Self-Trapping Bounds for Continuous-Time Nonlinear Quantum Walks on Path and Cycle Graphs
Abstract
We explore a continuous-time quantum walk starting at a single vertex on the discrete path and cycle with a cubic nonlinearity. Such nonlinearities arise in Bose-Einstein condensates described by the Gross-Pitaevskii equation or by nonlinear optical waveguide arrays. When the nonlinearity is sufficiently strong, the walker remains localized at its initial vertex, a phenomenon known as self-trapping. This contrasts with linear quantum walks, which are known for spreading quickly in one dimension. While self-trapping has been known numerically, we introduce an analytical method that proves self-trapping and yields a quantitative relationship between the nonlinearity coefficient and the trapping probability. We propose that this trapping can be used for timing in quantum state transfer, where a qubit is held at a node until it is ready to be transferred, and it can also be held again at the receiving node. This scheme can also be interpreted as a form of quantum memory, with the trap and transfer corresponding to the storage and release of quantum information.
Versions
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BibTeX
@misc{shi2026selftrapping,
title = {{Self-Trapping Bounds for Continuous-Time Nonlinear Quantum Walks on Path and Cycle Graphs}},
author = {Yujia Shi and Thomas G. Wong},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2605.20464},
eprint = {2605.20464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.20464},
}
RIS
TY - UNPB
TI - Self-Trapping Bounds for Continuous-Time Nonlinear Quantum Walks on Path and Cycle Graphs
AU - Yujia Shi
AU - Thomas G. Wong
PY - 2026
DA - 2026/08/13
PB - arXiv
DO - 10.48550/arxiv.2605.20464
UR - https://arxiv.org/abs/2605.20464
AB - We explore a continuous-time quantum walk starting at a single vertex on the discrete path and cycle with a cubic nonlinearity. Such nonlinearities arise in Bose-Einstein condensates described by the Gross-Pitaevskii equation or by nonlinear optical waveguide arrays. When the nonlinearity is sufficiently strong, the walker remains localized at its initial vertex, a phenomenon known as self-trapping. This contrasts with linear quantum walks, which are known for spreading quickly in one dimension. While self-trapping has been known numerically, we introduce an analytical method that proves self-trapping and yields a quantitative relationship between the nonlinearity coefficient and the trapping probability. We propose that this trapping can be used for timing in quantum state transfer, where a qubit is held at a node until it is ready to be transferred, and it can also be held again at the receiving node. This scheme can also be interpreted as a form of quantum memory, with the trap and transfer corresponding to the storage and release of quantum information.
ER -