Title: A Theory of Training Parameter-Shared Quantum Neural Networks from a Bayesian Perspective
Primary area: optimization
Keywords: Quantum Neural Network, Trainability, Bayesian Optimization, Parameter-Shared, Random Matrix Theory

Abstract: The objective function landscape of Quantum Neural Networks (QNNs) is both numerically and theoretically demonstrated to be highly non-convex, exhibiting numerous local optima. This raises an important question regarding the efficiency of training QNNs: can the optimization error systematically converge to a target threshold as the number of optimization iterations grows polynomially with the number of qubits $n$? In this work, we explore this question by proposing a theoretical framework from a Bayesian perspective. We focus on the trainability of Parameter-Shared QNNs (PS-QNNs), a widely used model for solving combinatorial optimization problems. Our first result shows that noise-free PS-QNNs with a depth of $\tilde{\mathcal{O}}\left(\sqrt{\log n}\right)$ can be trained efficiently. Furthermore, we demonstrate that if each quantum gate is influenced by a $q$-strength local Pauli channel, the noisy PS-QNN with a depth of $\mathcal{O}\left(\log n/\log(1/q)\right)$ can also be trained efficiently. These results provide valuable insights into the performance of QNNs, particularly in the context of the noisy intermediate-scale quantum era.

TL;DR: We rigorously provide the network depth at which parameter-shared quantum neural networks can be trained efficiently, resolving a long-standing open question.