Spectral Property of Magnetic Quantum Walk on Hypercube

• URL: http://arxiv.org/abs/2412.18401v2
• Date: Mon, 20 Jan 2025 00:56:17 GMT
• Title: Spectral Property of Magnetic Quantum Walk on Hypercube
• Authors: Ce Wang,
• Abstract summary: A quantum walk perturbed by a magnetic field can have spectral stability with respect to the magnetic potential. Our work might suggest that a quantum walk perturbed by a magnetic field can have spectral stability with respect to the magnetic potential.
• Score: 8.04586521562225
• License:
• 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.

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