Finding structural anomalies in graphs by means of quantum walks

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 except one of the spokes have loops, 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.

Figure 1

Star graph with an extra edge between vertices 1 and 2.