is an integral graph. Perfect state transfer Necessary conditions If a pair of vertices and on a graph admit perfect state transfer at time , then both and are periodic at time . Perfect state transfer on products of graphs Consider graphs and . If both and admit perfect state transfer at time , then their Cartesian product admits perfect state transfer at time . If either or admits perfect state transfer at time , then their disjoint union admits perfect state transfer at time . Perfect state transfer on walk-regular graphs If a walk-regular graph admits perfect state transfer, then all of its eigenvalues are integers. If is a graph in a homogeneous coherent algebra that admits perfect state transfer at time , such as e.g. a vertex-transitive graph or a graph in an association scheme, then all of the vertices on admit perfect state transfer at time . Moreover, a graph must have a perfect matching that admits perfect state transfer if it admits perfect state transfer between a pair of adjacent vertices and is a graph in a homogeneous coherent algebra. A regular edge-transitive graph cannot admit perfect state transfer between a pair of adjacent vertices, unless it is a disjoint union of copies of the complete graph . A strongly regular graph admits perfect state transfer if and only if it is the complement of the disjoint union of an even number of copies of . The only cubic distance-regular graph that admits perfect state transfer is the cubical graph. References External links Quantum Walk on arxiv.org CTQW on Wolfram Demonstrations Quantum walk Quantum mechanics