  The finite dihedral group generated by one rotation and one flip 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 conduct the model of discrete-time quantum walk on the
Cayley graph of the dihedral group by special coding mode. This construction
makes Fourier transformation can be used to carry out spectral analysis of the
dihedral quantum walk, i.e. the non-abelian case. Furthermore, the relation
between quantum walk without memory on the Cayley graph of the dihedral group
and quantum walk with memory on a cycle is discussed, so that we can explore
the potential of quantum walks without and with memory. Here, the numerical
simulation is carried out to verify the theoretical analysis results and other
properties of the proposed model are further studied.
