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.
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机译：作为一维整数晶格Z上的离散时间量子走势模型，最近由王某和YE构建的量子漫游[蔡王和小娟，量子伯努利噪音，量子信息处理15（2016）， 1897-1908]表现出相当不同的功能。在本文中，我们将这漫步扩展到更高的尺寸情况。更确切地说，对于一般的正整数D≥2，通过使用量子伯努利噪声，我们在D维整数z Z D上引入了一个离散时间量子的型号，我们称之为D维QBN步行。 D维QBN步行与由王和叶构成的量子行走共享相同的硬币空间，尽管它是后者的更高尺寸延伸。此外，我们证明，对于其初始状态的一系列选择，D维QBN步行具有D维标准高斯型的极限概率分布，其与通常的高尺寸量子的情况鲜明对比度。还获得了其他一些结果。
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