Hyperbolic color codes with constant rate and polynomial distance
Abstract
Recent advances in quantum hardware relax the strict geometric-locality constraints traditionally imposed on quantum error-correcting codes, motivating interest in high-rate quantum low-density parity-check (qLDPC) codes. At the same time, color codes provide a particularly rich setting for fault-tolerant quantum computation, underlying protocols such as single-shot error correction and self-correcting quantum computation. Hyperbolic color codes provide a class of high-rate qLDPC codes that also retain the structural features of color codes relevant to fault-tolerant quantum computation. However, previous constructions of hyperbolic color codes have achieved at most logarithmic code distance. In this work, we construct hyperbolic color codes with both constant rate and polynomial distance by building on arithmetic hyperbolic manifolds that support polynomial-distance hyperbolic toric codes. Our construction applies in arbitrary dimension D≥ 4. In even dimensions, the resulting type-D/2 color codes have constant encoding rate and polynomial distance, while in other cases the number of logical qubits and the code distance both exhibit polynomial scaling. We further derive explicit exponents for polynomial lower bounds as functions of the dimension and code type. These results establish a family of hyperbolic color codes simultaneously achieving constant rate and polynomial distance and providing a testbed to explore fault-tolerant quantum computation protocols that combine high-rate quantum codes with the structural advantages of color codes.
Related
- Denser Planar Color Codes
- 4D and 5D Layer Codes through Color Routing
- Constant Rate Codes with Fully Addressable Transversal T: Good Codes, Sparse Checks
- BARC codes: general polynomial framework for coherent-state superposition codes
- Circuit-Level Loss Performance of RHG and Foliated Floquet Color Codes in a Compound Photon--Atom Quantum Architecture
BibTeX
@misc{hasegawa2026hyperbolic,
title = {{Hyperbolic color codes with constant rate and polynomial distance}},
author = {Shun Hasegawa and Hayata Yamasaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16125},
eprint = {2609.16125},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16125},
}
RIS
TY - UNPB
TI - Hyperbolic color codes with constant rate and polynomial distance
AU - Shun Hasegawa
AU - Hayata Yamasaki
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.16125
UR - https://arxiv.org/abs/2609.16125
AB - Recent advances in quantum hardware relax the strict geometric-locality constraints traditionally imposed on quantum error-correcting codes, motivating interest in high-rate quantum low-density parity-check (qLDPC) codes. At the same time, color codes provide a particularly rich setting for fault-tolerant quantum computation, underlying protocols such as single-shot error correction and self-correcting quantum computation. Hyperbolic color codes provide a class of high-rate qLDPC codes that also retain the structural features of color codes relevant to fault-tolerant quantum computation. However, previous constructions of hyperbolic color codes have achieved at most logarithmic code distance. In this work, we construct hyperbolic color codes with both constant rate and polynomial distance by building on arithmetic hyperbolic manifolds that support polynomial-distance hyperbolic toric codes. Our construction applies in arbitrary dimension D≥ 4. In even dimensions, the resulting type-D/2 color codes have constant encoding rate and polynomial distance, while in other cases the number of logical qubits and the code distance both exhibit polynomial scaling. We further derive explicit exponents for polynomial lower bounds as functions of the dimension and code type. These results establish a family of hyperbolic color codes simultaneously achieving constant rate and polynomial distance and providing a testbed to explore fault-tolerant quantum computation protocols that combine high-rate quantum codes with the structural advantages of color codes.
ER -