Spectral Property of Magnetic Quantum Walk on Hypercube
Ce Wang
TL;DR
This work analyzes the spectral properties of a magnetic quantum walk on a hypercube, where a magnetic potential $\nu$ perturbs the walk through magnetic shift operators derived from quantum Bernoulli noises. The evolution operator $W^{(\nu)}$ is expressed in terms of a fixed coin operator system $\mathfrak{C}$ and unitary involutions $\Xi_j^{(\nu)}$, with a crucial relation $W^{(\nu)}(\widehat{Z}_{\sigma}^{(\nu)}\otimes u)=\widehat{Z}_{\sigma}^{(\nu)}\otimes(U_{\sigma}u)$ linking dynamics to $U_{\sigma}=\sum_j \mathcal{E}_{\sigma}(j)C_j$. The main contributions are the representations $\mathrm{Spec}^{(p)}(W^{(\nu)})=\bigcup_{\sigma} \mathrm{Spec}^{(p)}(U_{\sigma})$ and $\mathrm{Aev}(W^{(\nu)})=\bigcup_{\sigma} \mathrm{Aev}(U_{\sigma})$, and the result that these spectra are independent of the magnetic potential $\nu$, establishing spectral stability against magnetic perturbations. This highlights a robustness of the spectral properties under magnetic fields and connects the walk’s spectrum to the coin-operator structure, extending prior work on nu=0 and informing potential quantum-simulation applications.
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.