Quantum complexity of minimum cut

TL;DR

This paper characterizes the quantum query and time complexity of the minimum cut problem in undirected weighted graphs, providing algorithms and lower bounds that outperform classical approaches in various models.

Contribution

It introduces quantum algorithms for minimum cut with tight bounds and applies advanced techniques like graph sparsification and tree packing.

Findings

Quantum algorithms solve minimum cut with $ ilde O(n^{3/2}\sqrt{ au})$ queries.

Lower bounds match the quantum algorithm complexity, showing optimality.

Quantum complexity significantly improves over classical in certain models.

Abstract

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 $\tau$, we give a quantum algorithm to solve the minimum cut problem using $\tilde O(n^{3/2}\sqrt{\tau})$ queries and time. Moreover, for every integer $1 \le \tau \le n$ we give an example of a graph $G$ with edge weights $1$ and $\tau$ such that solving the minimum cut problem on $G$ requires $\Omega(n^{3/2}\sqrt{\tau})$ many queries to the adjacency matrix of $G$. These results contrast with the classical randomized case where $\Omega(n^2)$ 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 $\tilde \Theta(m)$. We show that the quantum query and time complexity are $\tilde O(\sqrt{mn\tau})$ and $\tilde O(\sqrt{mn\tau} + n^{3/2})$, respectively, where again the edge weights are between $1$ and $\tau$. For dense graphs we give lower bounds on the quantum query complexity of $\Omega(n^{3/2})$ for $\tau > 1$ and $\Omega(\tau n)$ for any $1 \leq \tau \leq n$. 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).