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arXiv:1004.0976 · New Journal of Physics

Tailoring discrete quantum walk dynamics via extended initial conditions

Germán J. de Valcárcel, Eugenio Roldán, A. Romanelli

Published Dec 13, 2010• 33 citations
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Abstract

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.

DOI
10.1088/1367-2630/12/12/123022

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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.
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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.
Two-dimensional continuous time quantum random walks (CTQRW) are physical processes where quantum particles simultaneously evolve in different permissible directions within discrete graphs. In order to force the quantum walkers (QWs) to evolve in such a fashion, one generally requires periodic potentials. Here, we demonstrate that two-dimensional CTQRW can be generated in free space by properly tailoring the initial wave functions. We analytically show that within a certain spatial region the arising probability distribution quantitatively resembles the probability pattern exhibited by a QW traversing a periodic lattice potential. These theoretical predictions were experimentally verified using classical laser light, appropriately shaped by a spatial light modulator. Expanding the presented results to the case of multiple walkers may open new possibilities in quantum information technology using bulk optics. (C) 2014 Optical Society of America
Quantum random walks are shown to have non-intuitive dynamics which makes them an attractive area of study for devising quantum algorithms for long-standing open problems as well as those arising in the field of quantum computing. In the case of continuous-time quantum random walks, such peculiar dynamics can arise from simple evolution operators closely resembling the quantum free-wave propagator. We investigate the divergence of quantum walk dynamics from the free-wave evolution and show that in order for continuous-time quantum walks to display their characteristic propagation, the state space must be discrete. This behavior rules out many continuous quantum systems as possible candidates for implementing continuous-time quantum random walks. Furthermore our findings demonstrate that the properties of the quantum walk are critically dependant on its initialization which in turn impacts any subsequent quantum algorithms based upon it.
We introduce the idea of a quantum walk with two particles and study it for the case of a discrete time walk on a line. We consider both separable and maximally entangled initial conditions, and show how the entanglement and the relative phase between the states describing the coin degree of freedom of each particle will influence the evolution of the quantum walk. In particular, these factors will have consequences for the distance between the particles and the probability of finding them at a given point, yielding results that cannot be obtained from a separable initial state, be it pure or mixed. Finally, we review briefly proposals for implementations.