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 out 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
Page edit log
- Victor V. Albert (2022-12-06) — most recent
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