An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$ to $t$. In this work we describe a quantum algorithm for the minimum $s{\operatorname{-}}t$ cut problem on undirected graphs. For an undirected graph with $n$ vertices, $m$ edges, and integral edge weights bounded by $W$, the algorithm computes with high probability the weight of a minimum $s{\operatorname{-}}t$ cut after $\widetilde O(\sqrt{m} n^{5/6} W^{1/3})$ queries to the adjacency list of $G$. For simple graphs this bound is always $\widetilde O(n^{11/6})$, even in the dense case when $m = \Omega(n^2)$. In contrast, a randomized algorithm must make $\Omega(m)$ queries to the adjacency list of a simple graph $G$ even to decide whether $s$ and $t$ are connected.

翻译：图中的一个$s{\operatorname{-}}t$最小割对应于移除后会使顶点$s$与$t$不连通的边的最小权重子集。寻找此类割是一个经典问题，且与寻找从$s$到$t$的最大流问题对偶。本文描述了一种针对无向图上最小$s{\operatorname{-}}t$割问题的量子算法。对于具有$n$个顶点、$m$条边且整数边权上界为$W$的无向图，该算法通过$\widetilde O(\sqrt{m} n^{5/6} W^{1/3})$次对$G$邻接表的查询，以高概率计算出最小$s{\operatorname{-}}t$割的权重。对于简单图，即使是在$m = \Omega(n^2)$的稠密情形下，该界始终为$\widetilde O(n^{11/6})$。相比之下，随机算法即便仅用于判定$s$与$t$是否连通，也必须对简单图$G$的邻接表进行$\Omega(m)$次查询。