Title: Quantum state transfer on graphs
Harvard University - Center of Mathematical Sciences and Applications
A quantum walk on a graph G describes the evolution of the quantum state of a particle on the graph, and is described by a discrete version of the Schrodinger equation involving a graph Hamiltonian on G. If u and v are two vertices of a graph, then we say that there is perfect state transfer from u to v if there is some time at which a particle starting at vertex u ends up at vertex v. Considerable research has been done in recent years on perfect state transfer in graphs, particularly in the case where the graph Hamiltonian is taken to be the adjacency matrix. In addition, one can include an energy potential on the vertex set, which amounts to adding a diagonal matrix to the Hamiltonian. I will present results showing how the potential can affect whether or not a graph admits perfect or approximately perfect state transfer. In particular, for paths of length greater than 4, there is no potential that can be chosen for which the path admits perfects state transfer. However, a potential can be chosen to achieve state transfer that is arbitrarily close to perfect. This applies more general to a large class of graphs with an involution.