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arXiv:1611.07457

Continuous-time limit of topological quantum walks

Radhakrishnan Balu, Daniel Castillo, George Siopsis, Christian Weedbrook

Published Nov 22, 2016• 0 citations
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Abstract

We derive the continuous-time limit of discrete quantum walks with topological phases. We show the existence of a continuous-time limit that preserves their topological phases. We consider both simple-step and split-step walks, and derive analytically equations of motion governing their behavior. We obtain simple analytical solutions showing the existence of bound states at the boundary of two phases, and solve the equations of motion numerically in the bulk.

DOI
10.48550/arxiv.1611.07457

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We study a position-dependent discrete-time quantum walk (QW) in one dimension, whose time-evolution operator is built up from two coin operators which are distinguished by phase factors from x≥0x\geq0x≥0 and x≤−1x\leq-1x≤−1. We call the QW the {\it complete two-phase QW} to discern from the two-phase QW with one defect\cite{endosan,maman}. Because of its localization properties, the two-phase QWs can be considered as an ideal mathematical model of topological insulators which are novel quantum states of matter characterized by topological invariants. Employing the complete two-phase QW, we present the stationary measure, and two kinds of limit theorems concerning {\it localization} and the {\it ballistic spreading}, which are the characteristic behaviors in the long-time limit of discrete-time QWs in one dimension. As a consequence, we obtain the mathematical expression of the whole picture of the asymptotic behavior of the walker, including dependences on initial states, in the long-time limit. We also clarify relevant symmetries, which are essential for topological insulators, of the complete two-phase QW, and then derive the topological invariant. Having established both mathematical rigorous results and the topological invariant of the complete two-phase QW, we provide solid arguments to understand localization of QWs in term of topological invariant. Furthermore, by applying a concept of {\it topological protections}, we clarify that localization of the two-phase QW with one defect, studied in the previous work\cite{endosan}, can be related to localization of the complete two-phase QW under symmetry preserving perturbations.
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