倾斜游走实现组合优化的超二次量子加速
Super-Quadratic Quantum Speedups for Combinatorial Optimization via Tilted Walks
- Global Technology Applied Research, JPMorganChase（摩根大通全球应用技术研究）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出量子倾斜游走框架，通过偏置判别矩阵实现精确组合优化，在相同假设下对条件化与搜索算法取得超二次加速，且无需基态制备，并在合成问题上验证优势。
中文摘要 AI 辅助
我们引入了量子倾斜游走，这是一个用于解决精确组合优化问题的量子算法框架。该框架应用一个倾斜哈密顿量的幂的平均值，该哈密顿量用目标函数对基础马尔可夫链（混合器）的判别矩阵进行偏置。我们的出发点是量子短路径算法，该算法制备此类哈密顿量的基态，并对某些组合优化问题获得相对于穷举搜索的超二次加速。最近，Le Gall 和 Tamaki（arXiv:2604.12131）开发了一种用于加权 MAX-E$k$-LIN2 和加权 MAX-$k$-CSP 的经典条件化与搜索算法。在相同假设下，他们的算法仅比量子短路径算法慢次二次级别。因此，现有的短路径算法并未相对于这一更强的经典基线建立超二次加速。对于最大化问题，我们给出了倾斜游走从具有较低目标值的起始状态初始化时，增加具有高目标值的目标态振幅的条件。该框架涵盖了条件化与搜索，并为相同问题实现了对其的超二次加速。虽然我们的框架将量子短路径算法作为特例恢复，但它既不需要基态制备，也不需要在基础混合器的基态中初始化。我们在一个合成优化问题上展示了这些优势，在该问题上，倾斜游走实现了超二次加速，而短路径算法则不能。
英文摘要
We introduce quantum tilted walks, a quantum algorithmic framework for solving exact combinatorial optimization problems. The framework applies an average of powers of a tilted Hamiltonian that biases the discriminant matrix of a base Markov chain (mixer) with the objective function. Our starting point is quantum short-path algorithms, which prepare the ground state of such a Hamiltonian and obtain super-quadratic speedups over exhaustive search for certain combinatorial optimization problems. Recently, Le Gall and Tamaki~(arXiv:2604.12131) developed a classical conditioning-and-search algorithm for weighted MAX-E$k$-LIN2 and weighted MAX-$k$-CSP. Under the same assumptions, their algorithm is only sub-quadratically slower than quantum short-path algorithms. Consequently, existing short-path algorithms do not establish a super-quadratic speedup over this stronger classical baseline. For maximization problems, we give conditions under which tilted walks increase the amplitude on the target state with high objective value when initialized from a starting state with lower objective value. This framework captures conditioning-and-search and yields super-quadratic speedups over it for the same problems. While our framework recovers quantum short-path algorithms as a special case, it neither requires ground-state preparation nor initialization in the ground state of the base mixer. We demonstrate these advantages on a synthetic optimization problem for which tilted walks achieve a super-quadratic speedup whereas the short-path algorithms do not.
