Perfect state transfer from quantum walk search algorithm

The perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices (sender and receiver) and show that starting the algorithm on the sender vertex the walk will reach the receiver vertex in $O(\sqrt{N})$ steps, where $N$ denotes the number of vertices of the graph. We focus on three models, namely the quantum walk on a star graph, complete graph with self-loops and Szegedy's walk with queries on a complete graph. While the walks on star graph and complete graph with self-loops allow for perfect state transfer for arbitrary number of vertices $N$, Szegedy's walk achieves state transfer with unit probability only in the limit of large $N$.