This study investigates the unitary equivalence of split-step quantum walks (SSQW). We consider a new class of quantum walks which includes all SSQWs. We show the explicit form of quantum walks in this class, and clarify their unitary equivalence classes. Unitary equivalence classes of Suzuki's SSQW are also given.
This study investigate the unitary equivalence classes of quantum walks on cycles. We show that unitary equivalence classes of quantum walks on a cycle with N vertices are parameterized by 2N real parameters. Moreover, the ranges of two of the parameters are restricted, and the ranges depend on the parity of N.