Perfect state transfer in Grover walks on association schemes and distance-regular graphs

TL;DR

This paper characterizes when perfect quantum state transfer occurs in Grover walks on association schemes and distance-regular graphs, providing necessary and sufficient conditions and specific classifications.

Contribution

It establishes a comprehensive framework for perfect state transfer in Grover walks on association schemes and distance-regular graphs, including new characterizations and classifications.

Findings

Necessary and sufficient conditions for perfect state transfer

Complete classifications for Hamming and Johnson schemes

Characterizations for distance-regular graphs of diameter 2 and 3

Abstract

This paper investigates perfect state transfer in Grover walks, a model of discrete-time quantum walks. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on graphs belonging to an association scheme. Our focus includes specific association schemes, namely the Hamming and Johnson schemes. We characterize all graphs on the classes of Hamming and Johnson schemes that exhibit perfect state transfer. Furthermore, we study perfect state transfer on distance-regular graphs. We provide complete characterizations for exhibiting perfect state transfer on distance-regular graphs of diameter $2$ and diameter $3$, as well as integral distance-regular graphs.