Quantum complexity of minimum cut

The minimum cut problem in an undirected and weighted graph $G$ is to find the minimum total weight of a set of edges whose removal disconnects $G$. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If $G$ has $n$ vertices and edge weights at least $1$ and at most $τ$, we give a quantum algorithm to solve the minimum cut problem using $\tilde O(n^{3/2}\sqrtτ)$ queries and time. Moreover, for every integer $1 \le τ\le n$ we give an example of a graph $G$ with edge weights $1$ and $τ$ such that solving the minimum cut problem on $G$ requires $Ω(n^{3/2}\sqrtτ)$ many queries to the adjacency matrix of $G$. These results contrast with the classical randomized case where $Ω(n^2)$ queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when $G$ has $m$ edges the classical randomized complexity of the minimum cut problem is $\tilde Θ(m)$. We show that the quantum query and time complexity are $\tilde O(\sqrt{mnτ})$ and $\tilde O(\sqrt{mnτ} + n^{3/2})$, respectively, where again the edge weights are between $1$ and $τ$. For dense graphs we give lower bounds on the quantum query complexity of $Ω(n^{3/2})$ for $τ> 1$ and $Ω(τn)$ for any $1 \leq τ\leq n$. Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger's tree packing technique (STOC 1996).