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arXiv:1212.3600 · New Journal of Physics

Understanding and controlling N -dimensional quantum walks via dispersion relations: application to the two-dimensional and three-dimensional Grover walks—diabolical points and more

Margarida Hinarejos, Armando Pérez, Eugenio Roldán, A. Romanelli, Germán J. de Valcárcel

Published Jul 23, 2013• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/15/7/073041

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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 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.
We suggest an alternative definition of NNN-dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This NNN-dimensional alternate quantum walk, AQW(N){\mathrm{AQW}}_{(N)}AQW(N)​, in contrast with the standard definition of the NNN-dimensional quantum walk, QW(N){\mathrm{QW}}_{(N)}QW(N)​, requires only a coin qubit. We discuss the quantum diffusion properties of AQW(2){\mathrm{AQW}}_{(2)}AQW(2)​ and AQW(3){\mathrm{AQW}}_{(3)}AQW(3)​ by analyzing their dispersion relations that reveal, in particular, the existence of diabolical points. This allows us to highlight interesting similarities with other well-known physical phenomena. We also demonstrate that AQW(3){\mathrm{AQW}}_{(3)}AQW(3)​ generates considerable genuine multipartite entanglement. Finally, we discuss the implementability of AQW(N){\mathrm{AQW}}_{(N)}AQW(N)​.
The Grover walk, which is related to Grover's search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.
We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.
We study the evolution of initially extended distributions in the coined quantum walk (QW) on the line.By analysing the dispersion relation of the process, continuous wave equations are derived whose form depends on the initial distribution shape.In particular, for a class of initial conditions, the evolution is dictated by the Schrödinger equation of a free particle.As that equation also governs paraxial optical diffraction, all of the phenomenology of the latter can be implemented in the QW.This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity.
This manuscript gathers and subsumes a long series of works on using QW to simulate transport phenomena. Quantum Walks (QWs) consist of single and isolated quantum systems, evolving in discrete or continuous time steps according to a causal, shift-invariant unitary evolution in discrete space. We start reminding some necessary fundamentals of linear algebra, including the definitions of Hilbert space, tensor state, the definition of linear operator and then we briefly present the principles of quantum mechanics on which this thesis is grounded. After having reviewed the literature of QWs and the main historical approaches to their study, we then move on to consider a new property of QWs, the plasticity. Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit. We show that such QWs can be used to quantum simulate a large class of physical phenomena described by transport equations. We investigate this new family of QWs in one and two spatial dimensions, showing that in two dimensions, the PDEs we can simulate are more general and include dispersive terms. We show that the above results do not need to rely on the grid and we prove that such QW-based quantum simulators can be defined on 2-complex simplicia, i.e. triangular lattices. Finally, we extend the above result to any arbitrary triangulation, proving that such QWs coincide in the continuous limit to a transport equation on a general curved surface, including the curved Dirac equation in 2+1 spacetime dimensions.