A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs
Simon Apers, Arinta Auza, Troy Lee
TL;DR
This work presents a quantum algorithm for the s-t minimum cut problem on dense undirected graphs, achieving sublinear query complexity by combining a quantum sparsification step with Grover-based edge learning to produce a contracted graph $G'$ that preserves all minimum s-t cuts. The algorithm follows the Rubinstein–Schramm–Weinberg framework in the cut-query model, but implements it in adjacency-list/adjacency-matrix models, yielding a total query bound of $\tilde{O}(\sqrt{m}\,n^{5/6}\,W^{1/3})$ (or $\tilde{O}(n^{11/6}W^{1/3})$ in the adjacency-matrix setting). The key ingredients are a quantum $\,\varepsilon$-cut sparsifier constructed in $\tilde{O}(\sqrt{mn}/\varepsilon)$ queries and a Grover-based procedure to learn the sparse contracted graph’s edges in $\tilde{O}(n\sqrt{m\varepsilon W})$ time, balanced by choosing $\varepsilon=(nW)^{-1/3}$. The result is a sublinear quantum-query method for exact s-t min-cut on dense graphs, with a matching classical lower bound showing that, in the randomized setting, linear queries are necessary for general connectivity questions, highlighting a meaningful quantum advantage in this domain.
Abstract
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 $m = Ω(n^2)$. In contrast, a randomized algorithm must make $Ω(m)$ queries to the adjacency list of a simple graph $G$ even to decide whether $s$ and $t$ are connected.