\([[2^r-1, 2^r-2r-1, 3]]\) 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
- Quantum Reed-Muller code — \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS codes are quantum Reed-Muller codes because they are formed from classical Hamming codes, which are equivalent to RM\((r-2,r)\).
- Small-distance block quantum code
Children
Cousins
- Hamming code — Quantum Hamming codes result from applying the CSS construction to Hamming codes.
- \([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code — \([[2^r-1, 2^r-2r-1, 3]]_p\) codes are prime-qudit extensions of \([[2^r-1, 2^r-2r-1, 3]]\) codes.
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
Page edit log
- Qingfeng (Kee) Wang (2022-01-07) — most recent
- Victor V. Albert (2021-12-30)
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