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

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

Description

Member of a family of self-dual CCS codes constructed from \([2^r-1,2^r-r-1,3]=C_X=C_Z\) Hamming codes and their duals the simplex codes. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix for a simplex code. The weight of each stabilizer generator is \(2^{r-1}\).

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 threshold 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 Hamming and simplex codes are both punctured RM codes.
\([[2^r-1, 2^r-2r-1, 3]]_p\) 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

\([2^r-1,2^r-r-1,3]\) Hamming code — Quantum Hamming codes result from applying the CSS construction to Hamming codes and their duals the simplex codes.
\([2^m-1,m,2^{m-1}]\) simplex code — Quantum Hamming codes result from applying the CSS construction to Hamming codes and their duals the simplex codes.
Concatenated qubit 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].
\([[4,2,2]]\) Four-qubit 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].
\([[6,2,2]]\) \(C_6\) 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].
Kitaev surface code — Quantum Hamming codes can be concatened with surface codes [5].
Concatenated qubit code — Quantum Hamming codes can be concatened with surface codes [5].
Quantum data-syndrome (QDS) code — Codes such as the quantum Hamming code can be expanded to QDS codes using almost any good binary linear code because their stabilizer generators all have the same weight [6].

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
[5]: M. Fang and D. Su, “Quantum memory based on concatenating surface codes and quantum Hamming codes”, (2024) arXiv:2407.16176
[6]: A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum Data-Syndrome Codes”, IEEE Journal on Selected Areas in Communications 38, 449 (2020) arXiv:1907.01393 DOI

Page edit log

Qingfeng (Kee) Wang (2022-01-07) — most recent
Victor V. Albert (2021-12-30)

Cite as:

“\([[2^r-1, 2^r-2r-1, 3]]\) 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.