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

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
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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}
}
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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.