Universal state transfer on graphs

A continuous-time quantum walk on a graph G is given by the unitary matrix U(t)=exp(−itA), where A is the adjacency matrix of G. We say G has pretty good state transfer between vertices a and b if for any ϵ>0, there is a time t, where the (a,b)-entry of U(t) satisfies |U(t)a,b|≥1−ϵ. This notion was introduced by Godsil (2011). The state transfer is perfect if the above holds for ϵ=0. In this work, we study a natural extension of this notion called universal state transfer wherein state transfer exists between every pair of vertices of the graph. We prove the following results about graphs with this stronger property: (i) Graphs with universal state transfer have distinct eigenvalues and flat eigenbasis (each eigenvector has entries which are equal in magnitude). (ii) The switching automorphism group of a graph with universal state transfer is abelian and its order divides the size of the graph. Moreover, if the state transfer is perfect, the switching automorphism group is cyclic. (iii) There is a family of complex oriented prime-length cycles which has universal pretty good state transfer. This provides a concrete example of a family of graphs with this universal property. (iv) There exists a class of graphs with real symmetric adjacency matrices which has universal pretty good state transfer. In contrast, Kay (2011) proved that no graph with real-valued adjacency matrix can have universal perfect state transfer. Finally, we provide a spectral characterization of universal perfect state transfer for graphs switching equivalent to circulants.