PERFECT STATE TRANSFER ON QUOTIENT GRAPHS
We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph C has perfect state transfer if and only if its quotient G/pi, under any equitable partition pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs a square(k)G(k)/pi(k) is isomorphic to the quotient, graph square(k)G(k)/pi, for some equitable partition pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.
perfect state transfer quotient graphs quantum walk equitable partitions