import sQUlearn
from sQUlearn import Executor, FidelityKernel, ProjectedQuantumKernel

# Define a quantum circuit for optimization
circuit = sQUlearn.Circuit()

# Define a QML model using the sQUlearn library
model = sQUlearn.QMLModel(circuit)

# Define a fidelity kernel for optimization
kernel = FidelityKernel(model)

# Define a projected quantum kernel for optimization
projected_kernel = ProjectedQuantumKernel(model)

# Create an executor for executing QML tasks on real quantum computers or simulators
executor = Executor()

# Define a function for optimizing the QML model using the executor
def optimize_model():
    # Train the QNN on a real quantum backend to enhance result accuracy
    model.train(executor)

    # Optimize the parameters to effectively counteract systematic errors inherent in the real quantum hardware
    model.optimize_parameters(executor)

    # Evaluate the Gram matrix on real quantum computers or a simulator backend with automatic session handling
    gram_matrix = executor.evaluate_gram_matrix(model)

    return gram_matrix

# Use the optimized model for quantum AI optimization
def quantum_ai_optimization():
    # Load the pre-trained adapter model
    from adapters import AutoAdapterModel
    model = AutoAdapterModel.from_pretrained("undefined")
    model.load_adapter("DaddyAloha/Bot-2", set_active=True)

    # Integrate the sQUlearn library with the adapter model
    quantum_model = sQUlearn.QMLModel(model)

    # Optimize the quantum model using the executor
    optimized_gram_matrix = optimize_model()

    # Use the optimized quantum model for AI optimization
    #...