Unitary equivalence classes of split-step quantum walks

In this study, we investigate the unitary equivalence of split-step quantum walks (SSQWs). Introduced by Kitagawa and generalized by Suzuki, SSQWs are two-state discrete-time quantum walks on a one-dimensional lattice. In this work, we define a new class of quantum walks that includes all SSQWs by making use of the rank conditions, which state that (1) the rank of operators representing the walk from vertex x to vertex x +/- 1 is less than or equal to 1 and that (2) the rank from vertex x to x (i.e., to itself) is less than or equal to 2. Initially, we abstractly define quantum walks in this class, then clarify by presenting the explicit form of such quantum walks. We also calculate their unitary equivalence classes and show that four real parameters are required for each vertex to parameterize the unitary equivalence classes. Further, we consider the unitary equivalence of Suzuki's SSQWs and prove that all parameters of Suzuki's SSQWs can be real. Finally, we discuss the chiral symmetry of SSQWs.

Automated summary

By utilizing rank conditions and presenting explicit forms, a new class of quantum walks including all SSQWs is investigated, with calculations of their unitary equivalence classes and parameterization. The unitary equivalence of Suzuki's SSQWs is considered, showing that all parameters can be real, and chiral symmetry of SSQWs is discussed.