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\([[12,1,3]]\) CE CSS code[1]

Description

Twelve-qubit constant-excitation (CE) CSS code that encodes one logical qubit with distance three. It is the smallest CE CSS code that corrects a single-qubit error [1]. Codewords lie in a fixed Hamming-weight subspace, making the code immune to coherent noise in the form of transversal \(Z\)-rotations.

One stabilizer tableau for the code is, up to Pauli frame [1], \begin{align} \begin{array}{cccccccccccc} X & X & X & X & I & I & I & I & I & I & I & I \\ I & I & X & X & X & X & I & I & I & I & I & I \\ I & I & I & I & I & I & X & X & X & X & I & I \\ I & I & I & I & I & I & I & I & X & X & X & X \\ Z & I & Z & I & Z & I & Z & I & Z & I & Z & I \\ Z & Z & I & I & I & I & I & I & I & I & I & I \\ I & I & Z & Z & I & I & I & I & I & I & I & I \\ I & I & I & I & Z & Z & I & I & I & I & I & I \\ I & I & I & I & I & I & Z & Z & I & I & I & I \\ I & I & I & I & I & I & I & I & Z & Z & I & I \\ I & I & I & I & I & I & I & I & I & I & Z & Z \end{array}~. \tag*{(1)}\end{align}

Protection

Corrects any single-qubit error (distance \(d=3\)). Protects from collective coherent noise in the form of transversal \(Z\)-rotations, since all codewords lie in the same Hamming-weight subspace [2,3].

Fault Tolerance

Fault-tolerant syndrome extraction using modified Shor and Steane methods adapted for CE codes: weight-\(2w\) stabilizers are measured using \(w\)-CE cat states, and zero-controlled NOT (\(\mathrm{C}_0 X\)) gates replace standard CNOT gates to preserve the constant-excitation structure [1].

Threshold

Pseudo-threshold of \(\sim 9.28 \times 10^{-4}\) (circuit-level stochastic noise) and \(\sim 5.98 \times 10^{-4}\) (with coherent corrections, \(\gamma = 0.01\)) under collective coherent noise [1].

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
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Qubit CSS code
Constant-excitation (CE) codeHamiltonian-based QECC Quantum
Concatenated qubit codeQubit Concatenated quantum QECC Quantum
This code is obtained by dual-rail concatenation of the \([[6,1,2]]\) CSS code [4; ID 50].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[12,1,3]]\) CE CSS code

References

[1]
C.-Y. Lai, P.-H. Liou, and Y. Ouyang, “Fault-Tolerant Quantum Error Correction for Constant-Excitation Stabilizer Codes under Coherent Noise”, (2025) arXiv:2507.10395
[2]
J. Hu, Q. Liang, N. Rengaswamy, and R. Calderbank, “CSS Codes that are Oblivious to Coherent Noise”, 2021 IEEE International Symposium on Information Theory (ISIT) 1481 (2021) DOI
[3]
J. Hu, Q. Liang, N. Rengaswamy, and R. Calderbank, “Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers”, IEEE Transactions on Information Theory 68, 1795 (2022) arXiv:2011.00197 DOI
[4]
Qiskit Community, “Qiskit QEC framework”, URL
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Zoo Code ID: css_12_1_3

Cite as:
“\([[12,1,3]]\) CE CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/css_12_1_3
BibTeX:
@incollection{eczoo_css_12_1_3, title={\([[12,1,3]]\) CE CSS code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/css_12_1_3} }
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Cite as:

“\([[12,1,3]]\) CE CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/css_12_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/12/css_12_1_3.yml.