Hamiltonian Expressibility for Ansatz Selection in Variational Quantum Algorithms

TL;DR

This paper investigates how the expressibility of quantum circuits, measured by Hamiltonian expressibility, affects the performance of variational quantum algorithms in solving different types of Hamiltonian problems, especially under noise.

Contribution

It introduces a Monte Carlo-based method to estimate Hamiltonian expressibility and analyzes its impact on solution quality across various circuit depths and problem types.

Findings

High expressibility circuits perform better for non-diagonal Hamiltonians in ideal conditions.

Low expressibility circuits are more effective for basis state solutions, especially in noisy environments.

Intermediate expressibility circuits can outperform others for certain superposition solutions under noise.

Abstract

In the context of Variational Quantum Algorithms (VQAs), selecting an appropriate ansatz is crucial for efficient problem-solving. Hamiltonian expressibility has been introduced as a metric to quantify a circuit's ability to uniformly explore the energy landscape associated with a Hamiltonian ground state search problem. However, its influence on solution quality remains largely unexplored. In this work, we estimate the Hamiltonian expressibility of a well-defined set of circuits applied to various Hamiltonians using a Monte Carlo-based approach. We analyze how ansatz depth influences expressibility and identify the most and least expressive circuits across different problem types. We then train each ansatz using the Variational Quantum Eigensolver (VQE) and analyze the correlation between solution quality and expressibility.Our results indicate that, under ideal or low-noise conditions and particularly for small-scale problems, ans\"atze with high Hamiltonian expressibility yield better performance for problems with non-diagonal Hamiltonians and superposition-state solutions. Conversely, circuits with low expressibility are more effective for problems whose solutions are basis states, including those defined by diagonal Hamiltonians. Under noisy conditions, low-expressibility circuits remain preferable for basis-state problems, while intermediate expressibility yields better results for some problems involving superposition-state solutions.