Time-series models play an important role in the research of economics and finance.The QBN-based walk is a new quantum walk model recently introduced in terms of quantum Bernoulli noise.In this paper, we construct a time-series model based on the QBN-based walk and illustrate its basic properties.
As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.
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 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.
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.
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.
Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal time. This paper introduces a new type of fermionic open quantum random walk by formulating it with QBN on single-excitation subspaces. Within this framework, we establish a rigorous criterion for irreducibility, which requires the conjunction of strong graph connectivity, algebraic irreducibility of the transition operators and nonvanishing transition amplitudes. For finite irreducible walks, we prove convergence to a unique invariant state and establish strong convergence under the additional condition of aperiodicity. The theoretical framework is validated through a comprehensive two-node example, which illustrates both the convergence behavior and the intrinsic quantum features of the model.