Seminari del Dipartimento di Matematica

Seminario del 2024

Filippo Girardi

Trained quantum neural networks are Gaussian processes

fisica matematica

interdisciplinare

probabilità

We study quantum neural networks trained on supervised learning problems in the limit of infinite width. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges to a Gaussian process whenever each output qubit is correlated only with few other output qubits. Then, we analytically characterize the training of the network via gradient descent. We prove that, whenever the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training is still a Gaussian process. These results generalize to the quantum setting a recent breakthrough in classical machine learning. Finally, we consider the statistical noise of the measurement at the output of the network and prove that a polynomial number of measurements is sufficient for all the previous results to hold and that the network can always be trained in polynomial time.