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arXiv:1512.07433 · Physical Review A

Electric quantum walks in two dimensions

Luis Alberto Bru, Margarida Hinarejos, Fernando Silva, Germán J. de Valcárcel, Eugenio Roldán

Published Mar 22, 2016• 22 citations
arXivHTMLPDF

Abstract

We study electric quantum walks in two dimensions considering Grover, alternate, Hadamard, and discrete Fourier transform (DFT) quantum walks. In the Grover walk, the behavior under an electric field is easy to summarize: when the field direction coincides with the $x$ or $y$ axes, it produces a transient trapping of the probability distribution along the direction of the field, while when it is directed along the diagonals, a perfect two-dimensional (2D) trapping is frustrated. The analysis of the alternate walk helps one to understand the behavior of the Grover walk, as both walks are partially equivalent; in particular, it helps one to understand the role played by the existence of conical intersections in the dispersion relations, as we show that when these are removed, a perfect 2D trapping can occur for suitable directions of the field. We complete our study with the electric DFT and Hadamard walks in 2D, showing that the latter can exhibit perfect 2D trapping.

DOI
10.1103/physreva.93.032333

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We study electric quantum walks in two dimensions considering Grover, alternate, Hadamard, and discrete Fourier transform (DFT) quantum walks. In the Grover walk, the behavior under an electric field is easy to summarize: when the field direction coincides with the xxx or yyy axes, it produces a transient trapping of the probability distribution along the direction of the field, while when it is directed along the diagonals, a perfect two-dimensional (2D) trapping is frustrated. The analysis of the alternate walk helps one to understand the behavior of the Grover walk, as both walks are partially equivalent; in particular, it helps one to understand the role played by the existence of conical intersections in the dispersion relations, as we show that when these are removed, a perfect 2D trapping can occur for suitable directions of the field. We complete our study with the electric DFT and Hadamard walks in 2D, showing that the latter can exhibit perfect 2D trapping.
The discrete quantum walk in N dimensions is analyzed from the perspective of its dispersion relations. This allows understanding known properties, as well as designing new ones when spatially extended initial conditions are considered. This is done by deriving wave equations in the continuum, which are generically of the Schrödinger type, and allows devising interesting behavior, such as ballistic propagation without deformation, or the generation of almost flat probability distributions, which is corroborated numerically. There are however special points where the energy surfaces display intersections and, near them, the dynamics is entirely different. Applications to the two- and three-dimensional Grover walks are presented.
We have recently proposed a two-dimensional quantum walk where the requirement of a higher dimensionality of the coin space is substituted with the alternance of the directions in which the walker can move [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. For a particular initial state of the coin, this walk is able to perfectly reproduce the spatial probability distribution of the nonlocalized case of the Grover walk. Here, we present a more detailed proof of this equivalence. We also extend the analysis to other initial states in order to provide a more complete picture of our walk. We show that this scheme outperforms the Grover walk in the generation of xxx-yyy spatial entanglement for any initial condition, with the maximum entanglement obtained in the case of the particular aforementioned state. Finally, the equivalence is generalized to wider classes of quantum walks and a limit theorem for the alternate walk in this context is presented.
Quantum random walk in a two-dimensional lattice with randomly distributed traps is investigated. Distributions of quantum walkers are evaluated dynamically for the cases of Hadamard, Fourier, and Grover coins, and quantum to classical transition is examined as a function of the density of the traps. It is shown that traps act as a serious and additional source of quantum decoherence. Furthermore, nontrivial temporal dependence of the standard deviation of the probability distribution of the walker is found when the trapping imperfections are introduced.
Quantum Walk (QW) has very different transport properties to its classical counterpart due to interference effects. Here we study the discrete-time quantum walk (DTQW) with on-site static/dynamic phase disorder following either binary or uniform distribution in both one and two dimensions. For one dimension, we consider the Hadamard coin; for two dimensions, we consider either a 2-level Hadamard coin (Hadamard walk) or a 4-level Grover coin (Grover walk) for the rotation in coin-space. We study the transport properties e.g. inverse participation ratio (IPR) and the standard deviation of the density function (σ) as well as the coin-position entanglement entropy (EE), due to the two types of phase disorders and the two types of coins. Our numerical simulations show that the dimensionality, the type of coins, and whether the disorder is static or dynamic play a pivotal role and lead to interesting behaviors of the DTQW. The distribution of the phase disorder has very minor effects on the quantum walk.
In this paper, we consider discrete-time quantum walks with moving shift (MS) and flip-flop shift (FF) on two-dimensional lattice Z2\mathbb{Z}^2Z2 and torus πN2=(Z/N)2π_N^2=(\mathbb{Z}/N)^2πN2​=(Z/N)2. Weak limit theorems for the Grover walks on Z2\mathbb{Z}^2Z2 with MS and FF were given by Watabe et al. and Higuchi et al., respectively. The existence of localization of the Grover walks on Z2\mathbb{Z}^2Z2 with MS and FF was shown by Inui et al. and Higuchi et al., respectively. Non-existence of localization of the Fourier walk with MS on Z2\mathbb{Z}^2Z2 was proved by Komatsu and Tate. Here our simple argument gave non-existence of localization of the Fourier walk with both MS and FF. Moreover we calculate eigenvalues and the corresponding eigenvectors of the (k1,k2)(k_1,k_2)(k1​,k2​)-space of the Fourier walks on πN2π_N^2πN2​ with MS and FF for some special initial conditions. The probability distributions are also obtained. Finally, we compute amplitudes of the Grover and Fourier walks on π22π_2^2π22​.
In this paper we focus our attention on a particle that follows a unidirectional quantum walk, an alternative version of the currently widespread discrete-time quantum walk on a line. Here the walker at each time step can either remain in place or move in a fixed direction, e.g., rightward or upward. While both formulations are essentially equivalent, the present approach leads us to consider discrete Fourier transforms, which eventually results in obtaining explicit expressions for the wave functions in terms of finite sums and allows the use of efficient algorithms based on the fast Fourier transform. The wave functions here obtained govern the probability of finding the particle at any given location but determine as well the exit-time probability of the walker from a fixed interval, which is also analyzed.