Quantum walks in higher dimensions

By T. D. Mackay, Stephen D. Bartlett, L. Stephenson, Barry C. Sanders. 2001

226 citations recorded by the source corpus.

We analyse the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the 'quantum coin toss' in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/eca09076ef1507ea76b84a880ea503ce12b06ada

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In Computational Theory and Mathematics

Citations in the corpus

Quantum walks: a comprehensive review (2012, background): “In [307], MacKay et al present numerical simulations of quantum walks in higher dimensions using separable and non-separable coin operators, Gottlieb et al [175] studied the convergence of coined quantum walks in R, and Dimcovic et al have put forward a general framework for describing discrete quan”
A new type of limit theorems for the one-dimensional quantum random walk (2002, background): “Very recently quantum random walks have been widely investigated by a number of groups in connection with the quantum computing, for exmples, [1, 3, 4, 5, 6, 8, 9, 13, 14].”
Universal quantum computation using the discrete-time quantum walk (2009, background): “rover coin, (4), operating on the incoming amplitude α. edges allow movement in both directions. A diﬀerent operator (coin) is needed at vertices with d>2 in order to act on the entire state space [25, 26]. The most symmetric of these is the Grover coin which can extend to any degree at a vertex,”
Quantum random walks: An introductory overview (2003, background): “…on the line the choice of the coin operator can have a serious impact on the shape of the resulting distribution, and this eﬀect is only ampliﬁed in higher dimensions (see [19] for examples of the eﬀects of various coins on a two-and three-dimensional grid) or on graphs of higher valence (degree).”