Benchmarking Variational Quantum Algorithms for Combinatorial Optimization in Practice

TL;DR

This paper benchmarks variational quantum algorithms for Max-Cut problems, comparing their performance with classical methods under realistic resource constraints to assess their practical viability.

Contribution

It provides a numerical analysis of the resource scaling and performance of shallow variational quantum algorithms for combinatorial optimization, with practical benchmarking insights.

Findings

Quantum algorithms outperform sampling at certain problem sizes.

Performance separation between quantum and greedy algorithms increases with problem size.

Correlation analysis reveals performance variability across instances.

Abstract

Variational quantum algorithms and, in particular, variants of the varational quantum eigensolver have been proposed to address combinatorial optimization (CO) problems. Using only shallow ansatz circuits, these approaches are deemed suitable for current noisy intermediate-scale quantum hardware. However, the resources required for training shallow variational quantum circuits often scale superpolynomially in problem size. In this study we numerically investigate what this scaling result means in practice for solving CO problems using Max-Cut as a benchmark. For fixed resources, we compare the average performance of training a shallow variational quantum circuit, sampling with replacement, and a greedy algorithm starting from the same initial point as the quantum algorithm. We identify a minimum problem size for which the quantum algorithm can consistently outperform sampling and, for each problem size, characterize the separation between the quantum algorithm and the greedy algorithm. Furthermore, we extend the average case analysis by investigating the correlation between the performance of the algorithms by instance. Our results provide a step towards meaningful benchmarks of variational quantum algorithms for CO problems for a realistic set of resources.