Qubit BCH code[1–5] 

Description

Qubit stabilizer code constructed from a self-orthogonal binary BCH code via the CSS construction, from a Hermitian self-orthogonal quaternary BCH code via the Hermitian construction, or by taking a Euclidean self-orthogonal BCH code over \(GF(2^m)\), converting it to a binary code, and applying the CSS construction.

Gates

Magic-state distillation protocols [6].

Threshold

Semi-analytical estimates of concatenated thresholds [7]. Qubit BCH codes are difficult to study numerically [8].

Notes

Qubit BCH codes for small \(n\) are tabulated in Ref. [5].

Parents

Cousins

  • Binary BCH code — Binary BCH codes are used to construct a subset of qubit BCH codes via the CSS construction.
  • Bose–Chaudhuri–Hocquenghem (BCH) code — BCH codes are used to construct qubit BCH codes via the CSS construction or the Hermitian construction.
  • Qubit CSS code — Some qubit BCH codes are CSS.
  • Hermitian qubit code — Hermitian self-orthogonal quaternary BCH codes are used to construct a subset of qubit BCH codes via the Hermitian construction.

References

[1]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]
M. Grassl, Th. Beth, and T. Pellizzari, “Codes for the quantum erasure channel”, Physical Review A 56, 33 (1997) arXiv:quant-ph/9610042 DOI
[3]
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
[4]
A. M. Steane, “Enlargement of Calderbank Shor Steane quantum codes”, (1998) arXiv:quant-ph/9802061
[5]
M. Grassl and T. Beth, “Quantum BCH Codes”, (1999) arXiv:quant-ph/9910060
[6]
Quantum Information and Computation 18, (2018) arXiv:1709.02789 DOI
[7]
A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
[8]
A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
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Zoo Code ID: quantum_bch

Cite as:
“Qubit BCH code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_bch
BibTeX:
@incollection{eczoo_quantum_bch, title={Qubit BCH code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_bch} }
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Permanent link:
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Cite as:

“Qubit BCH code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_bch

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/quantum_bch.yml.