Perfect state transfer on quotient graphs

We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/π, under any equitable partition π, 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 kGk/πk is isomorphic to the quotient graph kGk/π, for some equitable partition π. 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.