Perfect state transfer in Grover walks between states associated to vertices of a graph

TL;DR

This paper investigates conditions for perfect quantum state transfer in Grover walks on graphs, identifying specific graph classes and eigenvalue conditions that enable perfect transfer between vertex-associated states.

Contribution

It introduces a necessary eigenvalue condition for perfect state transfer in Grover walks and characterizes complete multipartite graphs where this transfer occurs.

Findings

Derived a necessary eigenvalue condition for perfect state transfer.

Identified complete multipartite graphs with equal partite set sizes supporting perfect transfer.

Determined the specific times at which perfect transfer occurs.

Abstract

We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, together with the time.