\([[15, 7, 3]]\) quantum Hamming code[13] 

Description

Self-dual quantum Hamming code that admits permutation-based CZ logical gates. The code is constructed using the CSS construction from the \([15,11,3]\) Hamming code and its \([15,4,8]\) dual code.

Transversal Gates

CNOT gate because it is a CSS code.Single-qubit Clifford operations applied transversally yield the corresponding Clifford gates on one of the logical qubits [4].Automorphism groups of the underlying classical codes can yield transversal Clifford gates when combined with qubit permutations [5; Sec. IV.A].Transversal CCZ gate [6].

Gates

CZ gates can be performed using qubit permutations, and a CCZ gate can be performed using four ancilla qubits [4].

Fault Tolerance

Clifford gates can be performed fault-tolerantly using two ancillary flag qubits, and a CCZ gate can be performed using four ancilla qubits [4].

Parent

Cousins

  • Perfect quantum code — \([[15, 7, 3]]\) quantum Hamming code is perfect as a CSS code, i.e., the number of its \(Z\)-type syndromes matches the number of \(X\)-type Pauli errors up to weight one [4].
  • \([[15,1,3]]\) quantum Reed-Muller code — Gauging six of the seven logical qubits of the \([[15,7,3]]\) code yields the \([[15,1,3]]\) code [7].

References

[1]
A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996) arXiv:quant-ph/9512032 DOI
[2]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[3]
Jim Harrington and Ben W. Reichardt, “Addressable multi-qubit logic via permutations,” Talk at Southwest Quantum Information and Technology (SQuInT) (2011).
[4]
R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
[5]
M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1302.1035 DOI
[6]
A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013) arXiv:1304.3709 DOI
[7]
A. Kubica and M. E. Beverland, “Universal transversal gates with color codes: A simplified approach”, Physical Review A 91, (2015) arXiv:1410.0069 DOI
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Zoo Code ID: stab_15_7_3

Cite as:
\([[15, 7, 3]]\) quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_15_7_3
BibTeX:
@incollection{eczoo_stab_15_7_3, title={\([[15, 7, 3]]\) quantum Hamming code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_15_7_3} }
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Cite as:

\([[15, 7, 3]]\) quantum Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_15_7_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/rm/stab_15_7_3.yml.