  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 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.
