Empirical performance bounds for quantum approximate optimization

TL;DR

This paper provides empirical performance bounds for QAOA on small MaxCut instances, revealing how approximation ratios improve with depth and identifying patterns for efficient heuristics, serving as a benchmark for quantum optimization.

Contribution

It offers the first comprehensive numerical analysis of QAOA performance on all small MaxCut instances, establishing empirical bounds and heuristics for quantum optimization.

Findings

QAOA exceeds classical approximation bounds on most small graphs.

Approximation ratio distributions narrow with increased QAOA depth.

Identified patterns in optimal parameters enable efficient heuristics.

Abstract

The quantum approximate optimization algorithm (QAOA) is a variational method for noisy, intermediate-scale quantum computers to solve combinatorial optimization problems. Quantifying performance bounds with respect to specific problem instances provides insight into when QAOA may be viable for solving real-world applications. Here, we solve every instance of MaxCut on non-isomorphic unweighted graphs with nine or fewer vertices by numerically simulating the pure-state dynamics of QAOA. Testing up to three layers of QAOA depth, we find that distributions of the approximation ratio narrow with increasing depth while the probability of recovering the maximum cut generally broadens. We find QAOA exceeds the Goemans-Williamson approximation ratio bound for most graphs. We also identify consistent patterns within the ensemble of optimized variational circuit parameters that offer highly efficient heuristics for solving MaxCut with QAOA. The resulting data set is presented as a benchmark for establishing empirical bounds on QAOA performance that may be used to test on-going experimental realizations.