\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code[1]

Description

CCS code constructed with a classical Hamming code \([2^r-1,2^r-1-r,3]=C_X=C_Z\) a.k.a. a first-order punctured Reed-Muller code RM\((r-2,r)\).

Protection

Protects against any single qubit error.

Transversal Gates

Pauli, Hadamard, and CNOT gates.

Decoding

Efficient decoder [2].

Fault Tolerance

Syndrome measurement can be done with two ancillary flag qubits [3].

Threshold

Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead [2].

Parents

Children

Cousins

References

[1]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]
H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, (2022) arXiv:2207.08826
[3]
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
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Zoo Code ID: quantum_hamming_css

Cite as:
\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_hamming_css
BibTeX:
@incollection{eczoo_quantum_hamming_css, title={\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_hamming_css} }
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https://errorcorrectionzoo.org/c/quantum_hamming_css

Cite as:

\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_hamming_css

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/rm/quantum_hamming_css.yml.