b mod r it's input b yeah yeah yeah yeah and b exactly you know like this is ideally what you want to do because the the problem with the graphical interface at quirk is that you you're limited with the qubits right you're limited with easy you can do it of course you can can work at the register level and whatnot it doesn't really matter because you can use this place and this place already do the job for you right decimal zero decimal one the symbol two three four blah blah blah etcetera um but i i wanna have more insights into the entanglement and that's something that i i don't know how to get into in the interface i i really don't know yeah i don't know maybe it's a bit of a a bit overblown to do like these but at least you know you can have a bit of a visual way of how the entanglement is evolving how things look like you know um we'll see we'll see if that becomes useful to explain algorithms or to explore algorithms i believe there's a nice way to explore a grouper's algorithm like these as well so anyway i'll leave it here thanks for joining was 45 minutes to the point we actually made it so i'm happy see you next time |
this is gonna be a bit more I'd say thinking about and investigating to be more about the quantum volume thing here because I as I came across this concept when I was taking a look at but one of the papers from rod that's using the said the the the nice for quantum computers thing can I find the paper Grover for nice or something like this yeah so um inspector they extensively talk about it but also quantum volume I thing in the I think in in in the Kiska textbook there's also a whole chapter about it which I haven't I haven't covered yet but we might as well do it now the the point for me doing this is that I see there's a bit of a so the IBM clearly uses these as a marketing thing where marketing or maybe maybe a way to communicate it's a single number metric that can be measured using a concrete protocol on near-term quantum computers of modest size the QV method quantifies the largest random circuit of equal width and depth that the computer successfully implements cone and computing systems with high fidelity operations high connectivity a large calibrated gate sense and circuitry writing tool chains are expected to have higher quantum volumes but there seems to be so here there's also gonna be a thing a bunch of so volume well quantum volume has a quantitative indicator the computing power of quantum processors processors and that's okay so this is the in is this the sort of the original paper where this is proposed or as this one the paper then there's the stuff from quantum this the the take from quantum computing report which I haven't read yet but I can quickly scan and this also this is the reason I decided to do these videos because I so today it treat from from Craig where he and he doesn't read often and he works at Google but it's he seems to make fun of this it's the best kind of volume announcement is typical of of nice news in that it ignores the real competition remember a PC with a fast state vector simulator easily has a quantum volume of seven billion supercomputers with specialized simulators get into the quadrillions [Music] also not giving for reporting lock lock to QV like like God intended it's a very strange they drive it that way presumably to make it some more and more impressive lock quantum volume or I like you for sure it is clearly better for a technical person standpoint let's try to understand a bit more these first and then let's see and let's try to take a look at different perspectives on this issue it's just a number really but I guess I guess it gets tricky when you think about okay should we use this number to measure how powerful a computer is right and and that's where the number shouldn't make sense so let's take a look at let's take a look at what IBM which seemed to be the ones that proposed these right because this is all so Jay can better come Gambetta is basically from IBM so this is the paper okay but the papers may be worth reading us also hmm the abstract saying the concept of quantum computing has inspired a whole new generation of scientists including physicists engineers and phenomenally change the landscape of infinite ontology with experimental demonstration stretching back from more than two decades the quantum computing community achieved a major milestone over the past few years the ability to build systems are stretching the limits for can be classic simulated and which enable cloud based research for a wide range of scientists as if it's the pullet Allen exploring early quantum systems while such noise in your inner right the frightening testbed for exploring the opportunities of ramifications you have the physician systems including quantum software cloud access benchmarks and quantum systems ok so this this is more than that but then there's a chapter and benchmarking quantum systems and I'm assuming this is gonna be there and marking systems ok benchmarking you're conserving new devices I see connectivity error rates Yates ed compilers and software stack performance quantum volume iBM has devised the benchmark called quantum volume that balances all of the ingredients of both ok so this is not pitched in originally it's pitched in here we believe this system agnostic metric provides a way to compare devices across different physical implementations and measure quality such as lower error rates that are ultimately necessary for practical on a computer but Follette in chronic embraces in randomized model circuits okay that's the actual paper that's the one I want [Music] algorithm okay so it's about five pages so anyway let's take a look at the let's I feel like I feel like we can start here probably good quantum volume is a single never met rate I can measure using it's a great protocol on YouTube current configures I read that so what's that what's the quantum volume protocol I give it protocol is you want to consider the following stems we should first import the relevant kiss get classes for the demonstration the Q function generates Q sequences it is well-known that quantum algorithms can be expressed as polynomial sized quantum circuits build from two cubed unitary Cades therefore a model circuit consists of D layers of random permutations of the cubed labels followed by random two cubic gates from su for when the circuit with M is odd one of the cutest Idol and each layer or precisely a key every circle with depth D and with M is a sequence of the layers of use each labeled by times T equals 1 to D and acting on M prima qubits each layer is specific using a uniformly random permutation of the M cubed indices and sampling you I think keep it saying be from the heart measure on what the hell am I reading in the for example has six cubits 2 0 1 3 5 7 10 we are going to look at subsets up to the full set each volume circuit will be depth equal to the number of so odd subsets after the full set each volume circuit will be depth equal to the number of cubes in the subset we generate the kind of volume sequences we start with a team with a small example so it doesn't take too long to run as an example we print the circuit correspond to the first QB sequence note that the ideal circuits are run on the first n qubits pass the first trial the transpiler to illustrate this circuit pages gates u 1 u 2 u 3 and C X so that's a there's a bunch of you three is control nazis three is control nodes simulate the ideal QE circuits step 2 so you're coming out with a circuit that is basically what I think they are going to is that with these control gates in here they basically you know they basically you cannot you can only do that if this connectivity so so I'd say if you cannot place an operation then you're gonna have a lower quantum volume then you simulate the ideal QV circuits then you calculate the heavy outputs maybe what they mean by this is the actual outputs and oh no the head to define when a model circuit U has been successfully implemented in practice we use the heavy output generation problem the ideal output distribution is these where it's an observable bit string consider the set of alpha probabilities given by the range sorted in ascending order the median of the set of probabilities is these and the heavy outputs are heavy outputs are X belong to this set such that okay so the heavy outputs are the outputs that are below there are over the medium the median properly right so they are the heavy ones in the interesting the heavy output generation problem is to produce a set of output strings such that more than two-thirds are heavy open areas from various depths and their probabilities so for cutely secularism [Music] okay so they find the noise model okay but that this is then yeah okay define the noise model with the Vandellas model for the simulator to simulate DK we add the polarizing or probabilities to the synod mu gates we can execute the QV sequences either using kiss key dears in i have a simulator or or an IBM q provided to obtain a list of experimental results calculate the average case fatality the average care fidelity between the m cubed ideal to meet her is you and the expand the execute you prima the observed distribution for an implementation and the probability of sampling a heavy output is this as an illustration will print the heavy output counts for various deaths chocolate the chip will depth the probability of observing a heavy out to play implementing a randomly selected depth d model is the ease the ease the cheerful deafness largest the usage that we are confident and we now convert the heavy outputs in the different trials and calculate the mean on a volume for up to six cubits and fifty trials with a cubed subset okay the quantum volunteers are within the depth of the model circle with equal importance and measures the largest square shaped model circuit a quantum computer can implement successfully on average the quantum volume is defined as lock - and here's the lock to the arc max mean will is a services for each depth for each depth released if the dev was successful or not and with all confidence interval for depth of successful the confidence interval most me over 97 45% with depth 3 greater than 2/3 confidence is your opponent not successful quantum volume is 8 yes - 2.3 mm okay so they basically with depth 3 this means so this means that they are generating 3x3 circuits if I understood all these blahblah here well what does it mean what is the key music wins pom pom pom pom pom this polynomial science crime circles were built from to keep it unitary kate's a model circuses of D layers of random permutations of the cubed labels a model circuit consists of T layers of random permutations of the cubed labels followed by random to qubit gates what is su for what is this when the circuit with the M is hot one of the key with societal and each layer does not winner I don't care more precisely a kewpie circuit with FDA and with M is a sequence of of the unitary gates u T of D layers mmm okay so each is each of this is a layer and then the each layer is you level top by x key and acting on on M Q eats each layer is preferred by choosing a uniformly random permutation a uniformly random permutation random permutation so that's around the permutation of the M cubed indices and sampling each this is physical and you a B prime acting on qubits and Ephraim's a hard measure on su4 what is these so but basically what this is doing is you're giving it at least list of qubits and then it's telling you to create a 50 QB circuit it's 50 model circles for each of these subsets of qubits that's what this thing does and what and so this is one of the circuits why that's one cubed stay idle because M is odd that's what they said here right the sacred wheeze so we in this case in this case they defined with us in like vertically so the amount of cubits and depth is like how far it goes I think so right so that's depth and that's we think that's why don't you say Cyril Nietzsche layer ok ok I guess the ideas odd is because then you cannot fill in all the stuff with with so you're gonna have to have one because you have like two qubit gates like control nuts and so you don't wanna basically for some reason you want to avoid you don't want to allow these for some reason why no then the whole idea is you simulators you seem simulators and then so then there's this whole thing with the heavy outputs in here wrote the idea resulted to a quantum volume fitter was this things to the heavy output so define when a model circuit you has been successfully implemented in practice we use the heavy opportun raishin prop the ideal output distribution as an observable bits okay so it's considered a set of ATO probabilities given by the range sorted in ascending order the median of the set of probabilities may be so the heavy outputs are then they are the ones that have probably be higher than the mediums than the median the heavy upon generation problem is to produce a set of output strings that more than two-thirds that more than two-thirds are heavy [Music] okay-y though as an assertion of printed outputs from various depths in their probabilities for trial 0 given have you put probably ideal okay I'm gonna try he grasped that completely the heavy outputs thing but basically it's a measure of how well does this really I think it's basically you know it's trying to be permissive right in terms of you want to sample the the circuit you want to sample the actual results and and then compare them to the ideal results but you don't want to be too picky so because of errors and whatnot so what you do is you you take the the heavy outputs and then and then what gate fatality why not we why are we now calculating the fatality chill depth we now convert the V outputs in the different trials and calculate the the mean and the error for planning graph and so the quantum volume is defined as the reason the the largest square shaped model circular quantum computer can implement successfully on average and that average success is if the confidence or if that experimental results of that qubit match the match the the ideal results with nine with ninety simple five percent confidence at least and the doll matching is done with a heavy with the heavy what we call the heavy what the heavy outputs okay that still a bit confusing but let's see let's see what one a competing report has to say the quantum metric the metric quantum volume has been in the news lately and this is a few more words of explanation this metric was first created by IBM in 2017 and modified in 2018 as a metric that would allow comparison of different quantum computers it has many valuable characteristics it is a well-documented straightforward of comparing different quantum computers that can be run on any gate level quantum computer not just once I use superconducting technology most importantly consider skewed count give it quality and other factors so it does not emphasize the number of cubes alone it is a good prefer used by quantum hardware engineers to measure their progress and development if they are able to come up with a new generation of hardware that increases the symmetric they are going to in the right direction however it is a poor tool however is if it is a poor tool for any users to use if they want to measure the goodness on a computer for solving their computational problems okay seems seems like I think I kind of agree I mean that takes what's obvious to me is that this takes account this takes into account but it does not really it does not really take into account connectivity does it because all honestly if you cannot connect like how is that how is kind of volume considerate taking into consideration connectivity for example because they clearly said that in here I know none here in the previous paper so I was coming from here and they basically benchmarking they said it takes into account the number of key weeks of course cuz the more qubits the bigger the depth and and and and though in the Wiis connectivity huggers are connected to one another matters at one extreme kids connected on a on a line would require significant overhead for any randomly selected gate between any random pair qubits at the other extreme if all qubits are connected to each other there is no additional overhead for a randomly selected gate between any random key we pay our at the hardware level concluding matters a lot and greatly influences matrix is just crosstalk fidelity etc it's important to strike a balance between connectivity and overhead for a given application but I don't really see directly how it's not included in metric error rates obviously because if the qubits are prone to errors and you're not gonna get successfully implemented circuits gated the choice of and performance of the underlying gate is important a large set of gates reduces the overhead to synthesize arbitrary gates or more quantum information but also regards for more complexity for calibration and stability but that's the same like error rates right this means if you if your gate set is really proud then you better be good at implementing those things and controlling those things if it gets a little then it means that for certain operations you're gonna have you're gonna have to basically pile a lot of combined a lot of other gates to create and synthesize another gate which this means you've got a chance of piling up errors compilers and software stack performance how is that even considered in there all of the ingredients are both compilers and software stack performance how is that how is that in included in there honestly it's not obvious to me but maybe anyway but I agreed it's not about how good the quantum computer will help you solve the problem it's it's like it's like I mean it's like with this paper here right so can I see the volumes over in say volume a valency of vol 16 right and so and so I I don't see any like this is this is like a volume that's twice the it's twice for Valencia right but if you take a look at these results and take a look at the dark blue versus the light blue right so here couple ends in our answer and the results are pretty similar to be honest here you've got an exception but it's not a solution to the problem anyway here as well I chose the round because I chose the wrong color here I've got a difference you've got a little bit of a difference or here I've got a big difference but it's not like in general the whole distribution in here and also even this case which is you know you've got another another a bit of a different gate set the reserve the art of M and the art of IX they done like it's not you know you can't see a big difference between these two processors and they've got a I would say a significantly different quantum volume so that seems to be a good point we will explain okay so the goodness of cone a bit of sign that quantum yeah I hope I haven't said something stupid but it seems like yeah it seems like this isn't right KL divergence will explain what kind of volume is not really appropriate for end-users in this brief but first we'll provide a simplified explanation of how these measures derive the quantum volume measurement involves testing a series of circuits with a square configuration by that we mean that the number of qubits equals the gate depth of the circuit for example a test may begin by trying out a two-by-two configuration this would mean a two cubed circuit with each gate going through two gate levels in these tests the gate sequence are alternating sequences of single qubit gates followed by two qubit gates the circuit is run run multiple times with the results measured and the results are analyzed with a certain statistical test to see how accurate the answer is there would be the theoretical result okay that's the heavy weights thing if test passes certain criteria then the test is repeated for a three by three configuration for why for configuration etc because qubits are imperfect each increased level gets harder and harder because the gate errors are stacking up and the test no longer passes the criteria once the larger square radius found that will pass the test the final volume is calculated as 2 to the power of n where n is the number of qubits and gate depth so a 4x4 Circuit would have a quantum volume of 16 a 500 of 32 a 6x6 of 64 etc for complete details on how these tests concluded will refer to you to the paper at the IPA being posted on arc your narky okay but that's basically that matches my understanding of these I'd say I've been to intuitive level I do see here I do ask myself probably or I see sense in in the question of why lock to is what we're taking a look at if you're calculating 2 to the power of n it's it's weird or I'm not so sure I understand okay but is it it's it's maybe a useful measure as in like to kind of quantify the errors or to quantify how how good a quantum computing quantum computer is but honestly honestly if the quantum computer would have almost no errors and that shouldn't be a factor okay but then you've got connectivity you've got a bunch of other factors they talked about here connectivity and gate set but again I don't see how criteria is tested in the gates had compilers the gate set maybe makes sense right the gates have kind of makes sense yeah our concerns what kind of volume okay the test is all based upon a square circuit configuration but very few quantum programs really have a square configuration some of the algorithms being developed for nice computer such as vq e NQ q aoa are wide and shallow this means that use a larger number of qubits belonging a few levels of gate depth or others may have much larger number of gate operations versus the number of qubits for example Sherlock Rossum can theoretically factor 2048 big numbers using about 4,100 logical qubits but it requires about eight point six times ten to the power of nine gate operations note that this is based upon logical or error-free qubits and not physical qubits and this is all based upon a square circuit yeah so we do not agree with calculating the quantum volume by using a formula of two to the power of a and we think this gives it they started the view of how fast the quantum computer performance is increasing let us explain this by an analogy let's suppose we are looking for office space and the landlord shows us two offices faces with this office base what I mentioned five by five or six by six you can either you can use either feet or meters as I mentioned depending upon and your what country you're in would you expect to pay twice as much for four six by six as for five by five no you could calculate the worth by looking at the square area in the to mind this is by six as well 44% more valuable than a five by five not hundred percent in a corner computing algorithm we do not think an end user would be able to increase their problem size by a hundred percent if they were provided new quantum computer that just had mmm one more qubit and one more gate level okay yeah but I mean that's a question of how you sell these right like the arguments here so far don't seem so much about the metric itself but about how it's used the focus for anyone developing a quantum computer should be how to make it the chief quantum advantage and solve problems better than a classical computer since classical computers are error free the equivalent quantum volume for a quantum programming program running on a quantum simulator in a classical period can be very high for example in 2019 Google run a quantum benchmark on the summit supercomputer that's six that successfully calculate the results of a 49 by 40 circuit so the equivalent TV for some it would be to the power 40 or okay that's that that's the same that's the same point that Craig makes in here so I mean I agree with these but I don't know where the critique is coming from though [Music] as in like is it that is it that not at all okay I get it it's like you you it's not it doesn't necessarily mean you're quantum computer or your keep you is better than another one with a lower quantum volume I think I think this is probably my personal opinion here would be that that's a metric that you know one can use to measure progress within your own development right so you're you're if the volume increases it's kind of what they're saying here it means you're going in the right direction but to compare quantum volumes even between GPUs from different companies it doesn't doesn't make much sense either because it depends it depends on I don't know it feels it it feels awkward mmm it doesn't feel like it makes it makes a lot of sense but I hey so this is the permutations of this system a model circuit okay so this is the original paper validating quantum certain computers using randomized model circuits assign to the single number metric or limit can be measured using a group protocol yeah the point is you cannot like the thing is once you reach numbers are go over like the 50 60 70 threshold like then you won't be able to run that anymore so what's the point like what's the point of that is it supposed to mark the the path towards I don't know a model sequences of D layers of random permutations of the cubic labels followed by random - cubic gates against that the two qubit gates here and this is the random permutations of the of the the different of the different cubed levels this means probably the different gates you wanna yeah that's even I mean that Daddy that's even the it's the point in here is like if you're if you're gonna measure like if you know a quantum computer at IBM can have the base gate say you WA you want you to you three and control not while another one will have another set of base gates yeah okay but I get it yeah you do that and then you still can it doesn't okay so it doesn't really matter that's the point of the measure if you're able to pull off a circle square circuit that speaking off then it means your quantum computing approaches or your your your keep you rocks right that's the idea but I don't know I really here is it as a good example like there's just a jump from 8 to 16 and there's no visible improvement in how the circuit runs this is the first thing that I can see here of how well the circuit runs these points in here is done seems super strong I get it the one with the simulation makes sense but that's the point the point is just to measure nice computers alright so like that that's why I find this beat yeah it's like yeah there's a point in that there's a point in that but I guess IBM people have a point as well when they say I mean I don't know if they say that but I guess it's the point is you want to try to measure in that direction right so are we getting better than the simulators but you won't get better than a simulator right because the simulator is really just error free so it's it's kind of limited by and you need to simulate it to calculate the quantum volume anyway I'm not so sure whether I should like you don't like it to be honest mmm quantum volume [Music] let's see this is called irons on stake on these ooh okay cool so several people asked me to comment on the recent announcement by Honeywell that they'll be that's a pretty fresh post against there's obvious that they'll soon have what they call the most powerful chronic computer I'm glad the Honeywell which many people might know as an air conditioning manufacturer has entered the race for trapped iron a quantum computing I wish them success I've known about what they were doing in part because through Potter my friend and colleague at UT Austin physics department took one year leave from UT to control to their Airport yeah I wanted to comment about one detail and honey was announcement namely the huge emphasis on quantum volume as a central metric for judging quantum computing progress and the basis for calling their own plant device the most powerful okay so they basically use the Quanah volume to then say that's gonna be the most powerful kind of computing to date one journalist asked me to explain why kind of volume is such an important measure I had to give her an honest answer I don't know whether it is yeah I kind of feel like the same I really don't know I can't can i endu retweet yeah I don't know really was invented a few years ago by a group at IBM according to one of their papers it can be defined roughly as ^ K where K is the largest number such that you can run a kqb random circuit with depth K and with any 20 connectivity and half at least say 2/3 prot tooth or 2/3 probabilities of measuring an answer that passes some statistical test in the paper they use would what the g10 and I named heavy output generation so that's from Scott okay so Google's linear cross entropy benchmark is similar I don't know why IBM takes the volume to be ^ K rather than K itself living data side though the idea was to invent a single goodness measure for chronic computers that can't be gamed either by building a huge number of qubits a ton MIT maintained nearly enough coherence what one might call the d-wave approach or her by building just one perfect cubed or by building qubits that behave well in isolation but unattractively know that they the any twenty consider karma makes things harder for architectures with nearest neighbor interactions only light to these Africans for conducting chips being built by Google regalian IBM itself you know the notion our offer researchers h-index define as the largest eight such that she's published h papers that garnered eighty citations each I didn't know what this quantum volume is basically an h-index for quantum computers it's an attempt to take several different yardsticks of experimental progress none terribly useful in isolation and combine them into one consumer indexing certainly I sympathize with the goal of broadening people's focus beyond but how many qubits does you have question mmm since the answer to our question is meaningless without fair information from that standpoint whenever seems like a clear step in our direction alas good hearts law states that as soon as a measure becomes a target it ceases to be a good measure yeah that happened years ago with the H index which now regular police academic hiring and promotion decisions mmm quantum volume is now looking to me like another example of good hearts law at work the position of Honeywell's PR seems to be that if they can if they can build a device that can apply six layers of gates or six cubits with full connectivity and good fidelity that will then count as the world's most powerful computer since it will have the largest volume one problem here is that such a device could be similarly by maintaining a vector of only 6,400 yields this is nowhere near Quanah supremacy between classical computers at some well-defined task which is necessary though not sufficient condition for doing anything useful okay I think but okay so thing of university of university that achieves an average faculty-student ratio of infinity by holding one class with your students it gets the best call only by exploiting the opposite effect in the scoring system so what's the alternative the policy I prefer simply to tell the world all your system specs as clearly as you can with no attempts made the bullet to bury the lead lead how many kids do you have with what coherence times reconnect 3d what are the one in two key we get Fidelity's yeah most importantly what's the main drawback of your system the spec that's the worse the one you must need to improve what prevents you from having a scalable quantum computer right now and are you going to tell me or will you make me Square appendix three point B in your paper or worse yet ask one of your competitors I confess that the answers to the above questions are hard to summarize in a single number unless you like concatenated binary encodings of them or something but they can in ineffably be ineffably combined to produce a progress metric that one of my postdocs suggests the calling quantum SCOTUS and which roughly costs to the number of expressions of wide-eyed surprise minus the number of grams okay I think that that I think we can leave it here quantum volume seems like an interesting idea it's a bit awkward in a couple of spots and it's definitely stupid to use it as a measure of power and themselves that's my most powerful computer computer I think that it's something I agree with so okay so that's kind of a like maybe here but not every tweet as in like sure a similarly as a quantum computer as well right so this is not the most powerful quantum computer it does I think as Scott puts it like the moving away from using qubits a single measure kind of makes feels like it makes sense but it's like I don't know I don't even know whether it was worth the video but I feel now it'd be better cuz I I didn't even in this paper I didn't really I didn't really go and do the stuff with the quantum volumes so here's a bill it's introduced quantum volume and KQ I don't know what KQ is hmm so what was that last mentioned we found out the Kon thief's solution music scene works on quantum volume 8 and very similarly on quantum volume 16 under K 1/4 solution using 0 on 40 beats works although not well and cube on QV 8 it performs much better but still with limited effectiveness I'm qubit 16 I wouldn't say that honestly it works better on TV 16 why how ok better as in like a slightly better but I don't think that is enough to call it slightly better how would you even say that the circuit is one of the largest K Q values reported to have been ran successfully on a solid-state quantum computer today what is KQ though it gives a measure of the capabilities of the machine independent of the algorithm in 2003 Steen proposed a similar measure for sundar words and needs and an error correction for using cue qubits and requiring K time steps but that's okay so that's just waste times depth the space time product KQ is a guideline to the required error rate which should below 1 / K times Q interesting so the gate Fidelity's are an interesting topic as well it's the KL or what is the KL taillight KL divergence of the data relative to expect properly distribution from pierced a simulation compares favorably to that of a uniform distribution what is KL KL divergence KL divergence as a measure of how one probabilities version is different from a second okay so I guess that's a okay how much something is different from the simulated probability distribution I guess that's what that's what the paper is using that data she sounded better mixed n values you can see difficult as possible using current processes for this particular problem differences in the decoherence criticism two processes result in the offer in the offensive elements of her position became more rapidly than the correct answer resulted in unexpectedly small decrease in overall success probability the processor with a smaller QV however we expect that emotional cases success probability will be more closely or more closely track the k a lot KL divergence okay um KL divergence here here from walking live election we should aquella leverages if use a better value for them you would suggest that the circle is small enough for both processors we but how is it calculated how do I calculate K L on of three cubits subdivided over outputs they can Avengers are the data they're using these to compare with the ideal Grover's result parents from pure state simulation comparison relief that of the uniform distribution with all output that is having equal probability so P expert okay okay with high noise levels so in the corner if this performs one [Music] version with all our profile is having equal probability so there is mister compared to this racial from Pearson compare severally to that of a uniform distribution figure 16 what is here 16 is 16 a uniform random output quantum algorithm performs well with all output values having go probability showing that quantum algorithm performs well although the overall quality of valance is superior to Aaronson to output values 0 in the 1 0 1 are more heavily weighted giving a slightly worse KL divergence the only one 1 0 1 0 1 1 and 1 0 1 yeah and 1 1 0 is 1 okay okay whatever so I think this basically clears out the stuff I don't I don't feel like I need to deke to this more all honestly even for the paper here Kelly maybe also the fight the gallery gates gallery houses krisily affero stuff the gates on devices light blue points are at the gate already on IBM I guess I fertilities just II it's related to the actual error rights right are the custodian are the equivalents is why blue points are the difficulty on on orange there of a key V key of a key have a QB eight and a lens at QB sixteen for each gate type we tested all possible mappings to the process topology collecting the results of 8,192 shots for each patterned top and bottom bar the maximum minimum values experimental results okay so you can see the T V 16 right and balanced yet the if fidelity is much higher the here's a big difference but still that doesn't have such a big influence in the final results all honestly while this initially Pierce's her demand reduction in require fidelity makes it a good trade-off for small problems as shown banks members also miss writing effective amplification for qubit search problems I don't know this whole thing here is it seems to work really but as I don't have access to Valencia and it were answer didn't seem to work well I'll have to know another table is really I think I feel just worries I did have access to I I think I do have access yeah so I should get similar results yeah KL divergence okay cool but I solved also a couple of stuff in here so I'm happy for that as well cool but that was probably an unnecessary long video about where I started with the quantum volume stuff and I just ended up losing myself with other things but anyway yeah good good good perfect |
quantum Fourier transform so qfg this is what we so this is where we left last time and what I am trying to do today is I'm gonna try to explore a bit more the actual problem because I think in the last video we just jumped into building the solution that was proposed here without really understanding what was I haven't in person I was I didn't have a good understanding of the problem so I want to just back up a little bit and then just just to remind you this is an example where we we're preparing a an input that we know it's gonna give us 0 0 1 as an answer and this is the actual so this is actually step number one step number two step number 3 issues like this is the actual way you build the QFT but I have the suspicion that as in most of the cases in quantum quantum algorithms is that those they exploit the structure of your initial state so the reason these steps work is because we we know that this has been prepared in a way that you know we can exploit certain certain properties of the state might be positive a negative uncertainty might be that the phases or the things are set up in a certain way I don't know the but I think that's and that's what so I want understand I'm not super I don't really like the way this is presented here I don't really like the fact that this is so that the example given is just they just prepare something which obviously has some rotations here which they are you know they're thought in inside right because those are exactly the same rotations were making here so I understand up for the sake of the example it's it's good but it's so basically I I've done two things I've opened up this morning basically I've done a bit of googling around and I also start taking a look at this block which is from which is written by Craig Guinea which is basically this block was shared I was not aware of the existence of this one of my subscribers of the channel shared that with me and I think that it's really uh I haven't really gone through all that at all just read a little bit the beginning and but I saw that there is a topic about how do you how do we prepare the states I think that it's definitely worth taking a look at because also I again ended up back a davits Cobb chick block where basically explains a bit the same thing and at least the problem is a bit clearer because here he explains it in a way that so we've got a function we've got a wave right and so I wait a second this was not here I think this is this was this was where did I find that I was reading something that's not what I'm looking for kana discrete Fourier transform let's just discrete Fourier transform I just found something this morning which was really eye-opening in terms of understanding the problem maybe this one yeah III uh yeah yeah this is this is this is it I mean because basically I understand I understood thanks to that what DFT really is right so the idea is imagine you've got and I was really not far away from that but I couldn't really grasp what was the de or like the Y was a discrete whatever so you have it makes you have that wave right so what you want to do at the end of the days you wouldn't know how like what are they what are the components that that make up to these so this is definitely not a sort of it maybe we can call it a pure way for it's it's sort of like it has has its own weird shape so it's it's obviously the result of composing one or more two or more waves so you want to find what are the think what are the the periods I think of those functions that make up for this one but the way you do that is first you sample so first and that's why it's called the screen I guess so first you take you you take your because that's sort of an like a continuous function and first what you do is you you based on on on a period right T you sample with equal equally spaced I think and that's that's kind of also what what we're gonna use in the corner I have the impression this is one of the reasons why the quantum algorithm uses sang the way these but let's let's go step by step so this is we basically sampled those points right and that's our input vector so that's our input so this is the input of the problem so you've got that sample you don't have the full thing you've got that sample and out of that's and out of the that sample you wanna you want to basically have a I don't know what you want to have but basically what you wanna have put a papa papa you wanna have I think here David explains it well so you wanna have is frequency filters blah blah blah so as the frequencies is what you want to have okay so we wanna have basically the frequencies of the of the waves that make up for that input exactly so before we what is the fridges form so you've got a vector and it maps into a new vector so this is our input that I you just you know checked in here right so this is this is your sampling of that and then we want to turn it into this and that's seems almost one-to-one to what the IBM guys you know to understand assume input signal consists of a true signal with frequency f10 and amplitude tool so it's described by this equation and a noise okay so here's an example of okay a true of a true signal that's exactly what I meant and in the sense that okay so this is explaining imagine that this is your input here right so this is your this is your input so imagine we know a priority that's made up of those two things off of this and then a little bit of noise and you know this plus these kind of make this signal right so and and why this is useful I guess one of the applications is if you're able to to get this and these out of out of just analyzing that and you can cancel the noise and do all this stuff like that right so that's kind of I guess how a lot of the stuff works now with noise cancelling or sort of basically signal processing whatever so um I guess what you want to have okay so that's the output you want to have is it correct after the FT so you wanna have two frequencies as a correct one frequency stand amplitude - I know the amplitudes okay look at this amplitude - an amplitude point three and this is what you have here so you've got point three you've got two so that's what you wanna get out I'm and those are the frequencies okay so this is the so here here you've got so here we've got the amplitude and here we've got we've got the frequencies so you've got frequency 10 and frequency 50 exactly I understand all that all those dots so that's what we want and that makes this approaching the problem much more understandable so we basically want to we want underst we want to get and how this is gonna be useful for Shor's algorithm I don't know and and but we'll see right and I think this is probably useful for other algorithms as well so we want to basically get that so we wanna we have we have this input signal and we want to understand which frequencies this is made now I wanted to for this video I wanna say I wanted to understand how can we represent that on how can we represent the inputs and the outputs like what's the concept and the quantum version of it because I think that's gonna help me so my idea is this is gonna help me understand the actual algorithm so this video we're gonna focus on understanding how the inputs are encoded and then that hopefully will make it easy to understand the algorithm so and I'm gonna I'm gonna I think here so here basically I'm gonna skip the first part where they were basically it is so they explain basically Shor's factoring algorithm sort of the overview I haven't really gone through this and a bit of background on the Fourier transform and then I guess the signals and signal processing and stuff like that um but I think this is where it's interesting probably so if you have a computer possible 32 states there is the thing that's the president computer for custom computers I'm quite a computer can rotated states through particular weighted combinations of the classical States called superposition is exactly okay so you can write down possible states of qubits this we know all this stuff so yeah so you have five cubits and you can basically write some things like like like this here right or like this here or like this here so a lot of fun combinations in this post when I say periodic state I mean a superposition where the weights of the classical states go like 0 0 0 0 not 0 0 0 0 0 not 0 ok so it seems like the because I'm guessing the and I'm making a big guess when he talks about periodic states it's this is about encoding the input because at the end of the day here and also here that's what it that's what it happens right so this is our input is and this is the period right one period two periods through third three periods four periods so we want to encode those things okay okay I understand so the the way that that he is suggesting to him to encode that is by saying that so we're gonna have a super position where the amplitudes are going to be the ones encoding the actual samples right so let's say this is 0 25 this is like zero 75 right so but they're gonna be equally spaced by by states where they'll have zero probability okay zero zero so four zeros and then one amplitude four zeros and one amplitude and so forth in other words the classical states that have nonzero weight should be evenly spaced and I'm I'm I bet you that this evenly spaced is gonna probably play a good trick and to understanding how the algorithm works um it's just my intuition is telling me that but um and they should all have their nonzero weight okay the same ah so that song wait a second I'm not completely right I thought that this was the value you were encoding but that would not make sense okay for example for example okay so an example so here we've got an example so if you've got this state 0 this state 5 or here has 0 5 10 15 20 25 and 30 is a periodic state it gives one divided by the square root of seven way to the States zero of blah blah blah and all the other ones are aren't given any weight so you see there's a difference of five in between in decimal okay so is a product state with period five okay okay yeah a more compact way to write the state is with summation notation at this point a certain subset of readers are probably thinking something something along the lines superposition that I bet the quantum computer is just secretly in one of those states okay let me just remove this the ask producers are real so we the specific quantum of pressure we care about in this boss is the QFT what the gift he does it's like it takes the weights of the states pretends they are the samples making up an audio file but they're all equal and your samples won't be equal figures at what the frequencies in that audio are then uses the strength then uses the strength of each frequency so maybe that's just the first step and then we're going to tweet the amplitudes to actually make them become the samples but that would be weird because the amplitudes I always sum up to one that sort of the probabilities figures aren't what the frequencies in the already are the strengths of each frequency as the new way it's defining the state of the computer a good predictor for whether you even slightly understood that last sentence is whether or not you mind just got blown hard it's all the time being blown hard and if you start with the classical state and apply giftee the resulting frequency spectrum looks completely flat the classical state all the action is in the faces not in the magnitudes by contrast the frequency spectrum of a periodic signal is definitely not flat like the spectrograms from earlier the frequency spectrum of blah blah blah if the current computer was really in just one of the classical states how could properly about the spacing between the possible states been getting into the output exactly so you see the spacing is crucial here the frequency spectrum tells a very clear story about what's really going on concrete example the green rectangle on the left is a chance display it's showing for each classical state the probability of the measuring this preposition at that point would return that state okay so that's exactly what we've done since it's separated by but do three four five six is a bit more nice probably not the same example um definitely not because we use five cubits here's more for the purpose of this post all that matters is that it can be done okay so we're that's that I'm happy that he's ignoring the QFT for the moment we're caring about the inputs and the outputs first so we understand how the algorithm is built the rectangle on the ride is a chance display of showing the probabilities of getting various outcomes when measuring the output state it has ten evenly spaced Peaks why ten because the number of frequency Peaks is behaving just like it did with really the input state spirit is ten so the frequency space output has ten Peaks just to check that if the opera's the number of Peaks equal to the period of the input okay before we continue I want to address why we're bothering with frequency space at all if we were rather if we are rather the period of the signal after the period of the signal why not just get it by looking directly at being put signal keep in mind that the green displays there is a non available in real life yeah sure so why don't we just sample the initial signals several times and watch for patterns why can't we figure out the period by sampling the input signal gee there sure are a lot of multiples of five in here the problem with noticing that a certain multiple keeps happening again and again is that as you'll see later in the problem we care about the signal where something from is going to have a random offset if we sample the number 213 that could be 50 times 4 or 13 times 2 or 13 or 10 times 21 with an offset of 3 okay let's just go see Peaks are resilient little imperfections preparing a prairie quantum state I've explained that that's probably the interesting part I've explained that if we had a priori quantum state with unknown period we could sample from the peaks in frequency space in order to learn something of the period but how do we prepare the periodic state in the first place first in any first in an easy case if the period we want is a power of two let's say 2 to the power 3 then preparing a periodic steady simple started with an N cubed quantum register initialize to 0 do nothing to the first 3 cubed and hit the rest of the qubits with a harem are gay each qubit with a harem our gate will transition from 0 to 2 this to this superposition putting the overall corner edges into the state damn that is a product state it's an cubed but can I pick the NI one if I pick if I'm just gonna start a new one I liked it I would like to stick to the example of three because maximization is easier so if I do this so there's an if I do that yeah okay yeah of course because you're leaving dot zero untouched here exactly so those are and now we have here so you see this like it's this one in between each right or in this case there's three in between each okay I'm sure entirely house is gonna so an alternative way to create a quantum state that has period aid is to hit every cubic with the harm or gate and the register we want to prepare into a register of size three then measure the order register and try it again if the result is in zero pretty crazy preparing all the periodic states instead of doing a three-beat edition like model age we can do a different model with some other number okay for example is it the progressive product quantum state with period 7 that applies a Fourier transform to show that there are seven peaks in the output state ok I'm gonna skip that for the moment I think we can dive into this later on in another video on how to prepare those states but I want to go a little bit farther without getting into the depth here we've got a simple example where we prepare let's do it like this so we prepare that with with period 1 ok ok ok those are other ways to prepare all the types of bring order periodic States figuring out periods from frequency samples but I think that's already the that's already the other part of the key of T that I wanted to avoid getting into before understanding you know or wait a second we're still in qft right history if you had periods from frequency samples here's a bit of a puzzle for you good it shows circular prepares the periodic stage using model addition like the last sections what I'm going to hide which modulus I'm using your job is if you're out the secret modulus three peaks so the R is 3 but remember when running an actual computation as I'm six preparing States with an on periods turning appeared into an extra square ruin put it all together okay but I didn't want to go all the way through the entire Shor's algorithm I wanted to understand why so the problem is that IFIF it feels I think this article that could be an interesting view on orthography but it kind of goes all the way through I really wanted to understand how PFT why QFT works and how does it work really um so what I get from here is that the periodic to predict states in computers so this is the input but still why why does this work if the amplitudes are all the same right what happens if what happens if we now do the actual this one so I'm gonna I'm gonna actually I'm gonna do it on this one so I'm just gonna I'm just gonna remove this and and do that so this is the and what if I replace this with that that's not what I was expecting that's not what I was expecting and that's why probably using different different learning material to learn one thing it's probably not a good idea always um good good good but I'm not giving up on this um so I kind of understand a little bit that concept I don't know if that's exactly how is this supposed to be done QFT how is that your sample I mean how can you how is that your encoded sample like I I don't get that right especially if we come back here so accent a quantum state and maps into the quantum state blah blah blah cubed Fourier transform so what does it kind of look like for a larger end yeah that's just a place the thing and I don't like that explanation because it's just trying to understand a little bit more how to prepare the state at all maybe that's just me just misleading so what I'm trying to understand is how do you prepare the initial state as your sample because I assume and I understand the sample is like in this case for example that right so how do you prepare that like doesn't make sense to me to me that those things are encoded in the amplitude because you would need to make sure that the all the amplitudes fulfill the rule of that if you square them and add them they are one so what makes sense to me is that those things are encoded in the and the faces right so you're basically so what it would make sense which is again maybe what was happening here okay I see this I was bashing on I was bashing as bashing on how this example was build but really that those things here I think so let me sorry let me just do this here was it like was it like a thing this was this was pie I think this was pie and a half and I think this was so the okay so the sample is encoded in in the that what is u1 give me a second what do you see one really gate as it is it a rotation on the z axis or one you want it's a cool to the okay it's an R sad okay equivalent to the rotation in that axis here so it's literally like you could use an RZ it's the same thing so it's so you're encoding that into the into the so say face okay but I'm curious if so if if that would be a valid thing to do I guess so if you have a bigger period I guess so I'm not so sure okay okay so but this is the this is the way you do it so you include your samples in this case is 0 3 5 4 - 0 3 5 4 here's a big here's kind of complex and that's actually interesting how do you okay so that was a bit of a nice journey I think we've really I mean we learned one thing which is I think that that actually makes sense it's good that's a good step um but I'm gonna I'm gonna do another video I do want to keep this video is too long so basically we're still figuring out how to encode that exactly is and why does this make any sense like why we're like how do we map between how do we map between like here we've got real numbers and then we are getting into imaginary imaginary numbers and then we'll see how that how the actual qft works and and and turns that into into the outcome that we want which are the frequencies studded off of you know of words that function made off cool perfect |
cool so i think we are live yes let me just see i have it on the phone as well i think we're live yeah we're live perfect there's gonna be a bit of a delay anyways if you're um i usually have the stream on my phone as well just to see the chat in in in real time yeah um yeah but it's basically that's basically it so people are signing up and so on yeah i mean i kind of i kind of what i can see here is um it's sort of the the concurrent people that are watching hi everyone there's like four five six seven people joining um so yeah i can kind of see that but i don't i i don't like to watch it all the time because i get i get distracted too often yeah yeah yeah but it's cool so i hope everyone can see i can see the game i think the music stopped because uh yeah but there is there is the sound and caesar i'm sorry that you can't see the sound the sound in this game is actually really good the the atmosphere that it creates is really um it really chill it's really nice it really kind of uh mysterious and i think it really kind of puts you into the like puzzle solving um puzzle solving mode so it's a pity but you'll be able to see the stuff afterwards and for for um everyone who is joining now and uh doesn't know caesar uh i mean uh you should know him probably uh so this is uh strange works and uh he's got a really nice hat and you should show me before the stream some other thing yeah nice yeah you're putting me you're putting me to shame because i have nothing prepared exactly yes it's a holiday it's a holiday season and there's there's actually a reason behind caesar being the first person in uh in this holiday special which is that this whole idea of the puzzles and everything that i've been doing in the past with quirk and and and other puzzle related stuff it really it was everything inspired by uh one of his was it like an introduction introductory workshop to quantum computing right and to your platform uh yes that's right that's right i i i i've done a lot of things and among them is sort of using our platform to introduce people to to quantum computer getting the first experience from knowing nothing but being on a fr being not been the important thing is that you're not afraid of quantum or coding but you don't have to know either and then to get them get introduce the basics qubits gates superpositions entanglement and an algorithm so daniel here he and yeah and you you gave you gave me this exercise where it was called the rubik's sphere uh we're not just chatting about this before the the before going live and uh it was really for me that was really eye-opening because i think it was and actually i think it was fairly simple to be honest the the the the puzzle itself because it was a you kind of try to office obfuscate the the state with um was it with what was it written was it written in um in kazam i think and yeah and then you had kind of had try i think the goal was different when you try to you had to put it in the one state i think or something like that that's right that's right it was um the idea was after i introduced the uh simple one qb gates to people i um i then assign them a simple task which is i give them a very obfuscated rotation on one cubit give them a set of uh limited gates and ask them to please put it in the ground state yeah exactly yeah it can be done by trial and error but for that people learn a lot of important concepts that they get some commute and so on so so it's a it's a good first task it's completely useless but such as the rubik's cube it's inspired from it's completely useless but one there's a lot about rotations and how complicated it is and that one and the other is not the same as the other way around so it's a good one it's a really awesome analogy but really that is what inspired the whole thing of you know hey you can make puzzles out of these right you can um and actually you can you can do puzzles that help you learn a lot of the stuff intuitively uh about quantum computing and stuff like these and when you think about like relative phases and interference and things like these um which interference is a is a concept that i always talk about because in most uh in most quantum computing editors that you see uh sort of circuit editors that you can see out there um you know you kind of have the the typical you've got the circuit you've got your state vector that tells you what the state is all the time um but you don't see because you don't see the calculations in between you don't see the interference happening right i mean you see after sort of you see the result and you see that maybe some states are not there or they have like a lower amplitude or not but you don't really see where is this actually happening and this is one of the things that quantum quantum odyssey really does well which you will see now in a second um and now i apologize because it can take it can take like a couple of seconds to um to load the puzzle just because i have my machine is not you know it's not a really awesome machine and it's got like obs it's got a bunch of things open but basically by the way the game actually um let me just put the code in the chat so if you want to buy the game the game is actually free to download and i pin the message with the link you can go there and download the game um you can play the uh the introductory material for free but then you've got like three different chapters that are behind like a paywall so you have to pay buy the game within the game so you can buy it in-game but then you can put this code that i'm typing in right now and you're gonna get 50 off uh we'll get the code we'll we'll keep the code um up and running i think we set it up for the first hundred people or something like that um but then we'll have another discount for after the stream as well if you guys want uh want to still buy the game and if you're buying this game sort of you're doing many many things so you're gonna be of course supporting the quantum odyssey team to develop the game farther because this game is really at its early stages you're going to be supporting the unitary fund as well because we'll donate part of the revenue to the unitary fund to be used as microgrants for for their projects and you will also of course be supporting my channel which is uh which is a huge thing uh for me so let me just pin these oh i can only pin one message yeah anyway it's it's you you should mention that what the uk phone does yeah yeah friends of mine so i think we should yeah so the unitary fine for those who don't uh who don't know it i mean i myself don't know absolutely everything that everything they're doing but it's uh it's sort of an entity that they uh just you know go around and fun uh fun projects which are uh you know open source and they are you know projects which would rather not get funding um and and they helped them with their micro grants to you know to kickstart and i think they funded already quite a lot of projects i mean uh the webpage is unitary.fund i think uh go check them out and you can also donate directly there if you want if you feel like donating more money uh they have like a donation button you can donate from you know as less as 10 50 euros dollars whatever to like as much as you want so and also right they they're doing they're donating for people that have ideas for open source projects in quantum computing exactly it's uh there's a lot of need for open source tools so if you're excited about to get started that's a very good place to do it you can apply if you have a good idea you think you should do what's missing you can you can write to them and apply for one of your friends as well and contribute to the open source community exactly yeah so it's always a good thing to do that's why i thought um that would be a nice thing to do in this kind of you know holiday season uh it's kind of the theme and i like to close the year with with some good uh some some good actions um let's maybe let me try to see if i can show you just quickly that's the main menu of the game um i tuned it down a bit in terms of the resolution and stuff so i hope that the quality still comes out good because otherwise my streaming setup was not working at all but you kind of you can actually uh so you have different things so you have the play the play which is basically it takes you to the actual main uh main area of the game um can you see that caesar on on the shares yeah so the stuff here in the bottom the introduction is what you get for free um and if you if you you know if you go here oh i think i shouldn't have done this now i have to wait a little bit because it probably loads the the chop there but we'll see so what this does is this um uh kind of takes you through a very basic introduction uh about quantum computing so like really the very basics assuming you have absolutely no idea um and it does so by walking you through uh small puzzles that kind of build up with time right so you're kind of building piece by piece yeah let's see if that if that loads yeah i couldn't i couldn't get it i couldn't get it better set up so but it should load in a minute and then we'll just go back to that yeah there you go so it's loading so you know this is what you what you will see for free i'll just i will be your guide as you familiarize yourself with how i'll just stop that you don't you cannot hear probably but uh caesar but it's just the voice that's like you know talking you through uh through the main puzzle um and everything else that was there uh in the main in the main menu so the three uh these three balls right which i am again like a physics noob but i think they're supposed to symbolize a quark or something like that um i really don't know uh so these these things in here they are chapter one the fundamental laws chapter two applied algorithms and chapter three thought experiments and so they all have like you know different degrees of complexity puzzles and stuff like that but you can only unlock them uh by paying so but let's go back to the mini and what you can do also with the free version which is really awesome is that you actually have access to an editor that allows you to create puzzles and to play around with the quantum circuit as well and so i've prepared some puzzles today for caesar here to try to solve life on the stream and see if we can put him to shame that uh that maybe he you know uh he's not as good as he thinks challenge challenge accepted i mean i'll give you first of all so um i'll start with with a really basic puzzle just so you get used to the interface okay and that should load faster because it's small it's a small uh example it's just because it's not the common it's not your typical um uh quantum circuit editor right so it's not that you have like the wires and you know you can just place the gates like that uh you see a bit of a different visualization so and it should load in a second it's a video you can't hear the music it's really i mean i'm using zoom so probably i should somehow share the audio as well but okay i prefer not to mess up with the setup you know yeah but as we as we as we wait for this to load uh how how is it maybe you can reveal a little bit of what's ahead for strange works at least in the coming in the coming months yeah so the trench resolution is to humanize quantum and for that what we means is build the tools needed for for developers um to to start contributing to quantum and quantum right now is thermal science and we're trying to make it easier for for it to become more engineering by building software tools for people that want to program quantum computers and we're going to have uh we have a community version that you can go try right now which is really awesome and uh that allows you to live in the browser without an installation to try um to try a different uh programming language programming languages and run your first circuits and whatnot and just tinker around create and share code and content and we're going to be having some big announcements next year that i cannot reveal yet i mean i've been i've been in touch with you guys for a while and and i i played with these with your platform myself and it's it's really promising so i can i can i can i can't wait to see what you have uh you know uh what you have planned for next year um but maybe once you're maybe once you're you know once you're like fully live and you've made the announcements and whatnot we should definitely have another chat i i i think there's so much to talk about in the platform that but anyway so can you see the puzzle i i see something cool so yeah it's funky so what you see here is what the game uh calls a computational map right so on the top it's like your input and um your your sort of your initial state right and these balls that are traveling kind of represent um somehow sort of this somewhat the state you're in right so and and as they move down they are evolving through the circuit and then at the end you have the final state so in in here you have a state where and we have two qubits i can open the circuit actually here so you can see that i have two qubits and they're all like here vertical right and then there's a gate which is kind of a secret gate which you're not supposed to know what it what what what's it called or whatever you kind of see some some colors in here in terms of what it's doing right um the colors basically what they do is they codify the face so actually they have a really cool encyclopedia so if you open this up you kind of have access to um a lot of this information in here about the computational map and and all this kind of stuff uh the color legend here there you go so so you have the colors right and you can also see sort of what's the interference patterns right so blue um blue and red i think they are the equivalent of like uh you know no no relative phase or like a relative phase of like 180 degrees and so they interfere right when you when when they clash so yeah so it's kind of like we won't be needing this for the game but uh for these puzzles but it's just nevertheless good to uh good to know that it's in here so and this is your circuit and the puzzle that we're gonna solve the real puzzle because this is just a test one you have the slots in here and you can you know pick an available gate and then place it whatever you want and the goal is that you need to kind of reach the state that you have that you see here at the end you see that it's a yellow one for the zero zero state and the green one for the one zero state um and also by the way the gates here the the icons for the gates are really cool because they try to visually show you what the effect of that gate is right so a z gate for example has a blue stripe and a red stripe indicating that it it adds a red face to the one component right and it leaves the other one untouched the y gate is a is a crossed it's got crossed lines because it actually also flips the amplitude um uh but it also it's got these lines colored because it also changes the phases of of those elements yeah so and the way that we're going to do this is yeah that's the puzzle and um you can tell me what are the moves that you want to do and i'll make them and then we'll see if you'll solve the puzzles i'm trying to understand the the notation what does the sk does like can you put it in just to try it between if you want i just want to know what it does so you wanted to keep it two or keep it one i'll put it on oh i see i see we'll put it here uh let's see if i can yes let's put it in one i just wanna i think i wanna i just wanna understand what's the effect of this okay all right so it's it's added these yeah i'm trying to understand what the colors mean yes okay you can take it back i think um it changes the one zero you change the one i see i change the color of it the goal the goal is you need to um so this is a given this this mesh in here it's something that it's we can attach and you have a green and a yellow and you have to get a yellow and a green right yes yes that that that i get and then um i see the quantum circuit there's a question mark and a blue box can you tell me what those are a bit i mean i know someone is hitting when it's not i just want to know what they represent a again a question mark oh yeah yeah i'm sorry this is the easiest one that's the puzzle that's the puzzle gate it's just it means you don't know what it is so that's what i've prepared for you they why why are they different is that they're the same no i think they're different because it's a two cubic gate and so it kind of pieces the two qubits um and so this square it just means that this is block it's i think it's just a way to indicate that uh that you you that the gate is a two cubic gate so it means it means nothing else it's just it's a two cubic game all right so let's try to put in uh the y gate on the first one so if we put the y gate on the first one let's see what it does ah that's fine so almost now you get now you've got the so now you've got the balls in the right uh so you've got balls in the right order but right before that like this gate is also changing their face right yeah i see i understand how it's done yeah yeah that's cool all right so um then exchange that i i'm trying i'm trying to know something i change the y for the x take it out and put the x on just see what it does [Music] yeah i want to see what faces are after this you got it you got it you got it that's good all right you got it so so basically the first thing i noticed was that the second digit doesn't doesn't change at all you see yeah i think we're training with zero zero and one zero only yeah so you have to get one qb get on the first one yeah it was just a matter of trying to hurt until we figured it out exactly yeah yeah um it has the game has some additional cues when you um i think when you hover one of the cubes you see some of the the numbers highlighted so it tells you what numbers you're affecting from the state whether it's the sort of the left most or the right most and stuff like that cool so okay so this is the um now now let's do something a bit more educational and i don't do the puzzles like i could just do something random right i could just yeah that's that's that's the serious mode now so i don't i don't just you know do random puzzles um that you just kind of have to somehow on computer something because that's also it's okay but it's maybe not as fun because you're not really learning something if you just do something random so the puzzle that i've prepared now is a puzzle that is supposed to teach you about interference and i know that you're an expert but nevertheless um i'm i picked this puzzle because i really want to show to the people watching that that this is one of the strengths of these visualizations of this game which is that you you can actually see the interference happening when the balls just uh crash and uh and they have like incompatible faces and stuff like that okay so and again like just as a reminder for everyone watching you have the uh the code in the chat if you want to go get the game you get 50 off and we are donating um a part of the revenue uh to the unitary fund which basically seeing a whole lot of black on this on the stream yeah uh it's it's just because i'm just reading some comments it's it's because the loading of the um the loading of the game takes a bit longer than i would like to and this is just because i have um i have not such a good computer um maybe maybe santa will will bring me some some stream streaming capable computer um okay so you should see the puzzle now in the stream as well because i see it in my in my phone so um okay here the situation is the following we've got two we've got a circuit that has two gates already fixed and you cannot move them right but we we see what gates are these so this is a um oh okay it's not available but this is a hadamard gate yeah so i i removed it from the available gate set so you cannot use a haramar gate to solve this puzzle um and you can only place operations in between the hotomars right so what we're seeing now is that we start at zero zero and by the way you can also slow down the the weight the the speed of the ball so it kind of lets you also uh you know take a look at what's happening and we can see is that what we can see is that when it goes through the second hallmark and i'll i'll do something now so you can see this so let me just zoom in for a second i'll stop this oh so there you go let me just speed things up again and you can see here that's what i meant by i chose interference right so the balls are splitting right and then um and you can see that when the red and the blue clash they they just interfere and they explode right so this is something that i usually don't see um in other circuit editors which i think makes these puzzles fun destructive interference yeah so that's destructive interference example exactly so the goal of the puzzle is to get the ball to be on the zero one state and totally blue yeah okay so i'm going to pay attention to it again just to see exactly where to do something here it's and if you if you can reason outlaw i know it's i know it's complicated but it's one of the things that i like to do with the audience here is that kind of show now it seems to be both zero and one the first cubit is the first thing to note and now it's coming together into four and i don't understand what the red does it's a little fast can you slow it down is there a way to slow it down and that the top part but the main thing is that it's going it's uh it's the zeros zero uh ball is going to zero on one so uh since this is one key it's a hard time this is quite standard it's a it's a superposition of 001 you see how that might get very often at the beginning of a quantum circuit because you want to create something quantum so you create a superposition now here it's on doing the hadamard game like i know this videos you mention it i'm trying to understand what the notation is so uh yes so i see i see so i think i guess the blue i guess the blue is one face and the red is a different face so the blue yeah the blue the blue means the blue means no phase exactly the blue is like zero degrees and the red is the is the one that interferes destructively with the blue so it's it's the uh 180 degrees uh-huh got it got it yeah so it's like up and down yeah so so here it's um yeah okay i see i see so let's see so we of course we want it we want to get it to the middle one and we need to change it so this means we have to do an operation between do we have do we we this means we have we need to do an operation between two qubits so we have to give it operations so so you know you have so we have only one qubit operations and the ones that you see here the z the y dx the z and the identity uh and then you have two slots only uh so you can only put two operations in a row max in terms of depth uh or just you know uh yeah you have basically these four slots where you can put stuff okay i'm trying to see all right let's let's put an x on the first q just to see what it does i want to see exactly so we'll put we'll put a next here also i'm unclear what the crossings happen ah okay so that didn't do anything yeah that the the the trick here is that doesn't it still doesn't put the balls where you want them at least not even one of them right right right [Music] i want to find an operation that changes the second there let me see zero all right so why don't instead of the x why don't you just put us that guy to change in which keeping i'll put in the first one first just to see i want to see i want to see on the side edition but i think i want to i'm going to go and i want after let's see this one on this annotation [Music] all right try to put it on the second one again sorry to put it in the second one okay cool uh sorry i'm sure we need more but i'm trying to understand yeah i'm trying to understand uh what all this stuff is doing so this is doing this one is doing nothing here yeah let's see that ah okay all right so let's let's uh i would like you to try the uh xk and then the z gauge yes the x gate in what in in in what key and after in the first qubit let's try that in the first given let's see how do i change this one didn't try the z-gate now mm-hmm after after the x it's funny because it's uh it we are so not used to this type of visualization that i like i had i had such a hard time understanding it's very strange uh huh yeah but but when you once they get used to it like once you get used to it it you know it really opens up uh your eyes all right change the set for the swap gate for the s it's it's the sd it's it's the sk8 is here yeah yeah what is that i guess i don't know it's not it's not ah i'm still struggling understanding what this uh yeah yeah so it's i i can i can give you a hint here so what's happening here right is the the s gate is uh it's changing the phase to green now when when the green hits the uh the next harmonic it will uh it basically splits its placed into a green and a yellow one so the inte the destructive interference that you you used to have before is not happening anymore because blue and green don't interfere destructively right yeah so so that's why you get this mixture of blues and greens and blues and yellows yeah yeah yeah so that's what's that's that's what's happening all right all right all right so and do we what are the gates we have at this one the five years we have so we have the z gate the y gate the x-gate the s-gate all right so let's try first s and then a y and q-bit one yeah first s yeah i know i need to change to give it too but i'm i'm trying to see yeah i mean i mean between you have to i think i think you're what the the problem you're having is these what the so these rays going down are not cubits right no that i understand that i understand yeah yeah but the the at the bottom there yes okay i see that's a face okay all right try let's try this let's try uh leave that there okay sorry uh it wasn't okay it was an escape right yeah i want to see if i can change a lot of things at the same time and then put a parameter s on the qb2 parallel to that one i just want to see when i i'm i'm i'm doing tomography to understand that's yeah that's the best approach i mean so the sk to put it to qb2 and and put it also on one and also so now this one adds up and it becomes red right okay all right i see so change the second one for an x the one the the here you have the key right so the cube qb2 and uh to be an x yeah i wanna see what it does let's see if it's okay so at least you get one ball going in the right direction [Music] yeah yeah okay so you get so you get you get you got at least one ball to hit the zero one state which is definitely what you want to but but they are not interfering correctly that's correct that's exactly right that's exactly right okay take out the swap take out the sk8 i just want to see what it does now but i see that this is uh sort of going one way i see what i'm changing now so now if they're interfering correctly but the end result it doesn't have the face that you want to have because you want the green phase um why don't why don't we try on y again instead of the x exactly where the x is just put that why don't we see again did we try that one already uh no i think you tried it in the first cube that's right let's try that yeah there you go yeah there you go right you got it you got it that's good there's a way you you could even solve it with two gates i think if you have the first the next gate and then you apply a thing an sk8 or something um you have to flip the face but it's but but the um the white gate is the sort of the most efficient way to solve that that's really cool perfect all right good good well done so now i have another puzzle that is um based on this one right so it's the same scheme but it's a bit different so let's let's give it a try as well let's see it's going back to the mini and uh again i apologize for those seeing the stream uh that might might be seeing a bit more black screens that they would like to i know that it's slow i know i'll get better at this um so this was interference one and now i have another puzzle called interference two and by the way the so they're gonna be adding a lot of new cool stuff to this game on the on the fly as well so they'll make this puzzle creation a bit more social as far as i know i hope i'm not saying anything wrong but the idea is that you can then challenge your friends and you know send puzzles to your friends um and stuff like that which um not only i think it's not only just for a competitive uh from a competitive perspective but also from an uh teaching perspective is a great tool now it's one of the things that and and if you go and buy the game with a 50 discount code by the way that it's on the chat you will have access to all the other content that also has um a really cool grover's module so it has a module that explains grover's algorithm and this is one of these unique explanations that it's nothing to do with the uh geometrical interpretation the typical one that you have where you have a vector that's approaching the solution but you have a an interpretation that is completely based on interference so you you kind of see and can explore how the interference patterns really play a role in grover's algorithm so anyway now um as you can see it's it's a similar scheme so we have two harmonar gates on the first qubit but now the target stat is different so we have instead of having a green bowl we have a bowl that is uh a bowl that is uh like it's a mixture of blue and green and another one here it's blue and yellow so it's a state that has a two faces it's on the second second one all right let's let's uh put on the qv2 position two let's try with the x xk so that's a good strategy because at least we know that it gets you it gets the balls in the place where they have to be right right and i remember that there was some uh some face to them so i just want to see what it was so now again go ahead let's see let me look at it again uh-huh so then we have the complete destructive interference there and before we try the ykk there which was the solution we had before like make it all green so i think the next day let's let me try and s after the x just right below that just also just below the x that this will change the face and we're going to let's see that so that changes the phase right before the hot mark right which then it's not exactly what you want because that's not good that's another solution to you that's another solution to the previous puzzle yeah i see that's cool okay so that one doesn't do it check it out and um give it as a z give it a set just to see what it does so this is that will switch we'll switch two things there and i want to see exactly what it's mentioning exactly so they both become red and they get split into these but then you're gonna have yeah um right all right okay i see so i i maybe maybe a little hint here and by the way the game has a really cool hidden mechanism which i haven't used for this but you can actually you can actually hold control and and i can type in like i can type in a uh like a hint in here for people solving the puzzle so um yeah okay but what you so what you're getting is now you're getting also destructive interference here and you don't want to now the the uh here is like what i really like about this visualization is that usually and actually let me see i have never tried that but there's a math overlay so this is still i think oh now you're talking earlier in your language so because the thing is that what would you um what you can see here right that sometimes is not obvious with the math unless you have a lot of practice with this is that you can see what what is this target face made of right you see it's made of blue and a green like because usually you just kind of have your amplitude your complex amplitude which kind of encodes both both things and you don't really you know you kind of see you can maybe think about that you see the angle of the phase but this visualization makes it easy to makes it easy to understand what are the ingredients you need to create the right interference right so if i drop if i drop the if i drop this gate we have i think that's a good starting that's a good move so here we have two blue faces right then we that that one was the one that they was just just when they're all blue yeah so you see the the there's a blue with blue that make a bigger blue and there's a blue with red which just destroys the whole thing so you got to try probably to make sure that this blue somehow splits into a green because the green will mix with the blue right right that would be an idea but i i don't know why don't we try a y gate right after the x i don't remember the way that i started yeah yeah let's try y get right after the x let's see what it does right also b2 yeah give it two okay let's see what happens oh why is it called kiwi too until because they put it there but it changes huh i see the problem i think with the y-gate is that it also flips the amplitudes of this year and the one which we don't want to because because we already had the balls in the right place okay so then and we try already yes which changes things um let's see instead of [Music] we tried x s yeah but we've only played with qb2 so i don't know maybe that is very true why don't we put a uh that's what the sk8 on the on the second part i mean just on the first qubit i don't think it's here you know let's see where it windows there all right i want to maybe zoom out a bit so i can see the whole thing yeah uh yeah but i think yeah i think you got it yeah there we go oh that's so you're you're because you're so the the thing is the the green face gets um gets split into a green and a yellow right um and this is i mean this is the i think isn't that the equivalent to the uh the complex phase right i think that's i think i i'm that's what i was trying to figure out at the beginning so what does the color means and yeah yeah exactly i think i think i think that's the idea so that's why they just then um add up and then you get these things here yep cool perfect puzzle solved so i have one last puzzle i don't know if you've got some time we can try one last more which is different than these ones um all right but this can be it can be complicated i think uh i did it i i did it myself trying to play with the game as well and it's uh let's see i think i think that is the puzzle tool i hope so but what do you think so far so what do you what do you think about the the visualization and the game and be be honest really be totally honest it's really unconventional so that's foreign i think it's fun what i will do is is do topography on all the things and take notes and then map the math of course you show me that you can just click on the thing and figure it out and then just because that's the easiest way yeah exactly i mean they have a really good encyclopedia as i said like that has a lot of this documentation as well in there and actually if you play the game like you follow the actual um the actual uh progress of uh you know from the easy to the heart levels you build up a lot of the knowledge already and so you get a lot of the intuition even if you're like a quantum if you're really like really well versed with quantum computing it's good to start from the very beginning because you get used to the game mechanics and the colors yeah which basically um help a lot there okay so there's there's the game loading okay so no that's not the one that i wanted uh but we can no that's not the one that i wanted that's easy come on sorry i'll just open the other one because that's too easy that's too easy the the other one is puzzle three which is the one that i uh that i wanted to play to play with which is uh let's see oh there we go it's already there solve puzzle three there you go the thing is you kind of see you see it's what i find really nice is that you see for example for the harder mark gate you're going to see how these balls split right which is what happens when you apply two harder mark gates in a row is that your superposition splits farther and you kind of you know you kind of see the state before because i think i think one oh wait a second i've got a question for you in the chat from uh amir says can cesar share more about what he means when he says perform tomography because you're saying you're you're doing demography yeah so so so so they so uh the the main the the interesting thing about a qubit is of course everybody is familiar with bits which is zeros on one and in quantum computing you basically you do not use imagine that zeros and ones are like this north poles on the south pole of of a sphere planet earth in quantum computing you you compete with the whole sphere essentially and one of the first thing you do and but uh because authenticity of quantum computing you cannot actually see the whole sphere you're gonna only see probabilities of north pole and south pole so for that so that means that once you have an unknown state um you cannot just look at it you know you actually have to do a lot of experiments to to try out what it is and uh and this is that's very similar to computerized tomography like when you go to the doctor like you look at it from different angles and you see what comes out so a very standard thing is like if you don't know a cubit you measure it and then you rotate it and you try again and you measure it and you do this many many times in all possible combinations and you build up statistics and then you're like oh then you gotta you get a you're pretty sure what it is after that so so it is just saying that you try a bunch of things and then figure out what what it really is yeah exactly i think that was it you know what the puzzle is that's that's the way you were trying to apply it it was more like i'm just it's just like trial trial and error and see and see what i but demographic is a systematic way you'll find out what it is so then you know exactly which gate you have to use so you spend a lot of time figuring out what exactly is happening and then they then you're gonna with math lesson clay after that all right let's look at this one huh yeah so this is really zoomed out so what we will see here is a puzzle where the target state is the the ground state so you want to go back to the zero zero let's see if i can zoom a bit um now as you can see what you get after this magic fancy gate in here is that you kind of get a really nice rainbow of faces yeah right so you're gonna have to and here i gave you plenty of space so um you have a lot of slots and here you also have the harmar gate to to use and you also can use control operations so um so you have a control gate here uh yeah and uh cool so it's gonna be let's start let's let let's start trying it seems like there's a lot of degrees of freedom here to work with all right let's try it so please cross there okay change let's say again everything splits up can you scroll up yes you see i have i think that's something that all right can we go to the control operations just put a nice one there yes so and you want to put a controller what is the control the first one what is the hc what's hc mean uh yeah so this is a so i this is not needed for this puzzle i think um it's a classical it's a classical coin toss uh i think based on one of the videos that the quantum odyssey team put up there it's a hadamard you think can think of it as a haramar that doesn't add uh the minus phase when the original state is a one state um and it just it's just a thing to show they use it in the game to show the difference between like a classical coin toss and a quantum coin toss that preserves information but i can't forget about this here it's not like i forgot to remove it so let's put the control and can you say what kind of control yes you can all right uh-huh that's funky i can i can't boost the speed up if you want see that again you can speed it up and uh maybe zoom out fully yeah okay oh i can pause it i didn't know that with space i can pause it so so now it's paused but i can i can put it yeah let it leave it running but show me the whole screen because i only see like the very top yeah oh you want to see what do you want to see you want to see that i want to see the bottom again just to remember what i want to say you need to get a big blue ball on the first laser all right okay all right can you swap the control on the y i just i want to see it yes yes yes it's really cool i think the way the controls uh are visualized here it's really cool because you can see you can see these the visual effects of you know which are the states that are uh swapping and whatnot so here you have [Music] between the zero one and the one one right because the control is so you see if i hover over the control you see that the there's some numbers that are blinking red yeah so that tells you what is the qubit uh effect on the the binary representation of the states oh so you got now a yellow and like all green yeah i don't know maybe it's not bad i to be honest i don't know i forgot totally like how how i did that so i don't even know the solution myself [Music] alright what are you thinking um open up open up your mind yeah yeah yeah can you can you leave it full because it helps me think if you leave it so so what i'm thinking is that so now i have yes i want to see the whole the the whole like uh chain all the way down that's the thing this year yeah like it's a trace again exactly yeah so you want to get all the you go we want to get a big blue ball in the first and the first sort of laser right so i'm assuming you're going to have to play with some you know you have to engineer some sort of interference so that you get you know you you kind of kill the the colors that you don't want right right right right right so if i if you want if you put up you can put a y and the at the very bottom of cuban one i want to see what that does at the very bottom like why one yes oh flower cube with one keep it on the one yeah or just i mean or just be low i can i can put it i can put it below so you can have more space and see yeah i just wanna get some type of thing oh this is my point is i wanna i i this is gonna swap of course this too and i wanna see exactly what it what the face of this is yeah ah hold on do you get no did you you we lost the control oh sorry you wanted the control gate yeah i wonder what we were having i mean let's let's leave it like what we're having and the the the previous the previous setup okay let's see what let's just leave it like that let's see what yeah all right this is what we have before and now i just want a single y gate below let's below all that do not delete what we have uh and keep it too yes at the bottom there somewhere let's just see what it does yeah uh-huh that's good okay so what do we what what do you think so what i'm trying to we're trying to see here is like because i want i need two things i need the i want to start uh okay so you want to start killing things right right i want to start killing things so i want to see i want to see what happens with the green ones as they go through them i mean these ones they they will never interfere right because they are like for interference you'll probably need a hammer gate but now you get a really good color scheme so yeah yeah i i i would almost say that now actually if you manage to interfere these with these like you almost have the puzzle solved yeah all right let's um let's bring the arms let's bring the weapons yeah okay so put a hadamard in a second let's see why so below everything like you know everything uh second cube in second cubicle uh oh that now you might get you now you might man if you saw if you solve the pencil just with these three moves like hat off let's see what happens oh okay not quite that point yeah not bad i mean so what's what's happened is it's turned but wait a second why is it let me pause the second before this okay so so we have like it's this just the one one state has a negative phase right so it makes sense that the first these two after the harm are willing will have negative interfe will have destructive interference and turn into a zero oh yeah and this will turn into locally a one i don't know sorry i i'm stepping in the thing is i have i forgot myself i forgot completely there okay but let me just slow this down you see that's that's what i like a lot about the game that you can see this ball splitting and that's where the pluses and the minuses go and so you're going to have so that's what you get to close okay okay this is uh and uh i i can just zoom out i like i like seeing the the the digits uh to remember exactly what everything is doing all right so i want i want to bring those two together that's zero zero one one all right can you put an x x k below what we just did on the right one so right here let's see i want to see what it does yeah let's speed it up that's not going to do it can now put a well we wait for that so let's put them closer together but it has not you you need constructive interference somehow yeah yeah some why don't you move the x to the first cubit i want to see i'm trying i'm still trying to understand oh sorry i uh i lost the gate so that's not gonna do it but i think that will that will just move balls around yeah yeah just that shifts the other ones okay instead of the the x there what happens if we put a mark there i think that can be interesting so let's put her on here i'll put it a bit lower so we can see the balls traveling yeah yeah yeah that makes me okay so now you okay so now let's zoom in let me zoom in because i'm this is not gonna do it but i wanna see it okay so wait a second uh oh yeah but it's not bad right because now i think you're almost there because i can make i need to find a way to make the blue and the red interfere yeah and and then remember the two blues that you have on the left side like on the the zero zero and zero one they gotta become both zeros you're like they're going to become a zero zero right right right right why don't you just put the highlight mark no no i when the element to the right i just want to see just move the one the bottom one let's move to the right yeah just now [Music] [Music] happens you saw oh oh almost yeah we're back to that one huh yeah it's because we aren't doing the the all the things with the alarm so the um delete oh sorry i deleted this one you can you can delete the bottom down the bottom one yeah delete the bottom one okay let's let me remember what this was doing [Music] okay so put put an s below everything now an answer why below middle and the second qubit instead of giving me though so here and something important for everybody to remember is that the quantum quantum mechanics don't commute so the order matters that's why i'm being given definitely did i pause the game or i know it's just long i don't like that i don't like that uh besides this there's some more comments in the chat loud i think uh he's the creator of the game hi loud he says zero zero and one one reminds me of the eprb paradox the state that's impossible to create without entanglement i don't know if he's trying to give you a clue but yep yeah yeah he's saying i mean for that so for that of course we have the control gate at the beginning this is 2k thing so let's have a control y at the beginning [Music] so why don't we do this since it's giving me a hint why don't you put a holla margaret before the control x on the left cubit before the column yeah just move and delete everything along his gave me a good hint there so to delete so what do you mean above that one they're right there yes put them put a hadamard there yeah how to mark the just here yes just there and delete stuff and delete the stuff below no no no the x the control you leave sorry the control i leave yeah i can leave the control roll scroll down let's let's look at it let's look at it down for now yeah no i know so uh can you put the x just an x gate here right all right let's let's wait for it to see what it does okay so we go for the same [Music] thing you still have yeah yeah i think i think that's not gonna that's not gonna cut it yet but i mean if i can oh now that's even weird that you have a lot of you have a lot of mixtures of faces i think you had a good you gotta you had a good shot there now let's go scroll down on the circuit and i want to start deleting stuff below and i want to see what it does yeah let's delete this the last the very last one [Music] and it won't be above it just about to get the right things [Music] don't you have the what are you trying to do with the hana mart here and the con because i think the control x if you if you wanted to do some sort of uh entanglement like wouldn't you put the hard mark on the on the on the kiwi that's going to be control you're totally right can you move it for me please yes you're absolutely right i don't know what you're trying to achieve yeah solaris yeah the thing is they come the comments come a bit uh [Laughter] and now we're back to scotland yeah they take out the hazard from a mother control it could be we need yeah let's look at it again so what is it what is now happening so now you have a mixture of weird stuff i liked it more before when you didn't have a mixture of faces so we have a face between zero we just have to change the second one i'm looking at it to see what was the effect just above the the those case we are okay let me follow them this is now we go to the control and we go to the x those are y gates and there and now you're doing the harmonics for the interference and you're getting a mixture i think these these y gates bother you here yeah let's take it out the y gate i think because now because they were avoiding the interference and now at least you're gonna get i think something [Music] also not no sorry let's see [Music] i'm just going to put them a bit more closer together so uh so they're easier for me to manage so here you have before i mean if you don't mind uh just like because i'm i'm curious about this as well so you had this before and i don't i think that was not that bad all right let's try the because now now you have at least you have a really clean face [Music] now did we try some wise below that i thought we didn't i think we didn't all right let's write what let's try a wire describe everything in qubit one first and then qv2 just to see what happens before the hadamard or after that after the harmonic okay so that's not bad like really i think that's and now you have a now you have an entire oh and now you have the epr like thing right don't you like zero zero one one yeah so now you just gotta undo that all right so one way will be can i change [Music] okay so uh what below everything that we have do uh x and a control next to each other the control being which keep it on the second one let's say so i'll try and then we try backwards yeah yeah we have we have this entangled state i'm trying to find what the correct circuit is yeah to to uh to undo the entanglement essentially following the creators uh yeah okay so now that's not bad at all all right so now you're in the z zero plus zero one mm-hmm then they put a hadamard just below the x hana mark just below the x and i no that was not good but it's not bad either i mean you have an equal superposition mm-hmm right so but i have the opposition between all things with all with all like positive faces yeah can you move the hadamard to the others to the second qubit yes and i think that's gonna make it i think that should make it alright let's see what it does yeah yeah yeah there you go you got it yeah you got it now you now the glasses gave you the power awesome man cool nice yeah it's uh yeah that was a bit i think i mean you know it's i think this one was maybe a little bit less uh educational i'm trying to think about what can you take out of these but it's it's definitely like the techniques that you develop to solve these puzzles that um kind of help you gain intuition in terms of you know the one thing that i do think the game is missing uh but i think that they're gonna add soon as well it's a way to see individual qubits um sort of i think it's going to be a view that will definitely add a lot of value to the puzzle solving because sometimes you want to take a look at the system right and sometimes just but when i found of the challenge was that i i i i think i think it'll be useful to do the very basic to like beginner examples just to get used to invitation yeah because i was still i'm like oh what was the red one there like it so yeah i think that very useful and then then you understand it and it's inner logic for for the for it and then the puzzles make more sense exactly you have to i think i think going through the basics of the game um i mean and the good thing is it's for free right so that part you can uh you know you can you can always download and play with and then uh i'll just say one more time before we finish the stream because i think we're done we're about like an hour in now so thanks for everyone who's been watching we've got like a lot of people watching so uh at least for my for my standards i'm really happy um you can download the game there's a link pinned in the chat in the video on youtube and there's also the the code below at the beginning of the chat that is going to give you 50 off and peter sure peter sure wishing you merry christmas merry christmas yeah so thank you very much everyone for watching and thanks caesar really for uh for being the for me it's been an honor really because as i said you were the inspiration for the whole thing you know the first time that we had the call and that you showed me the rubik's sphere um it really you know it really woke up all these interest in the puzzle uh puzzles world from that perspective and since then i've done a lot of other stuff as well with quirk but this is when i saw the game here as like that's the perfect thing to play with and build these puzzles this is great and keep up the good work on your channel cool man uh anything else you want to say about strange works no you want to keep it low-key and secret as you've always been which i really like you can go to quantum computer.com and create an account and start playing with circuits and programming languages if that picks your interest exactly quantum computing.com and create an account there uh so there is a comment in the chat just before we go someone's asking whether there's something even more complicated yeah you can make a lot of more complicated a lot louder the creator of the game just said in the chat that he once created the puzzle himself that it took himself six hours to solve so uh yeah so there's uh there's definitely a lot you can do with these now um this is the first episode of the holiday special with caesar i'm going to have more episodes coming up um i can already reveal that one episode is going to be with olivia lanes another episode is going to be with the eigenpros as well i don't know if you guys follow them in youtube uh definitely recommend that as well and uh yeah so uh looking forward they'll probably be in january we still have to fix the dates it's just because we're too close to the you know christmas stuff and so i think everyone should take a rest from you know youtube and uh all this stuff and just spend time with your friends and family uh and yeah uh cesar thanks a lot again for joining thanks thanks very much thank you enjoy the enjoy the conference what is it that you're talking to now afterwards uh um i am doing another um introduction to uh where does the power of quantum computer comes from tom before but is it like a public conference or something like okay cool thank then this team is lucky to have you there with the introduction and good luck uh yeah and we we stay in touch see you everyone and thanks for joining stopping the stream now |
Get the Game: <a href="https://link.xsolla.com/GO9BYp3T">https://link.xsolla.com/GO9BYp3T</a><br>50% off with code UNSYS50 |
The background music is distracting, but this sure looks like a fun way to analyze circuits., lol I think the background music adds a lot of atmosphere! ;) |
hi everyone i'm ethan hanson and i am doing a takeover of this channel you'll never see your beloved daniel again no more hahaha just kidding seriously though my name is ethan hanson i'm the quote the host of quantum computing now it's a podcast all about well probably guess corn computing I go over news basics and interviews and I'm doing a little crossover episode thing with Daniel so I'm doing his episode for him more or less it's gonna be different I'm not gonna show anything on the screen you're just gonna get to see sort of behind the scenes of what me making a podcast looks like and he's going to be doing an explanation on my channel or my podcast if you want to go over there and check that out again it's not going to be exactly what you're used to if you listen to my podcast but it'll be something cool and interesting and different and we're gonna just try it out I'm enjoying collaborating and I think that this is a great opportunity so what I'm going to do today is explain a bit about quantum key distribution which is something that I'm super fascinated in because it's sort of a that where the Venn diagram meets between quantum computing and data security both of which are I think are super cool but if you haven't heard the warnings in my podcast maybe this is the first time of you hearing my voice I should warn you I am so much not an expert in these things it's it's funny honestly so please take what I was saying with a grain of salt look it up on your own I'm going to give resources like where I found my information as well as further information if you want to learn on your own absolutely I recommend doing that learning on your own is the best way to learn I like explaining things cuz it helped me learn so yeah what that being said I'm gonna do my best to walk you through the basics while also walking myself through them so let's just jump into it I think the best place to start is the problem behind it so why do we even need key distribution let alone quantum key distribution so the problem is we as humans want to gossip behind people's backs online without them knowing what we're saying essentially I have a secret that I want to share with only one person and to help me explain this I have made popsicle stick representations of different people so here's me that's my channel logo I'm computing now if you didn't catch that earlier go check it out here is quantum intuition and this is my friend Tiffany important so let's say the gossip I want to share is behind Daniels back the number is 13 now it's very scandalous I want to tell me I want to tell Tiffany the number 13 and I only want my friend Tiffany who is definitely real and I definitely really have real friends okay sorry I was saying that I only want Tiffany to know the number 13 I don't want Daniel to know so turns out there are very interesting ways to do this involving modular arithmetic exponents and you end up with a shared secret so you have a key that's shared between just you and the person that you want to send information to so you can use that to encrypt but that's a little bit more in depth than what I'm going to do here if you want more information on that check out my podcast episode on quantum safe encryption which I have a link to in the description I think that does a good job of explaining it a bit more in depth but again if you want to know like exactly what's going on there's definitely better resources out there for now I'm going to start with a very very basic very simplified explanation of diffie-hellman key exchange if you want to look that up it's a di FF ER IE - h-e-l-l-o ma n i'm going to use it popsicle sticks and some Expo markers yeah there's a much better explanation from computerphile linked in the description and that one comes from someone who actually knows what he's talking about as opposed to me I don't know about other than a couple of computer science courses and reading things on Wikipedia yeah so for this demonstration I need everyone to pretend that addition can be performed but subtraction is impossible if you know what's going on behind diffie-hellman you'll know that what I'm doing is addition is modular exponentiation and subtraction is the inverse of that which you you just can't do unless you have all of the base information at least not in a reasonable amount of time yeah so it's in this scenario subtraction is impossible or not impossible because it takes a computer thousands of years to perform so there's explanation I have a number that I want to share Tiffani and first of all we need to have the key that we can encrypt it with so how encryption works is I have a number and in in this scenario we're going to say that multiplication you can do really fast but division you can't which is actually how it works in the real world you multiply two really big numbers together two really big prime numbers so that there's only one solution for dividing and unless you know one of those prime numbers it's really hard to find what the two prime numbers were but let's say my first prime number is 13 that's the one I want to multiply by I need to find a new prime number and we're gonna do that in a slightly skewed way because addition and subtraction rather than modular exponentiation yeah so I want to either just tell Tiffany hey numbers 13 but Daniel here might be standing right behind us right back here and he might be overhearing that so that's what's going on on the internet when you just broadcast everything I could say it I could shout it but then again over here so what I want to do is I want to say something isn't the number but that Tiffany can then use to figure out what the number is so what I'm going to do is we're going to start with a base number remember addition can be performed subtraction cannot here's my base number don't really care what it is we can call it X for now or blue so here's my blue base number everyone knows what blue is he's saying Daniel here everyone is able to see blue it's out in the open it's here in the middle and I have over here a secret color called green and what I'm gonna do is I am going to add my green to blue my grain and my blue and I am going to send that to Tiffany who is able to add her own color so I send this over to Tiffany Daniel can have it too for all I care doesn't really matter he can get all of these but because he cannot subtract all of these he doesn't know what to do with it he doesn't know that he can break the expo marker in the middle to turn it into blue and green we're all intents and purposes these are this is just one object but you can't do anything with sorry the exposure is not making this work nearly as well as I thought it would so Tiffany now has this she can add her own secret number to it red and then she can send this whole thing I actually she holds on to this and she's added on her red space she's gonna add on her red to her to the blue so she has this so she now has I wish I had another blue marker she now has blue and green as well as blue and red and she can add those two together so imagine another blue marker blue blue red green right and the order doesn't matter assume it's like addition where you can do it in whatever order and you'll get the same thing for 1 plus 2 is 3 2 plus 1 also 3 blue plus blue plus red pill screen is blue blue red green so now she can send she's got the whole thing but her thing was just blue and red so she is going to send that back over to me and now I can add on my blue and green so I get blue blue red green now we have the same key Daniel here in the middle saw the the two different parts go through but because he cannot he can't work this so that this is not the best example I highly recommend the computer file video he's as color mixing but I didn't have time to put together but that's probably a better explanation he is unable to come up with this key but now I have this key and Tiffany has this key and now I can say 13 the thing that I wanted to send to her I multiply that by this number to call it 17 13 times 17 is if I'm not mistaken so now I tell her 221 and because she knows the key she can divide 221 by 17 and get 13 so that is the basics of a difficult key exchange essentially you have your private key and you have the other person's private key that you don't know what those are but you take the shared the shared secret I guess not really a secret if it's shared but the base number you mix these two together you get your first secret the other person mixes theirs together yeah their first secret you mix all of it together you get the overall key you have to mix it in a certain way otherwise your person in the middle can figure out what you're saying behind their back and then you combine those together and you can use that as your key all right so it seems pretty airtight assuming take my word for it that in this analogy the real world equivalent of subtraction or breaking the expo markers and finding out what the different components is is impossible what's the problem the problem is that quantum computers can perform the subtraction in this analogy they can break the expo markers and see what's going on in between and then you can use that to pair out what the keys are and you can use that to you know read every signal message going back and forth every encrypted whatever you are looking at bank numbers transaction information what you're putting on your private Instagram account even hello anything that is encrypted dissy Hellman is so ubiquitous it's it's everywhere like they say in the computer file video every time you connect to a every time you connected to a server you're probably doing a difficult key exchange so because everywhere it's a big deal that quantum computers can break that there are so there's this thing called super singular elliptic curve I saw Janee cryptic cryptography which is a variation on diffie-hellman that is potentially quantum safe it hasn't been proven more information on that in my quantum safe encryption video but the whole point is that essentially it can be broken maybe not now because quantum computers are stone the noisy intermediate scale quantum era but at some point potentially can be broken so now what now what is quantum key distribution arm key distribution uses quantum properties of objects to encode information about the key that you're sending in a way that makes it just impossible to get the correct key without some prior information or without letting someone know that you are interfering with the the key sending its right so in this example it's absolutely possible for Daniel in the middle let's say he has a quantum computer because if anyone's gonna have a quantum computer it's gonna be him he's in the middle right he sits in the back here we're talking what he can hear everything we're talking back and forth and he leans in here's the information about her key takes it back to his quantum computer run some calculations and then he knows what her keas and now he knows exactly how we're encrypting everything so every time that we send a message between each other he can read that I'm saying 13 about him and we don't want that but we also don't know that it's going on there's no way to detect that under this scheme under the that's what the power of quantum key distribution is that you can detect if there's someone reading as you're trying to distribute keys and you can go oh no someone's reading that don't use that to contact me or to encrypt anything to me so here's the 30,000 foot overview of the bb-8 for protocol a quoi sorry a common quantum key distribution protocol and you can find out more information about this link in the show notes too IBM's quantum challenge recently finished up this is the first time that I had heard about bb-8 4 or 84 I never heard it pronounced before anyways is they had in their quantum challenge challenge 3 or exercise 3 was all about this you essentially got to create parts of it yourself and that was super helpful to do it hands-on into knowing how to do this intuitively you want to develop a quantum intuition sorry then that's a great way to do it I recommend going and checking that out okay so this is what you do you take your key so this that we would normally have built up piece by piece and scientific Hellman you take your whole key and you encode it in quantum measurement bases it cannot be distinguished with a measurement is to say they're not orthogonal so I take each bit in this and let's say that this is a 1 and this is a 0 and this is a one I choose random basis and those bases are non orthogonal which means it's not your 0 & 1 basis so if I have a 1 I'm not gonna put it like this I have a 0 I'm not gonna put it like this if I have a 0 then I so my two bases are 0 1 and plus 45 degrees minus 45 degrees and there's better explanations online but essentially if I have a I have a one that I want to encode and I have it in the zero one basis I put it as a one which I put down for Bloch sphere purposes if I have a zero in the up-down basis I put it in zero state if I have a one in the 45 degrees state which you can also look at as you've done a single Hadamard on it then that is your one state had Martin which becomes your - state if you know the cat notation for this it's - but that essentially looks like could this you're gonna go from 0 up to here and if you're in the zero state then you go to the plus state and that's how you encode your Basie's then you send all that any coded data you can do this with photon polarization which is why which is how they are doing it currently they can do this thousands of kilometers distance and do it for extended periods of time it's actually cool how not well established but how well tested this is and yet not a lot of people know about it but yeah he send all the encoded data so you've polarized your light and then the person on the other end is going to measure that encoded data also in random basis so they don't know if their basis their measurement bases are going to come out correct or not and so in this case let's say you are you have it measured in if you sent it in the zero one basis but it's actually being measured in the plus/minus basis now has a 50% probability of giving you the correct answer and so you might wonder how does that give you a consistent key between both parties get to that later so you measure all the encoded data in the random AC so sometimes they'll be correct sometimes they won't now you both ends announced essentially publicly all of the basis that you encoded and measured in and then you take the key and you only use the basis that it was that where they matched so you know that you got the right answer so that key now you've got it essentially built up both sides so we said this let's say that this one here was measured in the wrong basis we'd get rid of that now we have this this is now our key you'll notice it's smaller which means less secure we'll talk about that in a bit but now you can use this key to encrypt your data I can say this x 13 and tell that to Tiffany because the data has to be measured in order to read it and write if you've got something in a superposition you're collapsing that if there is an eavesdropper like Daniel's here in the middle right so back to our earlier example you are talking Daniel's back here let's say he we are using quantum key distribution he tries to listen in he tries to listen in we will know we can see him and go hey I see you don't use this key to communicate back and forth because he is make he's changing the states that were sending back and forth so he reads it Ferguson Tiffany tipping tech that tell me not to encrypt anything with that key because it was compromised so some issues with this like I already mentioned you're going to get smaller keys there's a lot of overhead so the two places where you get all of that overhead from first you discard all of the bits where you measured in the wrong basis because you're measuring in random basis so yeah you're encoding on random basis you're measuring a random basis you've got a 50/50 shot on either so you're only going to get the correct basis about 25% of the time so essentially if I want to send a key that is this many bits long I have to send one I have to send four times as many bits and then it's going to get collapsed down to just this which is a significant amount of overhead and then when you compare your bits of the final key right because we essentially announced that publicly because we're announcing that publicly actually I might have forgotten to mention this at the end the way that you determine whether someone was listening in is you compare bits in your public key and if they don't match up or if a certain number of the don't match up I guess a certain percentage don't match up then it's it means that you are Kia your he is not safe and you can't use it which is cool because you can't tell if someone is listening in but it also means that you are losing those bits that you're saying that you're putting out into the public because now anyone knows how those are in the key that helps me figure out what the rest of the key is what the rest of the key is but if I'm using the full key with those bits that I put out makes it much less secure than if I just truncate those bits and use a shorter amount for the actual encryption and decryption so you'll end up having to resend the key over and over if you just use four times because you'll if you just use the four times just decoherence and random noise from the environment will cause enough errors to bring it below your threshold of saying okay this key is safe so you'll have to send it over and over and over again which creates even more overhead then if you send something like ten times as long as you want so rather than just sending four times this you're sending ten times so in this analogy if I want to marker key I have to send twenty markers which is a significant commitment another issue is that you still need a classical channel that you trust what the Wikipedia article calls an authenticated classical channel because when you're sharing Basie's I might not actually be me right let's say we're sharing Macy's Tiffany and Tiffany goes okay these are the basis I had and then somehow cut off from the internet and Daniel decides to start wearing a mask that looks like me tel stiffly oh yeah those are the basis that I got go ahead and you can just use that key and we're good then they start communicating and it's actually back here actually Daniel the nefarious guy listening in on the conversation that Tiffany thinks she's having with me you do this with a sort of man in the middle attack where your Tiffin sending things to Daniel who's sending them to me back and forth and things just might not never match up but we'll never know about it we have Wikipedia puts this better than I could quantum key distribution is vulnerable to a man-in-the-middle attack when used without authentication to the same extent as any classical protocol since no known principle of quantum mechanics can distinguish friend from foe yeah these are just some of the problems the Wikipedia page lists a couple more it's very helpful that Wikipedia pages you can tell I suggest giving a quick read it's very readable Wikipedia page for being about quantum key distribution which I was pleasantly surprised about so what do we make of the claims that quantum key distribution is just unbreakable it's not unbreakable okay let's get that out on the table right now it's very fragile actually it's fragile in a good way and fragile in a bad way so because the quantum state is fragile eavesdroppers can be detected that's the whole reason they can be detected is because measuring the data changes the data as an article from quantum exchange which is company that works on quantum encryption and quantum key distribution said under their section about why it's unbreakable so you know take it with a grain of salt make sure you I heard to borrow a term from AP class AP History classes you got to check the point of view of the author's but anyways I say the security of quantum key distribution stems from the ability to detect any intrusions on the qkd transmission because of the unique and fragile properties of photons in a third party or eavesdropper who tries to read or copy the photons in any way will change the photon state the change will be detected by the endpoints alerting them that the key has been tampered with and must be discarded a new key is then transmitted that also means small perturbations can keep the key from reaching the endpoint or reaching that point in a coherent way in another sense it's fragile in that it hasn't been battle-hardened like our current implementations of diffie-hellman and ours they have right this marker example it's not what we actually do what we actually do is much much more complex and I guess the reason it's more complex I'm trying to say is that we've done this in the past this is what the idea started with and we realize hey we can break that this way and then we get a new iteration we haven't really had that with qkd but that being said work is being done right now to make it less fragile and less susceptible to attack if that work is carried out a quantum network could someday accompany the existing internet to provide more security exchange you've already seen that in places a couple places in China I know that there was a record for most distance traveled for quantum key exchange I believe it over 1200 kilometers and the reason for that is that scientists in China sent the photons up to a satellite and then back down to another base like base station which Space Station base station which is pretty cool honestly but it's just a proof of concept at this point but yeah if the work continues and if things go well there's a potential for a whole quantum Internet which won't replace the current Internet but we'll definitely supplement and that's what I have to say about quantum key distribution Thanks so that was even I hope that you enjoyed the episode a special episode and yeah we kind of it's one of the things we thought we'd do a bit of a switch you know for one episode see if that's something that you guys would enjoy something that even can usually do in his podcast cuz well it's all cast so I highly recommend that you go ahead and go to the quantum belly I think he has all the episodes listed in there but you can also get the podcast through your usual podcast source influences I guess but yeah there's a bunch of interviews and a bunch of other interesting topics in here that Ethan talks about and cool I hope you like it if you would like to see more of this stuff from time to time in the channel let me know it's been fine for us as well and for me especially and yes tuned for more |
so here basically I just started recapping the basic continuous variable quantum computing concepts just to help me refresh my mind in terms of you know to then approach in the next videos the actual Yellow Submarine project and then the researcher on it and I just realized that was not recording but now I'm recording so basically it just went through the introduction and a bit of the queue mode so yeah cue modes I remember as a physical system we can use quantum harmonic oscillator as a model that allows us to investigate the state of continuous variable quantum system and its associated cue modes in general for each state of the system there is an Associated cue mode therefore would make my distinction between state and chemo when there's no risk of confusion okay I was a bit confusing because the state in discrete corner in discrete variable computing the state is the stead of your whole system and then the qubits are part of the state there are multiple qubits that give you one state of the system whereas here they seem to be to indicate that each Q Morris and it's in a particular state in itself okay when there's no risk of mathematically if this system has a Hamiltonian H then it is described by the following okay the some of the different Hamiltonians okay nebula matched Katie knows the Hamiltonian of the case harmonic oscillator among n ki mode so literally it's the the system has a Hamilton which is the sum of all the Hamiltonians of the chemos it's made of from elementary quantum mechanics we know yeah elementary require mechanics the Hamiltonian of the quantum harmonic oscillator is given by its own unit mass [Music] in equation two well this is equation two when you're starting with the question for my friend whatever wk is probably a question five so this maybe is the time for I don't know should be starting from one probably in this equation tool think it's the frequency of the case mode the position X K in the momentum p k operators of the case mode obey the commutation relationship one so you've got here different components I see so basically you're defining Hamiltonian R as a function of frequency of position and momentum of that thing of that physical device which is then a well they call it quantum harmonic oscillator it just let me quickly check I'm not so sure you're probably you won't see that because I I can't switch tabs and so they when I'm recording but what is the quantum what is what is a quantum harmonic oscillator it's kinda mechanical analog of a classical harmonic oscillator because in the future potential can usually be approximated harmonic potential it's a model okay so it's it's a most important model system in quantum mechanics it is one of the few quantum mechanical systems for which an exact analytical solution is now so it's just a wave function is it that this is quantum mechanics I'm looking at these by the way I'll just roll out of here cuz I it's just a model it's just a way to model a wave functional what is that states whatever but it seems to be the case and of course it goes back to the fear I won't cut the appendix this is I mean a little bit for you so we introduced two new sets of operators defined in terms of X K and PK quadrature quadratures quadratures of a single chemo can are defined as follows okay so those are so those are called quadratures the position quadrature the momentum quadrature I assume that then can be I will thank me defined in terms of these quadratures okay they're called quadrature is because they are then here squared or what the quadrature operators obey the commutation relation since there are hermitian origin this is myself to dig into this is also appearing in the discrete version of computing they correspond to physical observables that can be measured okay so basically your can measure position and more momentum in or momentum okay but it's it makes it I have taught me it makes it much easier I mean I don't remember how did I react when I read all that stuff but I probably I'll go I'll go back and watch the video I think it had to be a lot of a lot of fungus now I'm gonna be both familiar with the Hamiltonian concepts with this whole things and after having that just at that and really having put the project side for some weeks now now it feels a bit more comfortable to read that so you've got position momentum oscillators the cue modes and onion and blah blah blah even if you don't really care about the actual formulas in here the quadrature control preseve the commutation relation since there whatever that means since there have each in their physical observables the luckiest there exists a special state that corresponds to the lowest energy instead of a quantum harmonic oscillator these days of lack hemostat also called the ground state and I guess it's pretty cheeky mode evaluate the studies about the girl because coming from classical mechanics we assume that a vacuum has no energy but in quantum mechanics the vacuum actually has an agenda we must take that into account in canoes are competing Connect bidding this is best in strength and also the specimens writing the analysis of a beam splitter if we send a forum in one port of the beams player but not the other port our analysis must strictly a new sport as a vacuum and if we require the use of the hamiltonian of either port Hamilton of the a new sport shall not be zero but we couldn't but won't be constrained by the Heisenberg uncertainty principle uh-huh interesting so it doesn't mean no energy okay Garson stays in operate certain class of states the so called Gaussian states are extremely important because they are easy to efficiently produce in the laboratory this is important because while we know the Gustin states are not enough for universal competition we still want to investigate what can we accomplish with them alone that we cannot accomplish classically in this section which reviews the describe crossing states and causing operators functions okay Gaussian functions found under the name of normal distributions in mathematics mathematics literature Gaussian functions occur in striking regularity throughout science we have already encountered a phenomena whose state is described as infraction the vacuum state a multivariate Gaussian function has the following form where X and x equals please because design a is an M for X positive definite definite matrix okay so no details we can have function in 1932 in a in a paper titled on the corner correction of thermodynamics a clipping work natural or weak may form a little function which is quasi-probability distribution for multiple particles for a single particle in the X space is the reaction has a flow form cannot solve this function from under state but the reason right understand what it means the interaction is a quasi-probability distribution describes the effects on quadratures if the solution to diffraction for a particular quantum state in the given way this has a form of a Gaussian function we say that such state is a Gaussian State okay this function belongs to Y the class of functions called quasi popular distributions okay so that's just kind of a background for now Gaussian States Glassons has occurred three main ways a spear coherent a spear squeezed or squeezed coherent States I remember there's something my question says are states of minimum uncertainty equally dispersed among the quadratures they are created by applying the displacement operator to a key mode in the vacuum state this means that the vacuum state is also a coherent state because it can be obtained by applying the displacement operator so you apply displacement gate to a vacuum and you get a coherent State this means that you get minimum uncertainty in both position and momentum squeezed States that refer to states where the quantum fluctuations in one culture are reduced at the expense of increasing uncertainty in the conjugate culture okay this is all this is saying is if we just assume we have its two cultures position and momentum so you squeeze if you squeeze a stay it basically means you're increasing your reducing uncertainty in one of those quadratures and one of those two they're called quadrature I guess yeah but increasing uncertainty in the other one squeezed coherent States refer to states they have displacement operator applied to the vacuum followed by application by definition of the key I guess I mean by application of squeezed game displacement operator applied to the vacuum Allah followed by the application of the chemo okay gasps an operator's an apprentice Casanova chosen for us and into a Garden State sense mentioned tool displacement is squeezing in the press in the preceding section now we'll operate on them and introduce two more the single more rotation and the two more minutes later okay displacement so the displacement operator increases the measure increases the measured position by alpha real and the measured momentum by imaginary alpha the imaginary part of alpha 4 Alpha being a real a complex number sorry it's action and the position quadrature is given by these and the multimeter is given by this in the matrix form taking these in place in the event matrix is one this placement of products as follows a real valued vector of the measured position elemental okay so yeah displacement so it adds or the real component of your complex number to the position and the imaginary in the complex component to the moment but I I know how this is increasing the uncertainty in one place and decreasing it in the other mmm this quizzical prayer another was the squeezing operator sorry the displacement is actually minimizes that minimizes the uncertainty okay at least that's what I miss do it here it's cooking citizens sink that plane misplaced my operator I was written what was written or that was just a coherent finish instead that's okay I said with minimal uncertainty equally dispersed among the quadratures you can create it by applying displacement to a vacuum okay the squeezing operator squeezing operator with squeezing parameter re this is the uncertainty United position or momentum all below the standard quantum limit while increasing density in the conjugate variable its actions rather than by these okay do this in a matrix form the squeezing operator acts as follows extreme ah yeah Valley factor yeah okay so that's basically squeezed reduces uncertainty in position or momentum and increases the in certainty in the other the rotation operator the rotation operator rotates quadratures in phase space by given okay angle specified from 0 to pi/2 pi its action and the position quadrature is the following and in the moment of quadrature it is the following ok ok they seem to be kind of like new earth sinus and cosinus in here is cosine is minus sine is okay the beamsplitter the Kingsley operator is two more operator that requires two key modes to operate upon it may be understood as a rotation between the two key modes these actions on the position quadratures are as follows and now I understand why this is why they say the beam splitter is entangling them because basically the effect on on the funky mode one is then actually depending on the position and the momentum of the chemo - and vice versa in the position of interesting of the position of of this of the position of your chemo - is down depending on the position amount of candy oh it's beautiful okay of make sense of the chemo one so without really understanding those things in concrete but you basically you see that the now they become dependents they become entangled that's why then people say that with the beam splitter you entangled two key modes in continuous variable computing which was be confusing by the time I think I was reading this and a similar thing happens to the momentum quadratures okay and then but the momentum is okay yeah this basically isn't that I think isn't this thing here the same that is that the rotation I see okay so it's kind of a conditional rotation it's funny because would you use to entangling what you used to entangle in in the discrete corner Konica pitting it's really also a condition a controlled operation right so you're controlling an X rotation on as that value not the other way is again yeah and so here you're controlling basically that's beautiful it's a rotation so that's like okay look at these so that's you're controlling a rotation on the position face rotation of the position as a function of that thing here which is then interesting there's a bit of an analogy in there okay in matrix form the beam splitter acts I can't really explain it into a technical technical details really but if there's really an analogy there that's interesting that's interesting so matrix form got it non-gaussian state operators will not have much to say about non Gaussian States for abundant I mentioned to non-gaussian operators and technical challenges on the implementation non-gaussian states states are simply where the wiener function does not result in a gaussian function we created when a cosmic rays are act upon Gaussian States the cousin operators they do not return Gus in statement given to us and state to cooperate and the cubic phase operator okay interesting nothing we're almost done with the Ignite measurements then non-gaussian appears our wants it to not return a constant state the cubic phase operator V whatever can be understood as the result of the following process prepared to key modes in the vacuum state apply a to mode squeeze of two mode squeeze operator on both commands it's the two most key squeeze of Prater something like that does not mean spirits in the other on both key modes then apply a large momentum displacement on the first chemo lastly perform a photon counting measurement on the first key mode this results in the second chemo behaving as if the cubic phase operator was applied to it and its state may be called a cubic same state which is non Gaussian state the action on the position quadrature it's given by action and the momentum cartridge is given by so it just affects the momentum that's basically what this is saying because this is is it seems like so you apply to creamer squeeze operator on both kilos then apply a large moment of displacement on the first key mode possible from for encounter measurement not the first key mode counting so you're counting if you're counting that's on the brand what kind of a single key mode okay the care operator be defined in terms of quadratures and it's called MK clear operator what the hell is this this definition is not even useful in terms of quadratures and it's called a number and it's called a number operator well then press we then proceeded to find the corporator with reference to the number of per area with proper okay do whatever that means okay okay I got a little more deep into this I guess but I can do this as I work through the actual the actual architecture in the actual coding houses but that basically gives me I think a rough idea and then measurements so that so it's just a bunch of weird operators in here and the measurements so you can do different measurements for incanting so you can't I guess the amount of photons and then you get like every goal and then that's real a homodyne detection now that inference are captured by it for the detector which converts fallens into electrons ever since intellect turns into electric currents right occurrence generally proportional to the number of photons we wish to attack the quadrature or the key mode under consideration to compress this the cumulus combined with the 56 the beam splitter then to output key modes of the beam splitter are the ones that are converted to the for occurrence and their difference is measured the quadrature of choice is measured by introducing effects and the beam splitter which is there to select the quadrature Hamelin detection is a projected cost in measurement that projects cousin stays in the cousin states this is saying that this is something you can use them to measure either the position or a hammer-on detection allows you to then measure what's the position or what's the momentum I think so we wish to detect a quadrature or the key mode under consideration it doesn't I don't care right now it but and then okay I'm detection interaction is a measurement of both the X and P quadrature single things with the results in them this resulted in certainty in the measurement of quadratures that's what ha isodyne Joseph's cousin says okay that's it we made it another 30 minutes it's cool yeah I don't didn't have much time but it was more of a recap I have I really kind of want to be deeper into this and I think I even checked sort of the some of those documentation and stuff but basically in the next videos I'm really gonna go through through the actual architecture of the QIO I example their building because it's referenced in other papers and then I'll probably dig deeper as needed but I think I mostly remember so you've got momentum you've got position those two things are real numbers at the end of the day and then you've got different fixed different ways to measure you can measure foreign counting I'm not sure what is this useful for and then you can either choose to measure position or momentum and you can choose or post at the same time but this creates some kind of uncertainty I don't know what it really means in this year maybe I said you don't get exact measurements maybe that's you just get a probability distribution or something like that I don't know this is something to take him to as well but cool so that's basically there's a lot of an analogy to the quantum to the discrete quantum computing it's just that you're using if it's really capital to a specific implementation because that's literally we're talking about forints right so so you've got momentum and position and this quadrature and there's different gates and whatnot then but I was it was it was nice to see how how the beam splitter creates entanglement because it basically relates they basically rotate it rotates a cue mode as a function of something else from the other two from the other cue mode so saying so then the one value depends on the other one as well so this is how you're kind of entangling me that's that's the equivalent of applying a hotter margate right and then that harem or gate is sort of a rotation but then that harm our game that that's relating you you basically after that you apply a control not on another cubed and then you know the value of one becomes dependent on the value of the other so pretty cool in other intuitive level I thinks it's good it's a good summary let's see now how much farther we have to take down but I think the next we don't do well really go back to so remember there are two different papers that were referencing here one was a paper to encode the actual graph and another paper that was actually based on a corner on a neural network design for continuous variable quantum computing and I'm basically gonna I think on a ticking to the second one first but I remember that I'd start with the other one and I didn't finish it so I'm gonna try to start with the other one see what happens cool it's nice it's perfect |
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