Self-dual Hamming-based CSS code that admits permutation-based CZ logical gates.
Single-qubit Clifford operations applied transversally yield the corresponding Clifford gates on one of the logical qubits . CNOT gate because it is a CSS code. Transversal CCZ gate .
CZ gates can be performed using qubit permutations, and a CCZ gate can be performed using four ancilla qubits .
Clifford gates can be performed fault-tolerantly using two ancillary flag qubits, and a CCZ gate can be performed using four ancilla qubits .
- 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 .
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- Jim Harrington and Ben W. Reichardt, “Addressable multi-qubit logic via permutations,” Talk at Southwest Quantum Information and Technology (SQuInT) (2011).
- R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
- 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
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- Victor V. Albert (2022-12-06) — most recent
“\([[15, 7, 3]]\) Hamming-based CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_15_7_3