\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code
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\((r-2,r)\).
Protects against any single qubit error.
Pauli, Hadamard, and CNOT gates.
Efficient decoder .
Syndrome measurement can be done with two ancillary flag qubits .Concatenations of Hamming-based CSS codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads .
Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead .
- 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\((r-2,r)\).
- \([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code — \([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code for \(p=2\) reduce to \([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS codes.
- Hamming code — Quantum Hamming codes result from applying the CSS construction to Hamming codes.
- A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
- H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, (2022) arXiv:2207.08826
- R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
“\([[2^r-1, 2^r-2r-1, 3]]\) Hamming-based CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_hamming_css