\([[2^r, 2^r-r-2, 3]]\) quantum Hamming code[1]
Description
A family of stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. Can be obtained from the CSS construction using a first-order \([2^r,r+1,2^{r-1}]\) RM code and a \([2^r,2^r-1,2]\) even-weight code [2].
Protection
Protects against any single qubit error.
Fault Tolerance
Concatenations of Hamming codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [3].
Parents
Cousins
- Perfect quantum code — Quantum Hamming codes saturate the asymptotic quantum Hamming bound.
- Hamming code — \([[2^r, 2^r-r-2, 3]]\) quantum Hamming codes are analogues of Hamming codes in that they saturate the asymptotic Hamming bound.
- Reed-Muller (RM) code — \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code can be obtained from the CSS construction using a first-order \([2^r,r+1,2^{r-1}]\) RM code and a \([2^r,2^r-1,2]\) even-weight code [2].
- Hamming 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. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
- [3]
- H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, (2022) arXiv:2207.08826
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:
“\([[2^r, 2^r-r-2, 3]]\) quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_hamming