Spectral Property of Magnetic Quantum Walk on Hypercube
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913655391518720 |
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| author | Wang, Ce |
| author_facet | Wang, Ce |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Property of Magnetic Quantum Walk on Hypercube Wang, Ce Quantum Physics Mathematical Physics Functional Analysis Probability Spectral Theory 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. |
| title | Spectral Property of Magnetic Quantum Walk on Hypercube |
| topic | Quantum Physics Mathematical Physics Functional Analysis Probability Spectral Theory |
| url | https://arxiv.org/abs/2412.18401 |