Quantum approximate optimization with Gaussian boson sampling

Hard optimization problems are often approached by finding approximate solutions. Here, we highlight the concept of proportional sampling and discuss how it can be used to improve the performance of stochastic algorithms for optimization. We introduce an NP-Hard problem called Max-Haf and show that Gaussian boson sampling (GBS) can be used to enhance any stochastic algorithm for this problem. These results are applied by enhancing the random search, simulated annealing, and greedy algorithms. With numerical simulations, we confirm that all algorithms are improved when employing GBS, and that GBS-enhanced random search performs the best despite being the one with the simplest underlying classical routine.