一种用于一般图中最大权完美匹配的量子缩放算法
A Quantum Scaling Algorithm for Maximum-Weight Perfect Matching in General Graphs
- University of California, Irvine（加州大学尔湾分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出首个在一般图最大权完美匹配问题上超越最佳经典组合算法的量子算法，运行时间为Õ(n m^{2/3} log W)，在稠密图中实现渐近加速。
AI中文摘要：
量子加速已被应用于许多基本图问题，包括匹配的大多数变体。然而，一个显著的例外是带整数边权的一般图中的最大权完美匹配（MWPM）问题，这可以说是最具挑战性的匹配变体。我们提出了一种用于一般图中MWPM的量子算法，其运行时间为 \\( \widetilde{O}(n m^{2/3}\log W) \\)，其中 \\( W \\) 是边权绝对值的上界。在稠密图情形下（即 \\( m\ge n^{3/2} \\)），这优于已知的最佳经典组合界 \\( \widetilde{O}(m\sqrt n \log W) \\)。据我们所知，这是首个在一般图MWPM问题上相较于最佳经典组合算法获得渐近改进的量子算法。我们方法的运行时间计入了QRAM初始化和访问开销（直至多对数因子），以及所有对数据结构的经典更新。在高层次上，我们的算法基于Duan、Pettie和Su的经典框架，但我们的算法需要用量子方法替换某些经典任务，对经典过程进行替代分析，并使用可在QRAM模型中有效实现的替代数据结构。
英文摘要：
Quantum speed-ups have been obtained for many fundamental graph problems, including most variants of matching. A notable exception, however, is the maximum-weight perfect matching (MWPM) problem in general graphs with integer edge weights, which is arguably the most challenging variant of matching. We present a quantum algorithm for MWPM in general graphs that runs in \( \widetilde{O}(n m^{2/3}\log W) \) time, where $W$ is an upper bound on the magnitude of the edge weights. This is an improvement over the best known classical combinatorial bound of \( \widetilde{O}(m\sqrt n \log W) \) in the dense regime, where $m\ge n^{3/2}$. To the best of our knowledge, this is the first quantum algorithm to obtain an asymptotic improvement over the best classical combinatorial algorithm for the MWPM problem in general graphs. The running time of our method accounts for QRAM initialization and access overheads up to polylogarithmic factors, as well as all classical updates to the data structures. At a high level, our algorithm is based on a classical framework due to Duan, Pettie, and Su, but our algorithm requires replacing certain classical tasks with quantum methods, alternative analysis of classical procedures, and the use of alternative data structures that can be implemented effectively in the QRAM model.
