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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 Gates

Applying transversal \(S\) and \(S^{\dagger}\), \(\sqrt{X}\), and Hadamard gates yields various logical gates [3].

Cousin

  • \([[9,1,3]]\) Shor code— Like the Shor code, the \([[8,2,2]]\) hyperbolic color code is a small code defined on the projective plane.

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum
Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum
Hyperbolic color code
Parents
Hyperbolic color code
The \([[8,2,2]]\) hyperbolic color code is defined on the projective plane.
Hermitian qubit codeStabilizer Hamiltonian-based Qubit QECC Quantum
The \([[8,2,2]]\) hyperbolic color code is Hermitian [2; ID 4926].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit 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. https://github.com/qiskit-community/qiskit-qec
[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.), 2024. https://errorcorrectionzoo.org/c/stab_8_2_2
BibTeX:
@incollection{eczoo_stab_8_2_2, title={\([[8,2,2]]\) hyperbolic color code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, 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.), 2024. https://errorcorrectionzoo.org/c/stab_8_2_2

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