Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups

TL;DR

This paper investigates continuous-time quantum walks on Cayley graphs of extraspecial groups, identifying conditions for perfect state transfer and fractional revival, and proving the absence of instantaneous uniform mixing in these graphs.

Contribution

It provides the first detailed analysis of quantum walk phenomena on Cayley graphs of extraspecial groups, including conditions for perfect state transfer and fractional revival.

Findings

Conditions for perfect state transfer and fractional revival are established.

Graphs admitting these phenomena are constructed using partial spreads.

No normal Cayley graph of an extraspecial group admits instantaneous uniform mixing.

Abstract

We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival, and use partial spreads to construct graphs on extraspecial $2$-groups admitting these various phenomena. Lastly, we use a result of Ada Chan to show that there is no normal Cayley graph of an extraspecial group that admits instantaneous uniform mixing.