Despite much recent work, the true promise and limitations of the Quantum Alternating Operator Ansatz (QAOA) are unclear. A critical question regarding QAOA is to what extent its performance scales with the input size of the problem instance, in particular the necessary growth in the number of QAOA rounds to reach a high approximation ratio. We present numerical evidence for an exponential speed-up of QAOA over Grover-style unstructured search in finding approximate solutions to constrained optimization problems. Our result provides a strong hint that QAOA is able to exploit the structure of an optimization problem and thus overcome the lower bound for unstructured search. To this end, we conduct a comprehensive numerical study on several Hamming-weight constrained optimization problems for which we include combinations of all standardly studied mixer and phase separator Hamiltonians (Ring mixer, Clique mixer, Objective Value phase separator) as well as quantum minimum-finding inspired Hamiltonians (Grover mixer, Threshold-based phase separator). We identify Clique-Objective-QAOA with an exponential speed-up over Grover-Threshold-QAOA and tie the latter's scaling to that of unstructured search, with all other QAOA combinations coming in at a distant third. Our result suggests that maximizing QAOA performance requires a judicious choice of mixer and phase separator, and should trigger further research into other QAOA variations.

翻译：尽管近期研究众多，但量子交替算子拟设（QAOA）的真实前景与局限性仍不明确。关于QAOA的关键问题在于其性能如何随问题实例输入规模扩展，特别是为达到高近似比所需QAOA轮次数的必要增长。我们提供了数值证据，证明QAOA在寻找约束优化问题的近似解时，相较于格罗弗式非结构化搜索具有指数级加速。该结果有力提示QAOA能够利用优化问题的结构，从而突破非结构化搜索的下界。为此，我们对多个汉明权重约束优化问题进行了全面数值研究，涵盖所有标准研究的混合器与相分离器哈密顿量组合（环形混合器、团簇混合器、目标值相分离器）以及受量子最小值搜索启发的哈密顿量（格罗弗混合器、阈值基相分离器）。我们发现团簇-目标-QAOA较格罗弗-阈值-QAOA具有指数级加速，并将后者的扩展性与非结构化搜索相关联，而其他QAOA组合的表现则明显落后。该结果表明，最大化QAOA性能需要审慎选择混合器与相分离器，并应推动对QAOA其他变体的进一步研究。