Perfect state transfer on quotient graphs

TL;DR

This paper investigates perfect quantum state transfer on quotient graphs, demonstrating new conditions and constructions that extend understanding of quantum walks and answer longstanding questions in the field.

Contribution

It proves that perfect state transfer on a graph is equivalent to its quotient under any equitable partition, and provides an algebraic description of a quantum walk construction by Feder.

Findings

Graphs with perfect state transfer can lack automorphisms swapping vertices.

The quotient of a Cartesian product of graphs is isomorphic to the product of their quotients.

The results answer a question posed by Godsil in 2011.

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 $G/\pi$, 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 $\Box_{k} G_{k}/\pi_{k}$ is isomorphic to the quotient graph $\Box_{k} G_{k}/\pi$, 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.