\([[8, 3, 3]]\) code[13] 

Description

Eight-qubit non-degenerate code that can be obtained from a modified CSS construction using the \([8,4,4]\) extended Hamming code and a \([8,7,2]\) even-weight code [3]. The modification introduces signs between the codewords.

See [4; Table 3.3] for its stabilizer generator matrix. The code's automorphism group is \(\text{A}\Gamma\text{L}(1,8)\) [5]. It is unique for its parameters, up to local equivalence [6][7; pg. 386].

Transversal Gates

Permutation-based gates [8; Sec. IV.D].No gates outside of the Pauli group were found in Ref. [9].

Parent

Cousin

References

[1]
D. Gottesman, “Class of quantum error-correcting codes saturating the quantum Hamming bound”, Physical Review A 54, 1862 (1996) arXiv:quant-ph/9604038 DOI
[2]
A. R. Calderbank et al., “Quantum Error Correction and Orthogonal Geometry”, Physical Review Letters 78, 405 (1997) arXiv:quant-ph/9605005 DOI
[3]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[4]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[5]
H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
[6]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[7]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[8]
M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1302.1035 DOI
[9]
H. Chen et al., “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_3_3

Cite as:
\([[8, 3, 3]]\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_8_3_3
BibTeX:
@incollection{eczoo_stab_8_3_3, title={\([[8, 3, 3]]\) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_8_3_3} }
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Cite as:

\([[8, 3, 3]]\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_8_3_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/stab_8_3_3.yml.