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.