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

TL;DR

This paper introduces a novel spectral mapping theorem for quantum walks with a shift operator of order three, applicable to triangulable graphs, expanding the understanding of eigenvalue relations in quantum walk dynamics.

Contribution

It presents a new spectral mapping theorem for quantum walks with a shift operator whose cube is the identity, specifically for triangulable graphs, extending previous theorems limited to involutive shift operators.

Findings

Derived a spectral mapping theorem for quantum walks with shift operator of order three.

Characterized triangulable graphs suitable for this quantum walk model.

Showed that eigenvalues of T-1/2 map onto the unit circle to determine eigenvalues of U.

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 $U$.