\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming 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

Latin rectangle encoder [2].Efficient decoder [3].

Fault Tolerance

Syndrome measurement can be done with two ancillary flag qubits [4].Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [3].

Threshold

Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead [3].

Parents

  • Quantum Reed-Muller code — \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming codes are quantum Reed-Muller codes because they are formed from classical Hamming codes, which are equivalent to RM\((r-2,r)\).
  • \([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code — \([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code for \(p=2\) reduce to \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming codes.

Children

Cousins

  • Hamming code — Quantum Hamming codes result from applying the CSS construction to Hamming codes.
  • Concatenated quantum code — Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [3].
  • \([[4,2,2]]\) CSS code — Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [3].
  • \([[6,2,2]]\) \(C_6\) code — Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [3].

References

[1]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]
A. M. Steane, “Fast fault-tolerant filtering of quantum codewords”, (2004) arXiv:quant-ph/0202036
[3]
H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[4]
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:
“\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming 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={\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming 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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Cite as:

“\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming 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/edit/main/codes/quantum/qubits/stabilizer/rm/quantum_hamming_css.yml.