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

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

Alternative names: \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code.

Description

A family of pure [2] non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound.

The family can be obtained from a modified CSS construction [3] 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 [3]. The modification introduces signs between the codewords.

Protection

Protects against any single qubit error.

Notes

The code is useful for entanglement distillation [4].

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 [3] 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 [3].
Projective geometry code— Gottesman codes are related to partial spreads in projective geometry [5].

Member of code lists

Primary Hierarchy

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 via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[3]: A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[4]: 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
[5]: J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini, and F. Pambianco, “The geometry of quantum codes”, Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial 6, 53 (2008) DOI

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

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