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arXiv:2211.07948 · Journal of Mathematical Physics

The uniform measure for quantum walk on hypercube: A quantum Bernoulli noises approach

Ce Wang

Published Nov 1, 2022• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0070451

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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.
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.
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.
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.
The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n)O(n\text{ }\text{log}\text{ }n)O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally nonuniform) distribution $\ensuremath{\pi}(x)$ in time O(n)O(n)O(n), and the stationary distribution is determined by the initial state of the walk. An explicit expression for $\ensuremath{\pi}(x)$ is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p\ensuremath{\approx}0.1$. A similar effect was previously reported for a quantum walk on an NNN-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps in obtaining---and preserving---a uniform distribution over the 2n{2}^{n}2n sites of the hypercube in the shortest possible time.
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.
Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.