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Hyperbolic surface code

Description

An extension of the Kitaev surface code construction to hyperbolic manifolds. Given a cellulation of a hyperbolic manifold of arbitrary dimension, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces.

Protection

Constructions (see code children below) have yielded distances scaling favorably with the number of qubits. The use of hyperbolic surfaces allows one to circumvent bounds on code parameters (such as the BPT bound) that are valid for lattice geometries.

Gates

Higher-dimensional hyperbolic surface codes can admit a cup product structure and can thus have logical gates in the Clifford hierarchy implemented by constant-depth Clifford circuits [1].

Decoding

Hastings decoder [2].

Cousins

  • Holographic tensor-network code— Both holographic tensor-network and hyperbolic surface codes utilize tessellations of hyperbolic surfaces. Encodings for the former are hyperbolically tiled tensor networks, while the latter is defined on hyperbolically tiled physical-qubit lattices.
  • Single-shot code— A 4D hyperbolic surface code can be decoded with the Hastings decoder [2] in time \(O(n\log n)\) and with a logical error scaling inverse polynomially with \(n\).
  • Hyperbolic color code— Hyperbolic color codes and hyperbolic surface codes are both defined on hyperbolic tilings.

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Homological code
Parents
Homological code
Hyperbolic surface code
Children
2D hyperbolic surface code
Guth-Lubotzky code
Freedman-Meyer-Luo code
Hurwitz surface code

References

[1]
N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology Invariants and Logical Quantum Operations”, Communications in Mathematical Physics 407, (2026) arXiv:2410.16250 DOI
[2]
M. B. Hastings, “Decoding in Hyperbolic Spaces: LDPC Codes With Linear Rate and Efficient Error Correction”, (2013) arXiv:1312.2546
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Zoo Code ID: hyperbolic_surface

Cite as:
“Hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hyperbolic_surface, arXiv:2606.11484
BibTeX:
@incollection{eczoo_hyperbolic_surface,
title={Hyperbolic surface code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/hyperbolic_surface}
}
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Permanent link:
https://errorcorrectionzoo.org/c/hyperbolic_surface

Cite as:

“Hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hyperbolic_surface, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/hyperbolic_surface.yml.