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\(((5,6,2))\) qubit code[1]

Description

Five-qubit cyclic CWS code detecting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[5,2,2]]\) code, the best five-qubit stabilizer code with the same distance [2].

Its codeword stabilizer consists of all cyclic shifts of \(ZXZII\). A standard-form CWS presentation uses the five-cycle graph together with the classical codewords \(00000\), \(11010\), \(01101\), \(10110\), \(01011\), and \(10101\) [2,3]. Its automorphism group is of size 3840 and given in Ref. [1] (see also [4; Corr. 18]).

Cousin

  • \([[4,2,2]]\) Four-qubit code— Tracing out any one qubit of the \(((5,6,2))\) code projector yields a \(((4,4,2))\) code; for this code, all five such partial traces are additive and therefore locally equivalent to the \([[4,2,2]]\) code [4; Thm. 8 and Corr. 18].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Codeword stabilized (CWS) codeQECC Quantum
\(((2m+1,3 \times 2^{2m-3},2))\) Rains codeSmall-distance block quantum QECC Quantum
Parents
\(((2m+1,3 \times 2^{2m-3},2))\) Rains code
The \(((5,6,2))\) code is the smallest nontrivial Rains code [4] (see also [6][5; Exam. 8]).
\(((n,1+n(q-1),2))_q\) union stabilizer codeSmall-distance block quantum QECC Quantum
The \(((5,6,2))\) code is the \(((n,1+n(q-1),2))_q\) union stabilizer code for \(n=5\) and \(q=2\) [7].
Cyclic quantum codeQECC Quantum
\(((5,6,2))\) qubit code

References

[1]
E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, “A Nonadditive Quantum Code”, Physical Review Letters 79, 953 (1997) arXiv:quant-ph/9703002 DOI
[2]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[3]
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
[4]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[5]
V. Aggarwal and A. R. Calderbank, “Boolean Functions, Projection Operators, and Quantum Error Correcting Codes”, IEEE Transactions on Information Theory 54, 1700 (2008) arXiv:cs/0610159 DOI
[6]
D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. Oh, “Graphical nonbinary quantum error-correcting codes”, Physical Review A 78, (2008) arXiv:0801.0831 DOI
[7]
V. Arvind, P. P. Kurur, and K. R. Parthasarathy, “Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group”, (2002) arXiv:quant-ph/0210097
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Zoo Code ID: qubit_5_6_2

Cite as:
“\(((5,6,2))\) qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qubit_5_6_2
BibTeX:
@incollection{eczoo_qubit_5_6_2, title={\(((5,6,2))\) qubit code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qubit_5_6_2} }
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Cite as:

“\(((5,6,2))\) qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qubit_5_6_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/5/qubit_5_6_2.yml.