Theory of overparametrization in quantum neural networks

The prospect of achieving quantum advantage with Quantum Neural Networks (QNNs) is exciting. Understanding how QNN properties affect the loss landscape is crucial to the design of scalable architectures. Here, we rigorously analyze the overparametrization phenomenon in QNNs with periodic structure. We define overparametrization as the regime where the QNN has more than a critical number of parameters Mc that allows it to explore all relevant directions in state space. Our main results show that the dimension of the Lie algebra obtained from the generators of the QNN is an upper bound for Mc, and for the maximal rank that the quantum Fisher information and Hessian matrices can reach. Underparametrized QNNs have spurious local minima that start disappearing when M> Mc. Thus, the overparametrization onset corresponds to a computational phase transition where the trainability is greatly improved by a more favorable landscape. We connect the notion of overparametrization to that of capacity, so that when a QNN is overparametrized its capacity achieves its maximum possible value. We run numerical simulations for eigensolver, compilation, and autoencoding applications to showcase the phase transition. Our results also apply to variational quantum algorithms and quantum optimal control.