Abstract
As a discrete-time quantum walk model on the one-dimensional integer lattice , 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 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice , 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.
Highlights
As quantum analogs of classical random walks, quantum walks [1] have found wide application in quantum information, quantum computing and many other fields [2,3]
Quantum Bernoulli noises refer to the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal time, and can provide an approach to the effects of environment on an open quantum system [7,8]
This just means that l 2 (Zd ) describes the position of the d-dimensional QBN walk, while H describes its internal degrees of freedom
Summary
As quantum analogs of classical random walks, quantum walks [1] have found wide application in quantum information, quantum computing and many other fields [2,3]. In 2016, by using quantum Bernoulli noises, Wang and Ye [9] introduced a discrete-time quantum walk model on the one-dimensional integer lattice Z, which we call the one-dimensional QBN walk below. Belonging to the category of unitary quantum walks, the one-dimensional QBN walk, exhibits quite different features It takes the space H of square integrable Bernoulli functionals as its coin space, has infinitely many internal degrees of freedom since H is infinite-dimensional. By taking the operators Cn , ε ∈ {−1, +1}d , n ≥ 0 as coin operators, we establish a model of discrete-time quantum walk on Zd , which we call the d-dimensional QBN walk Of this walk, we obtain a unitary representation in the function space l 2 Zd , H and a characterization in the tensor space l 2 Zd ⊗ H.
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