A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

Abstract

The conventional spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. This theorem gives most of the eigenvalues of the time evolution U by lifting the eigenvalues of an induced self-adjoint matrix T onto the unit circle on the complex plane. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs. Moreover, graphs we can consider for a quantum walk with such a shift operator is characterized by a triangulation. We call these graphs triangulable graphs in this paper. One of the differences between our spectral mapping theorem and the conventional one is that lifting the eigenvalues of T − 1/2 onto the unit circle gives most of the eigenvalues of U .

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

“…See Section 3 in [23] for more general claim and its proof. Relationship between U-spectra and T -spectra has been studied not only in the Grover walks but also in more general models [12,19,21]. We cite the result in [12], but the statement is slightly modified to fit our setting.…”