Benchmarking variational quantum algorithms for combinatorial optimization in practice

Abstract Variational quantum algorithms, and in particular variants of the variational quantum eigensolver, have been proposed as approaches to combinatorial optimization (CO) problems. With only shallow ansatz circuits, these methods are considered suitable for current noisy intermediate-scale quantum hardware. Yet, the resources required to train variational circuits often scale superpolynomially with problem size. In this study, we numerically investigate the practical implications of this scaling for CO problems, using Max-Cut and random QUBO instances as benchmarks. For a fixed computational budget, we compare the average performance of training shallow variational circuits, sampling with replacement, and greedy local search. We identify the minimum problem size at which the quantum algorithms consistently outperform random sampling, and for each size, we characterize their separation from greedy local search. Beyond average-case performance, we analyze correlations between algorithms across individual problem instances. These results strengthen the case for intuitive yet meaningful benchmarks of variational quantum algorithms for CO problems under realistic resource constraints.