Trained quantum neural networks are Gaussian processes
Filippo Girardi Giacomo De Palma
We study quantum neural networks made by parametric one-qubit gates and fixed
two-qubit gates in the limit of infinite width, where the generated function is
the expectation value of the sum of single-qubit observables over all the
qubits. First, we prove that the probability distribution of the function
generated by the untrained network with randomly initialized parameters
converges in distribution to a Gaussian process whenever each measured qubit is
correlated only with few other measured qubits. Then, we analytically
characterize the training of the network via gradient descent with square loss
on supervised learning problems. We prove that, as long as 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 still converges in distribution to a Gaussian process. 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.