  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.
