Assessing Quantum and Classical Approaches to Combinatorial Optimization: Testing Quadratic Speed-ups for Heuristic Algorithms

TL;DR

This paper critically examines the potential for quantum speed-ups in combinatorial optimization, finding current quantum heuristics do not outperform classical methods in tested scenarios, emphasizing the need for more rigorous benchmarking.

Contribution

It highlights the challenges in benchmarking quantum versus classical heuristics for combinatorial optimization and provides a detailed numerical analysis questioning the existence of quantum advantage.

Findings

Classical heuristics match quantum methods in tested cases.

Current quantum approaches do not show clear quadratic speed-up.

More rigorous numerical investigations are necessary.

Abstract

Many recent investigations conclude, based on asymptotic complexity analyses, that quantum computers could accelerate combinatorial optimization (CO) tasks relative to a purely classical computer. However, asymptotic analysis alone cannot support a credible claim of quantum advantage. Here, we highlight the challenges involved in benchmarking quantum and classical heuristics for combinatorial optimization (CO), with a focus on the Sherrington-Kirkpatrick problem. Whereas hope remains that a quadratic quantum advantage is possible,our numerical analysis casts doubt on the idea that current methods exhibit any quantum advantage at all. This doubt arises because even a simple classical approach can match with quantum methods we investigated. We conclude that more careful numerical investigations are needed to evaluate the potential for quantum advantage in CO, and we give some possible future directions for such investigations.