• Gaussian processes
• quantum computing
• quantum machine learning
• quantum neural networks

The goal of this thesis is to characterize the functions generated by quantum neural networks (QNNs) in the limit of infinite width of the network, i.e., of infinitely many qubits. The first result states that the probability distribution of the function generated by a QNN with randomly initialized parameters converges to a Gaussian process when each output qubit is correlated only with few other qubits. The second result analytically characterizes the training of a QNN via gradient descent. We prove that 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, generalizing to the quantum setting a recent breakthrough in classical machine learning. Therefore, a quantum advantage can be achieved when the mean and the covariance of such Gaussian process are hard to compute classically. Finally, to consider the real physical setting of the training, we have to take into account that the output of a QNN is the expectation value of a quantum observable, which can only be estimated via measurements. The last result of the thesis states that for a sufficiently large number of measurements, which is polynomial in the number of qubits, all the previous results are still valid. This implies the trainability of the QNN in polynomial time with respect to the number of qubits, a necessary condition for quantum speedups.