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arXiv · 1108.0339

Perfect state transfer on quotient graphs

R. Bachman·E. Fredette·J. Fuller·M. Landry·M. Opperman·C. Tamon·A. Tollefson
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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.

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R. Bachman, E. Fredette, J. Fuller, M. Landry, M. Opperman, C. Tamon, A. Tollefson. 2011-08-01. Perfect state transfer on quotient graphs. https://arxiv.org/abs/1108.0339

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