Finding structural anomalies in graphs by means of quantum walks

TL;DR

This paper investigates using quantum walks to detect structural anomalies like extra edges or loops in star graphs, demonstrating that quantum walks can efficiently identify such anomalies in O(√N) steps.

Contribution

The study shows that quantum walks can be effectively used to find structural anomalies in graphs, specifically in star graphs, with improved efficiency.

Findings

Quantum walks find extra edges or loops in star graphs in O(√N) steps.

Adjusting phase reflection can improve detection of certain anomalies.

Quantum walks can be used to identify structural anomalies under specific conditions.

Abstract

We explore the possibility of using quantum walks on graphs to find structural anomalies, such as extra edges or loops, on a graph. We focus our attention on star graphs, whose edges are like spokes coming out of a central hub. If there are $N$ spokes, we show that a quantum walk can find an extra edge connecting two of the spokes or a spoke with a loop on it in $O(\sqrt{N})$ steps. We initially find that if all of the spokes have loops except one, the walk will not find the spoke without a loop, but this can be fixed if we choose the phase with which the particle is reflected from the vertex without the loop. Consequently, quantum walks can, under some circumstances, be used to find structural anomalies in graphs.