Spectral Property of Magnetic Quantum Walk on Hypercube

TL;DR

This paper studies the spectral properties of a magnetic quantum walk on a hypercube, revealing that its spectrum remains unaffected by the magnetic potential, indicating spectral stability.

Contribution

It introduces a new model of magnetic quantum walk on hypercubes and analyzes its spectral characteristics, showing independence from magnetic potential.

Findings

Point-spectrum and approximate-spectrum are independent of magnetic potential.

Spectral stability of quantum walks under magnetic perturbations.

Representation of spectra in terms of coin operator system.

Abstract

In this paper, we introduce and investigate a model of magnetic quantum walk on a general hypercube. We first construct a set of unitary involutions associated with a magnetic potential $\nu$ by using quantum Bernoulli noises. And then, with these unitary involutions as the magnetic shift operators, we define the evolution operator $\mathsf{W}^{(\nu)}$ for the model, where $\nu$ is the magnetic potential. We examine the point-spectrum and approximate-spectrum of the evolution operator $\mathsf{W}^{(\nu)}$ and obtain their representations in terms of the coin operator system of the model. We show that the point-spectrum and approximate-spectrum of $\mathsf{W}^{(\nu)}$ are completely independent of the magnetic potential $\nu$ although $\mathsf{W}^{(\nu)}$ itself is dependent of the magnetic potential $\nu$. Our work might suggest that a quantum walk perturbed by a magnetic field can have spectral stability with respect to the magnetic potential.