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 τ if the uv-entry of |H(τ)u,v|=1. State transfer has been applied to key distribution in commercial cryptosystems, and it seems likely that other applications will be found. We offer a survey of some of the work on perfect state transfer and related questions. The emphasis is almost entirely on the mathematics.

► We survey the interactions between graph theory and perfect state transfer. ► Perfect state transfer is of interest in quantum computing. ► We include some new results and open questions.