Pretty good state transfer in Grover walks on abelian Cayley graphs

TL;DR

This paper investigates pretty good state transfer in Grover walks on graphs, providing a complete characterization for abelian Cayley graphs using Chebyshev polynomials, and identifying families with PGST but not perfect transfer.

Contribution

It offers a necessary and sufficient condition for PGST on abelian Cayley graphs, advancing understanding of quantum state transfer in these structures.

Findings

Characterization of PGST on abelian Cayley graphs

Identification of infinite families with PGST but no perfect transfer

Use of Chebyshev polynomials for analysis

Abstract

In this paper, we study pretty good state transfer (PGST) in Grover walks on graphs. We consider transfer of quantum states that are localized at the vertices of a graph and we use Chebyshev polynomials to analyze PGST between such states. In general, we find a necessary and sufficient condition for the occurrence of PGST on graphs. We then focus our analysis on abelian Cayley graphs and derive a necessary and sufficient condition for the occurrence of PGST on such graphs. Consequently, we obtain a complete characterization of PGST on unitary Cayley graphs. Our results yield infinite families of graphs that exhibit PGST but fail to exhibit perfect state transfer.