Topologically Protected Edge States in One-Dimensional Quantum Walks
Abstract
Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a ℤ- to ℤ2-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw--Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.
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BibTeX
@misc{maxey2026topologically,
title = {{Topologically Protected Edge States in One-Dimensional Quantum Walks}},
author = {Emily Maxey and Jacob Mansfield and Beth Thacker and Wade DeGottardi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05823},
eprint = {2609.05823},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05823},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Topologically Protected Edge States in One-Dimensional Quantum Walks
AU - Emily Maxey
AU - Jacob Mansfield
AU - Beth Thacker
AU - Wade DeGottardi
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05823
UR - https://arxiv.org/abs/2609.05823
AB - Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a ℤ- to ℤ_2-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw--Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.
ER -