Quantum Verification of Minimum Spanning Tree

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 $O^{\ast }(\sqrt{mn})$ 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 $O(n)$ calls to the weight oracle and $O(n+\sqrt{m}\log n)$ total work.