Localization of space-inhomogeneous three-state quantum walks

Abstract

Mathematical analysis on the existence of eigenvalues is essential because it is equivalent to the occurrence of localization, which is an exceptionally crucial property of quantum walks. We construct the method for eigenvalue problem via the transfer matrix for space-inhomogeneous n-state quantum walks in one dimension with n − 2 self-loops, which is an extension of the technique in a previous study (Quantum Inf. Process 20(5), 2021). By this method, we reveal the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state quantum walk with one defect whose time evolution varies in the negative part, positive part, and at the origin.

Localization in quantum walks with periodically arranged coin matrices

“…The localization phenomenon in the one-defect model has been used in quantum search algorithms [17,3,18], and the relationship between topological insulators and localization in the two-phase quantum walk has attracted much attention [19,20]. Also, the eigenvalue analysis of two-phase three-state quantum walks with one defect is investigated in [21], and the transfer matrix is also used in [22,23,24]. Furthermore, a previous study [15] showed that the method can be applied to a more general model, which satisfies the following conditions:…”