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\([[8, 3, 3]]\) Eight-qubit Gottesman code[1–4]

Description

Eight-qubit pure 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 [5; Table 3.3] for its stabilizer generator matrix. A stabilizer tableau for the code is given by [6; ID 6822] \begin{align} \begin{array}{cccccccc} X & Y & Y & X & Z & I & I & Z \\ Z & X & I & Z & X & I & Z & Z \\ X & Z & Z & I & I & X & Z & Z \\ Y & I & Y & Z & I & Z & X & Z \\ Y & Z & X & I & Z & Z & I & Y \end{array}~. \tag*{(1)}\end{align} The code’s automorphism group is \(\text{A}\Gamma\text{L}(1,8)\) [7]. It is unique for its parameters, up to equivalence [8][9; pg. 386].

Transversal and Permutation-Based Gates

Permutation-based gates [10; Sec. IV.D].No gates outside of the Pauli group were found in Ref. [11].

Gates

Logical Trotter circuits can be implemented via symplectic transvections [12], with explicit encoded parity circuits for this code given in Ref. [13; Sec. III.C].

Decoding

Weight-one lookup-table decoder built from the distinct syndromes of the 24 weight-one Pauli errors [13; Sec. III.A].Bitwise majority vote over repeated syndrome measurement rounds under phenomenological measurement noise [13; Sec. III.A].

Fault Tolerance

Chao-Reichardt flagged syndrome extraction using one syndrome ancilla and one flag qubit per stabilizer generator [14][13; Sec. III.B].Encoded Pauli rotations built from compute-rotate-uncompute parity circuits have circuit-level distance one in the logical \(Z\) sector [13; Sec. III.C]. A single phase-type fault on the parity ancilla propagates into a logical \(Z\) operator with trivial syndrome.Flag-conditioned recovery, \(Z\)-biased noise tailoring, CliNR resource verification, and flag postselection each suppress only the logical sector transverse to the rotation axis [13; Secs. III.D–III.F].

Code Capacity Threshold

\(4\%\) pseudo-threshold under depolarizing noise with a weight-one lookup-table decoder [13; Sec. III.A].

Threshold

\(1.5\times 10^{-3}\) pseudo-threshold for circuit-level depolarizing noise under one round of flagged error correction, assuming an ideal final round of weight-one correction [13; Sec. III.B].

Cousins

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum
Small-distance qubit stabilizer codeSmall-distance block quantum QECC Quantum
\([[2^r, 2^r-r-2, 3]]\) Gottesman code
Parents
\([[2^r, 2^r-r-2, 3]]\) Gottesman code
\([[8, 3, 3]]\) Eight-qubit Gottesman code

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, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “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]
R. Cleve and D. Gottesman, “Efficient computations of encodings for quantum error correction”, Physical Review A 56, 76 (1997) arXiv:quant-ph/9607030 DOI
[5]
D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
[6]
Qiskit Community, “Qiskit QEC framework”, URL
[7]
H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
[8]
S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
[9]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[10]
M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory 534 (2013) arXiv:1302.1035 DOI
[11]
H. Chen, M. Vasmer, N. P. Breuckmann, and E. Grant, “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022) arXiv:1912.10063 DOI
[12]
Z. Chen, J. O. Weinberg, and N. Rengaswamy, “Fault Tolerant Quantum Simulation via Symplectic Transvections”, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 158 (2025) arXiv:2504.11444 DOI
[13]
Z. Chen and N. Rengaswamy, “Towards Block-Level Fault-Tolerant Quantum Simulation on Small High-Rate Non-CSS Codes”, (2026) arXiv:2609.16159
[14]
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
[15]
R. L. Barnes, “Stabilizer Codes for Continuous-variable Quantum Error Correction”, (2004) arXiv:quant-ph/0405064
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Zoo Code ID: stab_8_3_3

Cite as:
“\([[8, 3, 3]]\) Eight-qubit Gottesman code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_3_3, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_8_3_3,
title={\([[8, 3, 3]]\) Eight-qubit Gottesman code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/stab_8_3_3}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_8_3_3

Cite as:

“\([[8, 3, 3]]\) Eight-qubit Gottesman code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_8_3_3, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/8/stab_8_3_3.yml.