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\(((9,12,3))\) qubit code[1]

Description

Nine-qubit cyclic CWS code correcting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[9,3,3]]\) code, the best nine-qubit stabilizer code with the same distance [2].

Its codeword stabilizer consists of all cyclic shifts of \(ZXZIIIIII\). In the coding-clique framework, it is realized by the nine-vertex loop graph \(L_9\); within a graph search [3], the realization \((L_9,12,3)\) is unique and there is no \((G,13,3)\) code on any nine-vertex graph.

The \(((9,12,3))\) qubit code can be combined to form an infinite family of distance-three qubit codes whose logical dimension is \(50\%\) larger than that of the optimal stabilizer code [4].

Decoding

Fault-tolerant scheme that converts the required POVM into 10 binary measurements whose redundancy is guaranteed by a classical code [5].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Codeword stabilized (CWS) codeQECC Quantum
Parents
Codeword stabilized (CWS) code
The \(((9,12,3))\) qubit code is a cyclic CWS code [6,7].
Cyclic quantum codeQECC Quantum
Small-distance block quantum codeQECC Quantum
The \(((9,12,3))\) qubit code can be combined to form an infinite family of distance-three qubit codes whose logical dimension is \(50\%\) larger than that of the optimal stabilizer code [4].
\(((9,12,3))\) qubit code

References

[1]
S. Yu, Q. Chen, C. H. Lai, and C. H. Oh, “Nonadditive Quantum Error-Correcting Code”, Physical Review Letters 101, (2008) arXiv:0704.2122 DOI
[2]
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[3]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[4]
S. Yu, Q. Chen, and C. H. Oh, “Two infinite families of nonadditive quantum error-correcting codes”, (2009) arXiv:0901.1935
[5]
Y. Ouyang, “Robust projective measurements through measuring code-inspired observables”, npj Quantum Information 10, (2024) arXiv:2402.04093 DOI
[6]
A. Cross, G. Smith, J. A. Smolin, and B. Zeng, “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
[7]
A. W. Cross. Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes. PhD thesis, Massachusetts Institute of Technology, 2008
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Zoo Code ID: qubit_9_12_3

Cite as:
“\(((9,12,3))\) qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qubit_9_12_3
BibTeX:
@incollection{eczoo_qubit_9_12_3, title={\(((9,12,3))\) qubit code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qubit_9_12_3} }
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Permanent link:
https://errorcorrectionzoo.org/c/qubit_9_12_3

Cite as:

“\(((9,12,3))\) qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qubit_9_12_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/9/qubit_9_12_3.yml.