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.
Open quantum walks (also known as open quantum random walks) are quantum analogs
of classical Markov chains in probability theory, and have potential application in quan-
tum information and quantum computation. Quantum Bernoulli noises are annihilation
and creation operators acting on Bernoulli functionals, and can be used as the environ-
ment of an open quantum system. In this paper, by using quantum Bernoulli noises as
the environment, we introduce an open quantum walk on a general higher-dimensional
integer lattice. We obtain a quantum channel representation of the walk, which shows
that the walk is indeed an open quantum walk. 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 quantum Bernoulli noises.