Perfect state transfer by means of discrete-time quantum walk search algorithms on highly symmetric graphs

TL;DR

This paper demonstrates that discrete-time quantum walk algorithms can achieve perfect state transfer between two vertices on highly symmetric graphs, such as star and complete graphs, with high efficiency and fidelity.

Contribution

It provides explicit calculations showing perfect state transfer using coined quantum walks on star and complete graphs, and extends results to Szegedy's walk with queries.

Findings

Perfect state transfer on star and complete graphs with $O(\sqrt{N})$ steps.

Achieves unit fidelity transfer with Szegedy's walk as $N$ grows large.

Demonstrates efficiency and scalability of quantum walk-based state transfer.

Abstract

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. It is shown by explicit calculation that for the coined quantum walks on star graph and complete graph with self-loops perfect state transfer between the sender and receiver vertex is achieved for arbitrary number of vertices $N$ in $O(\sqrt{N})$ steps of the walk. Finally, we show that Szegedy's walk with queries on complete graph allows for state transfer with unit fidelity in the limit of large $N$.