\([[2^r, 2^r-r-2, 3]]\) Gottesman code[1]
Alternative names: \([[2^r, 2^r-r-2, 3]]\) 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 \([2^r,r+1,2^{r-1}] = C_2^{\perp}\) first-order 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.Notes
The code is useful for entanglement distillation [3].Cousins
- Perfect quantum code— \([[2^r, 2^r-r-2, 3]]\) Gottesman codes saturate the asymptotic quantum Hamming bound.
- \([2^r-1,2^r-r-1,3]\) Hamming code— \([[2^r, 2^r-r-2, 3]]\) Gottesman codes are analogues of Hamming codes in that they saturate the asymptotic Hamming bound.
- \([2^m,m+1,2^{m-1}]\) First-order RM code— Gottesman codes can be obtained from a modified CSS construction [2] with a \([2^r,r+1,2^{r-1}] = C_2^{\perp}\) first-order RM code and a \([2^r,2^r-1,2] = C_1\) even-weight code [2].
Primary Hierarchy
Parents
\([[2^r, 2^r-r-2, 3]]\) Gottesman code
Children
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
- [3]
- C. A. Pattison, G. Baranes, J. P. B. Ataides, M. D. Lukin, and H. Zhou, “Fast quantum interconnects via constant-rate entanglement distillation”, (2024) arXiv:2408.15936
Page edit log
- Victor V. Albert (2022-12-04) — most recent
- Victor V. Albert (2022-07-20)
- Marianna Podzorova (2021-12-13)
- Victor V. Albert (2021-11-24)
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