Quantum walks in higher dimensions

Abstract

We analyze 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.

Decoherence in quantum walks – a review

“…Many of the basic results for quantum walks on the line and cycle, such as faster spreading, carry over to higher dimensional graphs. This was shown numerically for regular lattices of degree three and four in Mackay et al (2002). However, higher dimensional coins have a much wider set of possible types of coin operators, and further numerical study shows that the quantum speed up is not automatic for all possible choices.…”

“…Numerical studies are less constrained, due to the simplicity of the step-by-step evolution of quantum walks. Examples of quantum walks on various graphs of degree larger than two can be found in Mackay et al (2002), Tregenna et al (2003) and Carneiro et al (2005), and in the quantum walk search algorithm in Shenvi et al (2003) -see Section 4.1. Many of the basic results for quantum walks on the line and cycle, such as faster spreading, carry over to higher dimensional graphs.…”

“…First, the DFT (discrete Fourier transform) coin operator is unbiased, but asymmetric in that you cannot interchange the labels on the directions without changing the coin operator. First used in Mackay et al (2002), for d = 3 it looks like…”

“…Quantum walks using the DFT coin operator have interesting non-classical properties, which have been studied in Mackay et al (2002), Tregenna et al (2003), Carneiro et al (2005) and Krovi and Brun (2006a), but none of them have yet provided any quantum algorithms. The second commonly used coin operator is a highly symmetric coin based on Grover's diffusion operator (Grover 1996) with elements 2/d − δ ij , which in matrix form for d = 3 is…”

Continuous-Time Quantum Walks on the Symmetric Group

“…One of the topics that has recently received attention in the quantum computing community that highlights these differences is the the study of quantum computational variants of random walks, or quantum walks [1,3,5,6,8,9,11,15,18,19,23]. (A recent survey on quantum walks by Kempe [16] is an ideal starting point for background on quantum walks.)…”