Theory of overparametrization in quantum neural networks

TL;DR

This paper provides a rigorous analysis of overparametrization in quantum neural networks, showing how increasing parameters beyond a critical point improves trainability and capacity, with implications for quantum algorithms.

Contribution

It establishes a theoretical framework linking overparametrization to Lie algebra dimension and Fisher information, revealing a phase transition in QNN trainability and capacity.

Findings

Overparametrization reduces local minima in the loss landscape.

A critical parameter number $M_c$ bounds QNN capacity and Fisher information.

Numerical simulations confirm the phase transition in various applications.

Abstract

The prospect of achieving quantum advantage with Quantum Neural Networks (QNNs) is exciting. Understanding how QNN properties (e.g., the number of parameters $M$) affect the loss landscape is crucial to the design of scalable QNN 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 $M_c$ 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 $M_c$, and for the maximal rank that the quantum Fisher information and Hessian matrices can reach. Underparametrized QNNs have spurious local minima in the loss landscape that start disappearing when $M\geq M_c$. Thus, the overparametrization onset corresponds to a computational phase transition where the QNN trainability is greatly improved by a more favorable landscape. We then connect the notion of overparametrization to the QNN 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 overparametrization computational phase transition. We note that our results also apply to variational quantum algorithms and quantum optimal control.