Efficient measure for the expressivity of variational quantum algorithms

TL;DR

This paper introduces a new method using covering numbers to quantitatively measure the expressivity of variational quantum algorithms, accounting for system noise and guiding their design.

Contribution

It provides an upper bound on VQAs expressivity based on quantum gates and observables, and analyzes how noise impacts expressivity on near-term quantum devices.

Findings

Expressivity is bounded by the number of gates and observables.

Expressivity decays exponentially with circuit depth under noise.

Higher expressivity correlates with better generalization and VQE accuracy.

Abstract

The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigen-solvers (VQEs) heavily depends on the expressivity of the employed ansatze. Namely, a simple ansatze is insufficient to capture the optimal solution, while an intricate ansatze leads to the hardness of the trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. In particular, we first exhibit how the expressivity of VQAs with an arbitrary ansatze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our work opens the avenue for quantitative understanding of the expressivity of VQAs.