We construct and analyze a message-passing algorithm for random constraint satisfaction problems (CSPs) at large clause density, generalizing work of El Alaoui, Montanari, and Sellke for Maximum Cut [arXiv:2111.06813] through a connection between random CSPs and mean-field Ising spin glasses. For CSPs with even predicates, the algorithm asymptotically solves a stochastic optimal control problem dual to an extended Parisi variational principle. This gives an optimal fraction of satisfied constraints among algorithms obstructed by the branching overlap gap property of Huang and Sellke [arXiv:2110.07847], notably including the Quantum Approximate Optimization Algorithm and all quantum circuits on a bounded-degree architecture of up to $\epsilon \cdot \log n$ depth.

翻译：我们构造并分析了一种用于随机约束满足问题（CSPs）在大子句密度下的消息传递算法，通过随机CSPs与平均场伊辛自旋玻璃之间的联系，推广了El Alaoui、Montanari和Sellke关于最大割问题的工作[arXiv:2111.06813]。对于具有偶数谓词的CSPs，该算法渐近地求解了一个与扩展的Parisi变分原理对偶的随机最优控制问题。这给出了在Huang和Sellke[arXiv:2110.07847]的分支重叠间隙性质所阻碍的算法中，能够满足约束的最优比例，其中特别包括量子近似优化算法以及所有在度有界架构上深度不超过$\epsilon \cdot \log n$的量子电路。