05C50 Online Seminar: Xiaohong Zhang

Let M be a Hermitian matrix associated to a graph X on n vertices. For any time t \geq 0, the transition matrix of the continuous quantum walk on X relative to M at time t is given by U(t)=exp(itM). For two vertices a and b of X, if there is some time t such that U(t)e_a=\gamma e_b for some scalar \gamma, then we say there is perfect state transfer from a to b at time t if a\neq b, and say the walk is periodic at vertex a at time t if a=b. In this talk, we will see how the field where the entries of M lie in influences state transfer properties. We will solve an interesting eigenvalue problem that arises from the study of quantum walks: which oriented Cayley graph has all its eigenvalues integer multiple of \sqrt{\Delta} for some square-free integer \Delta. This generalizes a result of Bridges and Mena on when a Cayley graph has only integer eigenvalues.

This talk will be recorded and the speaker's slides will be made available in the main website.