In this paper we study discrete-time quantum walks on Cayley graphs corresponding to Dihedral groups, which are graphs with both directed and undirected edges. We consider the walks with coins that are one-parameter continuous deformation of the Grover matrix and can be written as linear combinations of certain permutation matrices. We show that the walks are periodic only for coins that are permutation or negative of a permutation matrix. Finally, we investigate the localization property of the walks through numerical simulations and observe that the walks localize for a wide range of coins for different sizes of the graphs.

翻译：本文研究二面体群对应Cayley图上的离散时间量子游走，其中Cayley图包含有向边和无向边。我们考虑采用Grover矩阵的一参数连续变形作为硬币，这些硬币可表示为特定置换矩阵的线性组合。研究表明，仅当硬币为置换矩阵或其负矩阵时，游走才具有周期性。最后，通过数值模拟探究游走的局域化特性，发现对于不同规模的图，在较宽泛的硬币参数范围内游走均出现局域化现象。