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arXiv · 2412.18401

Spectral Property of Magnetic Quantum Walk on Hypercube

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

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Ce Wang. 2025-01-20. Spectral Property of Magnetic Quantum Walk on Hypercube. https://doi.org/10.1063/5.0232056

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