Connes Distance in Discrete-Time Quantum Walks
Abstract
We study the geometric spread of discrete-time quantum walks using Connes' distance formula. We introduce its discrete analogue by adapting the continuous spectral metric to the lattice, with the Dirac operator replaced by the unitary step operator. Focusing first on a (2+1)-dimensional quantum walk, we evaluate the dynamical spread of localized wavepackets to compare propagation in free space against that under a uniform magnetic field. We find that while the free walk shows linear ballistic growth, gauge-induced phases in the magnetic walk constrain the admissible test operators, suppressing the long-time geometric expansion. Furthermore, we apply this discrete framework to a (1+1)-dimensional plastic quantum walk simulating an inhomogeneous spatial metric. The results demonstrate that the discrete Connes distance naturally captures the underlying spatially varying metric, providing a robust measure of the wavepacket's spread that adapts directly to the local simulated geometry.
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BibTeX
@misc{trabelsi2026connes,
title = {{Connes Distance in Discrete-Time Quantum Walks}},
author = {Rayan Trabelsi and A. F. Reyes-Lega and Pablo Arrighi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31488},
eprint = {2609.31488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31488},
}
RIS
TY - UNPB
TI - Connes Distance in Discrete-Time Quantum Walks
AU - Rayan Trabelsi
AU - A. F. Reyes-Lega
AU - Pablo Arrighi
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31488
UR - https://arxiv.org/abs/2609.31488
AB - We study the geometric spread of discrete-time quantum walks using Connes' distance formula. We introduce its discrete analogue by adapting the continuous spectral metric to the lattice, with the Dirac operator replaced by the unitary step operator. Focusing first on a (2+1)-dimensional quantum walk, we evaluate the dynamical spread of localized wavepackets to compare propagation in free space against that under a uniform magnetic field. We find that while the free walk shows linear ballistic growth, gauge-induced phases in the magnetic walk constrain the admissible test operators, suppressing the long-time geometric expansion. Furthermore, we apply this discrete framework to a (1+1)-dimensional plastic quantum walk simulating an inhomogeneous spatial metric. The results demonstrate that the discrete Connes distance naturally captures the underlying spatially varying metric, providing a robust measure of the wavepacket's spread that adapts directly to the local simulated geometry.
ER -