Pretty good state transfer on Cayley graphs over dihedral groups

The transition matrix of a graph Γ with the adjacency matrix A is defined by H ( t ) ≔ exp ( − i t A ) , where t ∈ R and i = − 1 . The graph is said to admit a pretty good state transfer between a pair of vertices u and v if for any e > 0 , there is a time t such that | e v t H ( t ) e u | ≥ 1 − e . The state transfer is perfect if the above inequality holds for e = 0 . Perfect (pretty good) state transfer on graphs has received extensive attention recently due to their significant applications in quantum information processing and quantum computations. In this paper, we study pretty good state transfer on Cayley graphs over dihedral groups. We find that if n is a power of 2, then Cay ( D n , S ) exhibits pretty good state transfer for some subset S in D n , some concrete constructions are provided. We also show that this is basically the only case for a non-integral Cayley graph Cay ( D n , S ) to have PGST.