text: 382,121,1512458719925264386,1362475407874867200,Max Hunter Gordon,"New paper out today! <LINK> We show how to prepare the covariance matrix on a quantum computer to do quantum principal component analysis: use an ensemble average density matrix! Thanks to @MvsCerezo , @LCincio and @ColesQuantum for a great collaboration! 1/n Principal component analysis (PCA) is a common technique in data analysis and machine learning. Our work provides a simple method to prepare the covariance matrix on a quantum computer, a key step in quantum PCA, using the ensemble average density matrix. 2/n <LINK> We find our method is equivalent to doing ‘PCA without centering’ which we interpret as doing PCA on a symmetrized dataset. We rigorously analyze the difference between these two approaches. 3/n <LINK> We argue that PCA is natural on quantum data (with complex amplitudes) and in this case there is no difference between PCA and PCA on symmetrized data! For classical datasets one can bound the deviation of the spectrum obtained with our method from that of standard PCA. 4/n <LINK> To explore the performance of our method on classical data we do PCA on the MNIST data set. We find that our method for constructing the covariance matrix to calculate the principal components gives practically equivalent results to conventional PCA. 5/n <LINK> We also look at how our approach performs on doing PCA of H2 and BeH2 ground states and make some pretty plots!  Our work paves the way for the implementation of quantum PCA on near-term quantum computers. 6/n <LINK>",https://arxiv.org/abs/2204.03495,"Principal component analysis (PCA) is a dimensionality reduction method in data analysis that involves diagonalizing the covariance matrix of the dataset. Recently, quantum algorithms have been formulated for PCA based on diagonalizing a density matrix. These algorithms assume that the covariance matrix can be encoded in a density matrix, but a concrete protocol for this encoding has been lacking. Our work aims to address this gap. Assuming amplitude encoding of the data, with the data given by the ensemble $\{p_i,| \psi_i \rangle\}$, then one can easily prepare the ensemble average density matrix $\overline{\rho} = \sum_i p_i |\psi_i\rangle \langle \psi_i |$. We first show that $\overline{\rho}$ is precisely the covariance matrix whenever the dataset is centered. For quantum datasets, we exploit global phase symmetry to argue that there always exists a centered dataset consistent with $\overline{\rho}$, and hence $\overline{\rho}$ can always be interpreted as a covariance matrix. This provides a simple means for preparing the covariance matrix for arbitrary quantum datasets or centered classical datasets. For uncentered classical datasets, our method is so-called ""PCA without centering"", which we interpret as PCA on a symmetrized dataset. We argue that this closely corresponds to standard PCA, and we derive equations and inequalities that bound the deviation of the spectrum obtained with our method from that of standard PCA. We numerically illustrate our method for the MNIST handwritten digit dataset. We also argue that PCA on quantum datasets is natural and meaningful, and we numerically implement our method for molecular ground-state datasets. ",Covariance matrix preparation for quantum principal component analysis,6,"['New paper out today!\n<LINK>\nWe show how to prepare the covariance matrix on a quantum computer to do quantum principal component analysis: use an ensemble average density matrix! \nThanks to @MvsCerezo , @LCincio  and  @ColesQuantum for a great collaboration! 1/n', 'Principal component analysis (PCA) is a common technique in data analysis and machine learning.\nOur work provides a simple method to prepare the covariance matrix on a quantum computer, a key step in quantum PCA, using the ensemble average density matrix. 2/n https://t.co/Iufr3Nwmv5', 'We find our method is equivalent to doing ‘PCA without centering’ which we interpret as doing PCA on a symmetrized dataset. We rigorously analyze the difference between these two approaches. 3/n https://t.co/KrNfnh0Ind', 'We argue that PCA is natural on quantum data (with complex amplitudes) and in this case there is no difference between PCA and PCA on symmetrized data! For classical datasets one can bound the deviation of the spectrum obtained with our method from that of standard PCA. 4/n https://t.co/ldiGfbeZA1', 'To explore the performance of our method on classical data we do PCA on the MNIST data set. We find that our method for constructing the covariance matrix to calculate the principal components gives practically equivalent results to conventional PCA. 5/n https://t.co/zfRpzaOtzs', 'We also look at how our approach performs on doing PCA of H2 and BeH2 ground states and make some pretty plots! \n\nOur work paves the way for the implementation of quantum PCA  on near-term quantum computers. 6/n https://t.co/jASuN1F0LV']",22,04,1489