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arXiv:1704.07976

Unitary equivalent classes of one-dimensional quantum walks II

Hiromichi Ohno

Published Apr 26, 2017• 0 citations
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Abstract

This study investigated the unitary equivalent classes of one-dimensional quantum walks. We determined the unitary equivalent classes of one-dimensional quantum walks, two-phase quantum walks with one defect, complete two-phase quantum walks, one-dimensional quantum walks with one defect and translation-invariant one-dimensional quantum walks. The unitary equivalent classes of one-dimensional quantum walks with initial states were also considered.

DOI
10.48550/arxiv.1704.07976

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We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way. The classification largely matches the one recently obtained (arXiv:1611.04439) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of sites. Similarly, if walks are considered equivalent when they differ only by adding a trivial walk, i.e., one that allows no jumps between cells, then the classification collapses also to the general one. The indices of the general classification can be computed in practice only for walks closely related to some translation invariant ones. We prove a completed collection of simple formulas in terms of winding numbers of band structures covering all symmetry types. Furthermore, we determine the strength of the locality conditions, and show that the continuity of the band structure, which is a minimal requirement for topological classifications in terms of winding numbers to make sense, implies the compactness of the commutator of the walk with a half-space projection, a condition which was also the basis of the general theory. In order to apply the theory to the joining of large but finite bulk pieces, one needs to determine the asymptotic behaviour of a stationary Schrödinger equation. We show exponential behaviour, and give a practical method for computing the decay constants.
This study is motivated by the previous work [14]. We treat 3 types of the one-dimensional quantum walks (QWs), whose time evolutions are described by diagonal unitary matrices except at one defected point. In this paper, we call the QW defined by diagonal unitary matrices, ``the diagonal QW'', and we consider the stationary distributions of general 2-state diagonal QW with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW with one defect. One of the purposes of our study is to characterize the QWs by the stationary measure, which may lead to answer the basic and natural question, ``What are stationary measures for one-dimensional QWs?''. In order to analyze the stationary distribution, we focus on the corresponding eigenvalue problems and the definition of the stationary measure.
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