Quantum Verification of Minimum Spanning Tree

TL;DR

This paper presents a quantum algorithm that efficiently verifies a minimum spanning tree in a weighted graph using significantly fewer oracle calls than previous methods, reducing verification complexity.

Contribution

It introduces a quantum verification method for minimum spanning trees that requires only O(n) oracle calls, improving over prior approaches.

Findings

Verification uses O(n) oracle calls

Total work is O(n + √m log n)

Significantly reduces verification complexity

Abstract

Previous studies has shown that for a weighted undirected graph having $n$ vertices and $m$ edges, a minimal weight spanning tree can be found with $O^*(\sqrt{mn})$ calls to the weight oracle. The present note shows that a given spanning tree can be verified to be a minimal weight spanning tree with only $O(n\bigr)$ calls to the weight oracle and $O(n+\sqrt{m}\log n)$ total work.