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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].

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.

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Qubit stabilizer code
Galois-qudit BCH codeStabilizer Hamiltonian-based QECC Quantum
Galois-qudit BCH codes for \(q=2\) reduce to qubit BCH codes.
Qubit BCH code

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:
https://errorcorrectionzoo.org/c/quantum_bch

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.