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\([[8,2,2]]\) hyperbolic color code[1]

Description

An \([[8,2,2]]\) hyperbolic color code defined on the projective plane. It is a self-dual CSS code.

A stabilizer tableau for the code is given by [2; ID 4926] \begin{align} \begin{array}{cccccccc} Z & Z & Z & Z & I & I & I & I \\ X & X & X & X & I & I & I & I \\ Z & Z & I & I & Z & Z & I & I \\ X & X & I & I & X & X & I & I \\ Z & Z & I & I & I & I & Z & Z \\ X & X & I & I & I & I & X & X \end{array}~. \tag*{(1)}\end{align}

Transversal and Permutation-Based Gates

Applying transversal \(S\) and \(S^{\dagger}\), \(\sqrt{X}\), and Hadamard gates yields various logical gates [3]. For instance, the physical gate \(S^{\dagger}_{2}S^{\dagger}_{3}S_{4}S_{5}S_{6}S_{7}CZ_{01}\) implements the logical action \(\bar{S}_{0} \bar{Z}_{1} \overline{CZ}_{01}\).

Cousins

  • \([[9,1,3]]\) Shor code— Like the Shor code, the \([[8,2,2]]\) hyperbolic color code is a small code defined on the projective plane.
  • \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code— Up to qubit permutation, the \([[8,2,2]]\) hyperbolic color code is the symplectic double of a local-Clifford-equivalent version of the \([[4,1,2]]\) LNCY code. This version is obtained by applying the phase gate \(S\) to every qubit, so its stabilizer group is generated by \(YYII\), \(IIYY\), and \(ZZZZ\).

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
Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Color codeQubit CSS Stabilizer Hamiltonian-based QECC Quantum
Hyperbolic color code
Parents
Hyperbolic color code
The \([[8,2,2]]\) hyperbolic color code is defined on the projective plane.
Self-dual CSS codeQubit CSS Stabilizer Hamiltonian-based QECC Quantum
Small-distance qubit stabilizer codeQubit Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum
\([[8,2,2]]\) hyperbolic color code

References

[1]
C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
[2]
Qiskit Community, “Qiskit QEC framework”, URL
[3]
H. Chen, M. Vasmer, N. P. Breuckmann, and E. Grant, “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022) arXiv:1912.10063 DOI
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Zoo Code ID: stab_8_2_2

Cite as:
“\([[8,2,2]]\) hyperbolic color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_2_2, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_8_2_2,
title={\([[8,2,2]]\) hyperbolic color 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/stab_8_2_2}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_8_2_2

Cite as:

“\([[8,2,2]]\) hyperbolic color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_2_2, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/8/stab_8_2_2.yml.