Geometry-optimized hyperbolic codes for modular fault-tolerant quantum architectures
Abstract
Quantum fault tolerance is a prerequisite to harness the full potential of quantum computers. However, achieving fault tolerance remains a major challenge since it requires optimizing tradeoffs between several intertwined theoretical and experimental parameters, including the encoding rate, code distance, parity-check weights, code planarity, and routing. Surface codes remain strong candidates for fault tolerance due to their simple geometric structure, established decoding methods, locality, and experimental attainability. In this work, we present finite families of hyperbolic surface codes that leverage the optimal efficiency scaling kd2/n = C (logk)2 attainable by surface codes with bounded local geometry. We show that optimizing the periodic identifications can double the code distance and thereby increase the code's efficiency by a factor of four, at fixed qubit count, encoding rate, and parity-check weight. We also address the routing problem of hyperbolic surface codes by introducing a topology-aware algorithm that partitions a hyperbolic code into bounded planar modules while reducing long-range communications demand. Circuit-level Monte Carlo simulations yield finite-size threshold estimates for both modular and monolithic layouts, showing that tripling the long-range gate error probability mildly lowers, but does not eliminate, the estimated error-correction thresholds. Together, these results strengthen the case for hyperbolic surface codes as structured and highly efficient large-scale quantum error correction codes for future fault-tolerant architectures.
Related
- Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua
- Quantum Geometry in Hyperbolic Band Theory
- A paradoxical route from hyperbolic geometry to proper proximality
- Global recovery of Lorentzian principal geometry from the hyperbolic Dirichlet-to-Neumann map
- Geometry of Unimodular Random Hyperbolic Polyhedra
BibTeX
@misc{mahmoud2026geometryoptimized,
title = {{Geometry-optimized hyperbolic codes for modular fault-tolerant quantum architectures}},
author = {Ahmed Adel Mahmoud and Steven Rayan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10948},
eprint = {2610.10948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10948},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Geometry-optimized hyperbolic codes for modular fault-tolerant quantum architectures
AU - Ahmed Adel Mahmoud
AU - Steven Rayan
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10948
UR - https://arxiv.org/abs/2610.10948
AB - Quantum fault tolerance is a prerequisite to harness the full potential of quantum computers. However, achieving fault tolerance remains a major challenge since it requires optimizing tradeoffs between several intertwined theoretical and experimental parameters, including the encoding rate, code distance, parity-check weights, code planarity, and routing. Surface codes remain strong candidates for fault tolerance due to their simple geometric structure, established decoding methods, locality, and experimental attainability. In this work, we present finite families of hyperbolic surface codes that leverage the optimal efficiency scaling kd^2/n = C (logk)^2 attainable by surface codes with bounded local geometry. We show that optimizing the periodic identifications can double the code distance and thereby increase the code's efficiency by a factor of four, at fixed qubit count, encoding rate, and parity-check weight. We also address the routing problem of hyperbolic surface codes by introducing a topology-aware algorithm that partitions a hyperbolic code into bounded planar modules while reducing long-range communications demand. Circuit-level Monte Carlo simulations yield finite-size threshold estimates for both modular and monolithic layouts, showing that tripling the long-range gate error probability mildly lowers, but does not eliminate, the estimated error-correction thresholds. Together, these results strengthen the case for hyperbolic surface codes as structured and highly efficient large-scale quantum error correction codes for future fault-tolerant architectures.
ER -