State Transfer on Graphs

TL;DR

This paper surveys mathematical research on perfect state transfer in graphs, a quantum computing concept involving the operator exp(itA) and specific vertex transfer conditions.

Contribution

It provides a comprehensive overview of mathematical results related to perfect state transfer in graphs, highlighting key developments and open questions.

Findings

Summary of conditions for perfect state transfer

Identification of graph classes with perfect state transfer

Discussion of mathematical techniques used in the field

Abstract

If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the operator $\exp(itA)$. We say that we have perfect state transfer in $X$ from the vertex $u$ to the vertex $v$ at time $\tau$ if the $uv$-entry of $|H(\tau)_{u,v}|=1$. This concept has potential applications in quantum computing. We offer a survey of some of the work on perfect state transfer and related questions. The emphasis is almost entirely on the mathematics.