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.
Highlights
Quantum walks (QWs) are becoming widespread in physics
It evidences the role played by the dispersion relations as anticipated: For distributions whose Discrete Fourier Transform (DFT) is centered around some k0, the local variations of ω around k0 determine the type of wave equation controlling the QW dynamics
The use of the dispersion relations is the central argument behind this view as its analysis allows to get much insight into the propagation properties of the walk, even allowing for the tailoring of the initial distribution in order to reach a desired asymptotic distribution, as we have demonstrated
Summary
Quantum walks (QWs) are becoming widespread in physics. Originally introduced as quantum versions of random classical processes [1] and quantum cellular automata [2], they have been deeply investigated, specially in connection with quantum information science [3,4,5,6,7,8,9], and have been recently shown to constitute a universal model for quantum computation [10, 11]. Some of us have been insisting in the interest of analyzing the dynamics of coined QWs from the perspective of their dispersion relation [13, 40], which has been used for purposes different to ours (see, e.g., [41, 42]) We think this is an interesting approach that is suitable for non localized initial conditions, i.e., when the initial state is described by a probability distribution extending on a finite region in the lattice. In order to avoid possible confusions, to conclude this initial part we state that the ordering of the coin base elements we will be using in the matrix representations of operators and kets is |1+ , |1− , . . . |N+ , |N−
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