\([[8, 3, 3]]\) Eight-qubit Gottesman code

\([[8, 3, 3]]\) Eight-qubit Gottesman code[1–3]

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 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

\([[2^r, 2^r-r-2, 3]]\) Gottesman code

Cousins

\([8,4,4]\) extended Hamming code — The \([[8, 3, 3]]\) code is obtained via a modified CSS construction from the \([8,4,4]\) extended Hamming code.
\([[8, 2:1, 3]]\) hybrid stabilizer code — \([[8, 2:1, 3]]\) hybrid stabilizer code is obtained from the \([[8,3,3]]\) Gottesman code by using one of its logical qubits as a classical bit.

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

Page edit log

Feroz Ahmad (2024-03-14) — most recent
Victor V. Albert (2023-11-28)

Cite as:

“\([[8, 3, 3]]\) Eight-qubit Gottesman 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.