Fig. 2: Three-qubit entanglement generation.

a Quantum circuit for entanglement generation between three nuclear spins, one nitrogen, and two carbon isotopes. This is equivalent to the quantum circuit of encoding in three-qubit quantum error correction in Fig. 3. H, X, and Y, respectively, denote Hadamard, Pauli-X, and Pauli-Y gates. b Pulse sequence in the experiment. Nuclear spin manipulation is performed by radiofrequency (RF), and the geometric phase manipulation between the electron (e), nitrogen (N), and carbon (C) is performed by microwaves (MW) using the GRAPE algorithm. RO, Init, E_y, and A₁, respectively, denote readout, initialize and two orbital excited states. c Conceptual diagram of the holonomic controlled-phase gate in the encoding. The π phase is given to \({|110\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\) and \({|101\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\). D₀ is the zero-field splitting and HF_N is the hyperfine (HF) splitting between the electron and the nitrogen. d Classical evaluation of the GHZ entanglement generation for three qubits. Correlations are confirmed by measuring the z axis for all nitrogen and carbon nuclear spins. Since the quantum correlation of three qubits can be measured with the same technique as the quantum correlation of two qubits, it was not measured in consideration of the experimental time.