Perfect state transfer in graphs related to linear groups in two dimensions

TL;DR

This paper constructs specific graphs derived from linear groups over finite fields, demonstrating that quantum walks on these networks enable perfect state transfer, which is significant for quantum communication.

Contribution

It introduces new families of graphs based on linear groups that facilitate perfect quantum state transfer, expanding the understanding of quantum network design.

Findings

Graphs from $ ext{SL}(2,q)$, $ ext{GL}(2,q)$, and $ ext{GU}(2,q^2)$ support perfect state transfer.

Quantum walks on these graphs exhibit ideal transfer properties.

The construction applies to graphs with $q$ an odd prime power.

Abstract

We construct families of graphs from linear groups $\mathrm{SL}(2,q)$, $\mathrm{GL}(2,q)$ and $\mathrm{GU}(2,q^2)$, where $q$ is an odd prime power, with the property that the continuous-time quantum walks on the associated networks of qubits admit perfect state transfer.