We present a quantum algorithm for sampling an edge on a path between two nodes s and t in an undirected graph given as an adjacency matrix, and show that this can be done in query complexity that is asymptotically the same, up to log factors, as the query complexity of detecting a path between s and t. We use this path sampling algorithm as a subroutine for st-path finding and st-cut-set finding algorithms in some specific cases. Our main technical contribution is an algorithm for generating a quantum state that is proportional to the positive witness vector of a span program.

翻译：我们提出一种量子算法，用于从无向图的邻接矩阵表示中，采样节点s与t之间路径上的某条边。研究表明，该算法的查询复杂度与检测s与t间是否存在路径的查询复杂度渐进等价（相差不超过对数因子）。我们将此路径采样算法作为子程序，应用于特定情况下的st路径查找与st割集查找问题。本文的主要技术贡献在于：提出一种生成与跨度程序正见证向量成正比的量子态的算法。