Geometry Dependence of Error Thresholds in Two-Dimensional Toric Codes
Abstract
We investigate how the lattice geometry influences the error thresholds of two-dimensional toric codes. In the presence of bit- or phase-flip errors, the toric code maps onto the two-dimensional random-bond Ising model (RBIM). We determine the critical behaviour of the RBIM using replica-exchange Monte Carlo simulations. While previous studies have explored thresholds for general lattice geometries under suboptimal decoders such as minimum-weight perfect matching, the rapid development of near-optimal decoders makes resolving the ultimate, maximum-likelihood code capacity relevant. We compute these optimal error thresholds on the Nishimori line for the square, honeycomb, triangular, dice, and kagome lattices. Owing to lattice duality, the phase-flip threshold on a given lattice is equivalent to the bit-flip threshold on its dual. We find that the optimal thresholds of dual-lattice pairs display a characteristic duality-driven splitting around the self-dual square-lattice, mirroring the qualitative behaviour observed for suboptimal decoders. While the average coordination number has the strongest impact on the error threshold, our results demonstrate that the detailed arrangement of vertices and plaquettes also plays a significant role in determining its precise value.
Related
- Topological Quantum Error-Correcting Thresholds of a Symmetry-Enriched Toric Code
- Error-Adaptive Quasi-local Decoding of the Toric Code
- Kaluza-Klein tower thresholds and scheme dependence of the species scale
- Quantum Error Correction in Nearly Critical Toric Codes
- Homological Thresholds in Randomly Monitored Quantum Error-Correcting Codes
BibTeX
@misc{lessing2026geometry,
title = {{Geometry Dependence of Error Thresholds in Two-Dimensional Toric Codes}},
author = {D. Lessing and A. Langheld and C. Krämer and J. A. Koziol and K. P. Schmidt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07579},
eprint = {2609.07579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07579},
}
RIS
TY - UNPB
TI - Geometry Dependence of Error Thresholds in Two-Dimensional Toric Codes
AU - D. Lessing
AU - A. Langheld
AU - C. Krämer
AU - J. A. Koziol
AU - K. P. Schmidt
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07579
UR - https://arxiv.org/abs/2609.07579
AB - We investigate how the lattice geometry influences the error thresholds of two-dimensional toric codes. In the presence of bit- or phase-flip errors, the toric code maps onto the two-dimensional random-bond Ising model (RBIM). We determine the critical behaviour of the RBIM using replica-exchange Monte Carlo simulations. While previous studies have explored thresholds for general lattice geometries under suboptimal decoders such as minimum-weight perfect matching, the rapid development of near-optimal decoders makes resolving the ultimate, maximum-likelihood code capacity relevant. We compute these optimal error thresholds on the Nishimori line for the square, honeycomb, triangular, dice, and kagome lattices. Owing to lattice duality, the phase-flip threshold on a given lattice is equivalent to the bit-flip threshold on its dual. We find that the optimal thresholds of dual-lattice pairs display a characteristic duality-driven splitting around the self-dual square-lattice, mirroring the qualitative behaviour observed for suboptimal decoders. While the average coordination number has the strongest impact on the error threshold, our results demonstrate that the detailed arrangement of vertices and plaquettes also plays a significant role in determining its precise value.
ER -