Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Wang Ce School of Mathematics and Statistics, Huazhong University of Science and Technology Wang Caishi School of Mathematics and Statistics, Northwest Normal University

arXiv subject: Mathematical Physics (math-ph) arXiv subject: Probability (math.PR) mathscidoc:2205.54001

Entropy, 2020.4
As a discrete-time quantum walk model on the one-dimensional integer lattice Z, the quantum walk recently constructed byWang and Ye [CaishiWang 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.
quantum Bernoulli noises; quantum walk; quantum white noises; quantum probability
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  title={Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises},
  author={Wang Ce, and Wang Caishi},
Wang Ce, and Wang Caishi. Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises. 2020. In Entropy. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517120856828467183.
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