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arXiv:2412.18401 · Journal of Mathematical Physics

Spectral property of magnetic quantum walk on hypercube

Ce Wang

Published Jan 1, 2025• 1 citation
arXivHTMLPDF

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 W(ν) for the model, where ν is the magnetic potential. We examine the point-spectrum and approximate-spectrum of the evolution operator 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 W(ν) are completely independent of the magnetic potential ν although 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.

DOI
10.1063/5.0232056

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A magnetic quantum walk is a quantum walk coupling to (or perturbed by) a magnetic field, which plays an active role in describing the motion of a particle in a magnetic field. In this paper, we introduce and investigate a discrete-time magnetic quantum walk in the framework of Bernoulli functionals. We first introduce magnetic shift operators in terms of the annihilation and creation operators on Bernoulli functionals, and then use them to construct the desired walk. After proving some technical results, we establish a formula for determining probability distributions of the walk. Finally, we show that the walk has probability distributions that are completely independent of the magnetic potential when its initial state meets some mild conditions.
We study disorder-free localization in a spin-coupled quantum walk on the hypercube within the quantum Bernoulli noises framework. We derive a general criterion for disorder-free localization: it occurs if and only if some spectral subspace of a reduced unitary evolution contains a vector whose position marginal probability is non-uniform. For the Grover walk with distance-dependent coupling φσ=∣σ∣π/(n+1)φ_σ=|σ|π/(n+1)φσ​=∣σ∣π/(n+1) in dimension d=4d=4d=4, we explicitly construct a non-uniform fixed point in the all +1+1+1 spin configuration. This implies that disorder-free localization occurs in our model.
In this paper, we present a quantum Bernoulli noises approach to quantum walks on hypercubes. We first obtain an alternative description of a general hypercube, and then, based on the alternative description, we find that the operators ∂k*+∂k behave actually as the shift operators, where ∂k and ∂k* are the annihilation and creation operators acting on Bernoulli functionals, respectively. With the above-mentioned operators as the shift operators on the position space, we introduce a discrete-time quantum walk model on a general hypercube and obtain an explicit formula for calculating its probability distribution at any time. We also establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution. Finally, we show that the walk produces the uniform measure as its stationary measure on the hypercube provided its initial state satisfies some mild conditions. Some other results are also proven.
We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators U(C)U(C)U(C) on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix C∈U(3)C\in U(3)C∈U(3), and study their spectral properties. For circulant unitary coin matrices CCC, we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for U(C)U(C)U(C). This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices CCC which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of U(C)U(C)U(C) is pure point.
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In this paper, we introduce a generalized continuous-time quantum walk (CTQW) on a hypercube, which can be interpreted as a CTQW with a “coin”. We examine its properties from a perspective of probability distribution and spectral theory, and obtain formulae for calculating its probability distributions and spectra, respectively. In particular, we prove that, for a wide range of initial states, the walk produces the uniform measure on the hypercube as its a stationary measure. Finally, we show that the time-reversal symmetry of the walk can be described directly by a unitary operator. Some other results are also proven.