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

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: quantum_hamming

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
BibTeX:
@incollection{eczoo_quantum_hamming, title={\([[2^r, 2^r-r-2, 3]]\) quantum Hamming code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_hamming} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/quantum_hamming

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

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