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

The finite dihedral group generated by one rotation and reflection is the simplest case of non-abelian group. Cayley graphs are diagrammatic counterparts groups. In this paper, much attention given to graph Considering characteristics elements in group, we propose a model three-state discrete-time quantum walk (DTQW) on Caylay with Grover coin. We derive analytic expressions for position probability distribution long-time limit return starting from origin. It shown that localization effect governed size underlying coin operator initial state. also numerically investigate properties proposed via time-averaged at designated position. abundant phenomena DTQW can help community better understand develop new algorithms.