Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins

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 (real) linear combinations of permutation matrices of order three. 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.