Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks

TL;DR

This paper establishes a fundamental trade-off in deep quantum neural networks between the efficiency of gradient measurement and the network's expressivity, proposing a new ansatz that optimally balances this trade-off.

Contribution

It rigorously proves a trade-off between gradient measurement efficiency and expressivity in deep QNNs and introduces the stabilizer-logical product ansatz (SLPA) that achieves this bound.

Findings

SLPA reduces sample complexity for training

SLPA maintains accuracy and trainability

Trade-off guides QNN design

Abstract

Quantum neural networks (QNNs) require an efficient training algorithm to achieve practical quantum advantages. A promising approach is gradient-based optimization, where gradients are estimated by quantum measurements. However, QNNs currently lack general quantum algorithms for efficiently measuring gradients, which limits their scalability. To elucidate the fundamental limits and potentials of efficient gradient estimation, we rigorously prove a trade-off between gradient measurement efficiency (the mean number of simultaneously measurable gradient components) and expressivity in deep QNNs. This trade-off indicates that more expressive QNNs require higher measurement costs per parameter for gradient estimation, while reducing QNN expressivity to suit a given task can increase gradient measurement efficiency. We further propose a general QNN ansatz called the stabilizer-logical product ansatz (SLPA), which achieves the trade-off upper bound by exploiting the symmetric structure of the quantum circuit. Numerical experiments show that the SLPA drastically reduces the sample complexity needed for training while maintaining accuracy and trainability compared to well-designed circuits based on the parameter-shift method.