A Query-Efficient Quantum Algorithm for Maximum Matching on General Graphs

TL;DR

This paper introduces a quantum algorithm for maximum matching in general graphs that improves query complexity over previous methods, utilizing classical algorithms combined with quantum guessing techniques.

Contribution

It presents the first quantum algorithms for maximum matching on general graphs with improved query complexities, extending bipartite graph results.

Findings

O(n^{7/4}) queries in the matrix model

O(n^{3/4}(m+n)^{1/2}) queries in the list model

Combines classical maximum matching algorithms with quantum guessing methods

Abstract

We design quantum algorithms for maximum matching. Working in the query model, in both adjacency matrix and adjacency list settings, we improve on the best known algorithms for general graphs, matching previously obtained results for bipartite graphs. In particular, for a graph with $n$ nodes and $m$ edges, our algorithm makes $O(n^{7/4})$ queries in the matrix model and $O(n^{3/4}(m+n)^{1/2})$ queries in the list model. Our approach combines Gabow's classical maximum matching algorithm [Gabow, Fundamenta Informaticae, '17] with the guessing tree method of Beigi and Taghavi [Beigi and Taghavi, Quantum, '20].