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Quantum Information Processing
Published in: Quantum Information Processing 3/2021

01-03-2021

Three-state quantum walk on the Cayley Graph of the Dihedral Group

Authors: Ying Liu, Jia-bin Yuan, Wen-jing Dai, Dan Li

Published in: Quantum Information Processing | Issue 3/2021

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Abstract

The finite dihedral group generated by one rotation and one reflection is the simplest case of the non-abelian group. Cayley graphs are diagrammatic counterparts of groups. In this paper, much attention is given to the Cayley graph of the dihedral group. Considering the characteristics of the elements in the dihedral group, we propose a model of three-state discrete-time quantum walk (DTQW) on the Caylay graph of the dihedral group with Grover coin. We derive analytic expressions for the position probability distribution and the long-time limit of the return probability starting from the origin. It is shown that the localization effect is governed by the size of the underlying dihedral group, coin operator and initial state. We also numerically investigate the properties of the proposed model via the probability distribution and the time-averaged probability at the designated position. The abundant phenomena of three-state Grover DTQW on the Caylay graph of the dihedral group can help the community to better understand and to develop new quantum algorithms.
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Metadata
Title
Three-state quantum walk on the Cayley Graph of the Dihedral Group
Authors
Ying Liu
Jia-bin Yuan
Wen-jing Dai
Dan Li
Publication date
01-03-2021
Publisher
Springer US
Published in
Quantum Information Processing / Issue 3/2021
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-021-03042-y

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