Description
Self-dual Hamming-based CSS code that admits permutation-based CZ logical gates.
Transversal Gates
Single-qubit Clifford operations applied transversally yield the corresponding Clifford gates on one of the logical qubits [3]. CNOT gate because it is a CSS code. Transversal CCZ gate [4].
Gates
CZ gates can be performed using qubit permutations, and a CCZ gate can be performed using four ancilla qubits [3].
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 [3].
Parent
Cousin
- Perfect quantum code — \([[15, 7, 3]]\) Hamming-based CSS 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 [3].
References
- [1]
- A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
- [2]
- Jim Harrington and Ben W. Reichardt, “Addressable multi-qubit logic via permutations,” Talk at Southwest Quantum Information and Technology (SQuInT) (2011).
- [3]
- R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
- [4]
- 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
Page edit log
- Victor V. Albert (2022-12-06) — most recent
Cite as:
“\([[15, 7, 3]]\) Hamming-based CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_15_7_3