State Transfer on Graphs

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 $\mathrm{\mid }H(\tau {)}_{u,v}\mathrm{\mid }=1$. 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.