A Model of Discrete-Time Quantum Walk in Terms of Bernoulli Functionals

Caishi Wang, Xiaojuan Ye

Caishi Wang, Ce Wang, Suling Ren +1 more

Ce Wang, Caishi Wang

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.

Andrew M. Childs

Zhenzhen Yang, Caishi Wang

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.

Philippe Biane