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International Journal of Quantum Information

Probabilistic stability of magnetic quantum walk

Zheng Yang, Caishi Wang, Lixia Zhang, Suling Ren

Published Feb 21, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749925500066

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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.
As a unitary quantum walk with infinitely many internal degrees of freedom, the quantum walk in terms of quantum Bernoulli noise (recently introduced by Wang and Ye) shows a rather classical asymptotic behavior, which is quite different from the case of the usual quantum walks with a finite number of internal degrees of freedom. In this paper, we further examine the structure of the walk. By using the Fourier transform on the state space of the walk, we obtain a formula that links the moments of the walk’s probability distributions directly with annihilation and creation operators on Bernoulli functionals. We also prove some other results on the structure of the walk. Finally, as an application of these results, we establish a quantum central limit theorem for the annihilation and creation operators themselves.
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.
In this paper, we introduce and study a model of discrete-time quantum walk based on quantum Bernoulli noises, which can be viewed as a perturbation of a model existing in the literature. We first construct a collection of self-adjoint unitary operators via quantum Bernoulli noises. Then, with these operators as the shift operators, we introduce a model of discrete-time quantum walk. We examine basic properties of the model via its shift operators and establish a formula for calculating its probability distributions. Finally, we investigate stationary measures of the model, and explicitly obtain a family of such measures.
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.
Let [Formula: see text] be a nonnegative integer and [Formula: see text] the power set of the set [Formula: see text]. Then there is an adjacency relation in [Formula: see text] such that [Formula: see text] together with the relation forms a regular graph. In this paper, we propose a model of continuous-time magnetic quantum walk (MQW) on the graph [Formula: see text], and investigate its properties from a viewpoint of probability and quantum information. We first introduce a magnetic Laplacian [Formula: see text] on the graph [Formula: see text] and examine its spectrum. And then, with [Formula: see text] as the Hamiltonian, we construct our model of continuous-time MQW on the graph [Formula: see text]. We find that the model has probability distributions that are completely independent of the magnetic potential at all times. And we show that it has perfect state transfer at time [Formula: see text] when the magnetic potential satisfies some mild conditions. Some other interesting results are also obtained.
In this paper, we consider limit probability distributions of the quantum walk recently introduced by Wang and Ye (C.S. Wang and X.J. Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Inf. Process. 15 (2016), no. 5, 1897-1908). We first establish several technical theorems, which themselves are also interesting. Then, by using these theorems, we prove that, for a wide range of choices of the initial state, the above-mentioned quantum walk has a limit probability distribution of standard Gauss type, which actually gives a new limit theorem for the walk.