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

Efficient decoder [2].

Fault Tolerance

Syndrome measurement can be done with two ancillary flag qubits [3].Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [2,4]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [5].

Threshold

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

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 \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [2,4]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [5]. Quantum Hamming codes can also be concatenated with surface codes [6].
  • \([[4,2,2]]\) Four-qubit code— Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [2,4]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [5].
  • \([[6,2,2]]\) \(C_6\) code— Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [2,4]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [5].
  • Kitaev surface code— Quantum Hamming codes can be concatenated with surface codes [6].
  • 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 [7].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Quantum rainbow code
Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum
Quantum Reed-Muller codeCSS Stabilizer Hamiltonian-based QECC Quantum
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 codeCSS Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum
\([[2^r-1, 2^r-2r-1, 3]]_p\) prime-qudit CSS code for \(p=2\) reduce to \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming codes.
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code
Children
\([[7,1,3]]\) Steane code
\([[15, 7, 3]]\) quantum Hamming code

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”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[3]
R. Chao and B. W. Reichardt, “Quantum Error Correction with Only Two Extra Qubits”, Physical Review Letters 121, (2018) arXiv:1705.02329 DOI
[4]
S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
[5]
C. Gidney and T. Bergamaschi, “A Constant Rate Quantum Computer on a Line”, (2025) arXiv:2502.16132
[6]
M. Fang and D. Su, “Quantum memory based on concatenating surface codes and quantum Hamming codes”, (2025) arXiv:2407.16176
[7]
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
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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.