Universality in perfect state transfer

A continuous-time quantum walk on a graph is a matrix valued function exp( iAt) over the reals, where A is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices u, v, there is a time where the (v, u) entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (2014) [3]. Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non prime-power order where some edges are missing. (C) 2017 Elsevier Inc. All rights reserved.