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arXiv:2007.07398 · Quantum Information and Computation

Localization does not occur for the Fourier walk on the multi-dimensional lattice

Akihiro Narimatsu

Published Mar 22, 2021• 1 citation
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Abstract

The existence of localization for the Grover walk on the multi-dimensional lattice is known. This paper gives some conditions for the existence of localization for the space-homogeneous quantum walks. We also prove that localization does not occur for the Fourier walk on the multi-dimensional lattice.
DOI
10.26421/qic21.5-6-3

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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.
Abstract We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved.
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