paper

Perfect state transfer on quotient graphs

arXiv:1108.0339

Abstract

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 / I ¨ $ , under any equitable partition $I ¨ $ , 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 $□_{k} G_{k} / I ¨ _{k}$ is isomorphic to the quotient graph $□_{k} G_{k} / I ¨ $ , for some equitable partition $I ¨ $ . 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.

20 pages, 10 figures