Qubit BCH code[15] 

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

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 et al., “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
[9]
P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
[10]
L. Ioffe and M. Mézard, “Asymmetric quantum error-correcting codes”, Physical Review A 75, (2007) arXiv:quant-ph/0606107 DOI
[11]
G. G. La Guardia, “New families of asymmetric quantum BCH codes”, Quantum Information and Computation 11, 239 (2011) DOI
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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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“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.