Spectral Property of Magnetic Quantum Walk on Hypercube

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.