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.

12 pages

References in corpus (9)

• Quantum Mechanics helps in searching for a needle in a haystack
• Quantum random walks - an introductory overview
• Quantum walks: a comprehensive review
• Decoherence in Quantum Walks on the Hypercube
• Quantum walks in external gauge fields
• Two-dimensional quantum walk under artificial magnetic field
• Singular continuous Cantor spectrum for magnetic quantum walks
• The uniform measure for quantum walk on hypercube: a quantum Bernoulli noises approach
• Localization for magnetic quantum walks