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

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.

Automated summary

The paper focuses on the three-state discrete-time quantum walk on the Caylay graph of the dihedral group, deriving analytic expressions for position probability distribution and long-time limit of the return probability. It is found that the localization effect is determined by the size of the underlying group, coin operator, and initial state. Numerical investigations are conducted on the model's properties through probability distribution and time-averaged probability at a specific position, showcasing the rich phenomena of three-state Grover DTQW on the Caylay graph of the dihedral group to aid in understanding and developing new quantum algorithms within the community.