\([[2^r, 2^r-r-2, 3]]\) Gottesman code[1] 

Also known as \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code.

Description

A family of non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound.

The family can be obtained from a modified CSS construction [2] with a \([2^r,r+1,2^{r-1}] = C_2^{\perp}\) RM code and a \([2^r,2^r-1,2] = C_1\) even-weight code [2]. The modification introduces signs between the codewords.

Protection

Protects against any single qubit error.

Parents

Child

Cousins

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. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
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Zoo Code ID: quantum_hamming

Cite as:
\([[2^r, 2^r-r-2, 3]]\) Gottesman code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_hamming
BibTeX:
@incollection{eczoo_quantum_hamming, title={\([[2^r, 2^r-r-2, 3]]\) Gottesman code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_hamming} }
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Permanent link:
https://errorcorrectionzoo.org/c/quantum_hamming

Cite as:

\([[2^r, 2^r-r-2, 3]]\) Gottesman code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_hamming

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