The SAT problem is a prototypical NP-complete problem of fundamental importance in computational complexity theory with many applications in science and engineering; as such, it has long served as an essential benchmark for classical and quantum algorithms. This study shows numerical evidence for a quadratic speedup of the Grover Quantum Approximate Optimization Algorithm (G-QAOA) over random sampling for finding all solutions to 3-SAT (All-SAT) and Max-SAT problems. G-QAOA is less resource-intensive and more adaptable for these problems than Grover’s algorithm, and it surpasses conventional QAOA in its ability to sample all solutions. We show these benefits by classical simulations of many-round G-QAOA on thousands of random 3-SAT instances. We also observe G-QAOA advantages on the IonQ Aria quantum computer for small instances, finding that current hardware suffices to determine and sample all solutions. Interestingly, a single-angle-pair constraint that uses the same pair of angles at each G-QAOA round greatly reduces the classical computational overhead of optimizing the G-QAOA angles while preserving its quadratic speedup. We also find parameter clustering of the angles. The single-angle-pair protocol and parameter clustering significantly reduce obstacles to classical optimization of the G-QAOA angles.

中文翻译：

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用于 3-SAT 的 Grover-QAOA：二次加速、公平采样和参数聚类


SAT 问题是典型的 NP 完备问题，在计算复杂性理论中具有根本重要性，在科学和工程中有许多应用;因此，它长期以来一直是经典和量子算法的重要基准。这项研究显示了 Grover 量子近似优化算法 （G-QAOA） 在随机采样上二次加速的数值证据，用于查找 3-SAT （All-SAT） 和 Max-SAT 问题的所有解。与 Grover 的算法相比，G-QAOA 的资源密集度更低，对这些问题的适应性更强，并且在对所有解决方案进行采样的能力方面超过了传统的 QAOA。我们通过在数千个随机 3-SAT 实例上进行多轮 G-QAOA 的经典模拟来展示这些优势。我们还观察到 G-QAOA 在 IonQ Aria 量子计算机上对于小型实例的优势，发现当前的硬件足以确定和采样所有解决方案。有趣的是，在每一轮 G-QAOA 回合中使用相同对角的单角度对约束大大减少了优化 G-QAOA 角度的经典计算开销，同时保持了其二次加速。我们还发现了角度的参数聚类。单角度对协议和参数聚类显著减少了 G-QAOA 角度经典优化的障碍。