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The holonomic approach to controlling (nitrogen-vacancy) NV-center qubitsprovides an elegant way of theoretically devising universal quantum gates thatoperate on qubits via calculable microwave pulses. There is, however, a lack ofsimulated results from the theory of holonomic control of quantum registerswith more than two qubits describing the transition between the dark states. Inlight of this, we have been experimenting with the IBM Quantum Experiencetechnology to determine the capabilities of simulating holonomic control ofNV-centers for three qubits describing an eight-level system that produces anon-Abelian geometric phase. The tunability of the geometric phase via thedetuning frequency is demonstrated through the high fidelity (about 80%) of3-qubit off-resonant holonomic gates over the on-resonant ones. The transitionbetween the dark states shows the alignment of the gate dark state with thequbits initial state hence decoherence of the multi-qubit system iswell-controlled through a 0.33pi rotation. The electron return probability canexhibit spin-orbit coupling-like behavior as observed in topological materialsbased on the extra geometric phase.

Holonomic control of a three-qubits system in an NV center using a near-term quantum computer