  The holonomic approach to controlling (nitrogen-vacancy) NV-center qubits
provides an elegant way of theoretically devising universal quantum gates that
operate on qubits via calculable microwave pulses. There is, however, a lack of
simulated results from the theory of holonomic control of quantum registers
with more than two qubits describing the transition between the dark states. In
light of this, we have been experimenting with the IBM Quantum Experience
technology to determine the capabilities of simulating holonomic control of
NV-centers for three qubits describing an eight-level system that produces a
non-Abelian geometric phase. The tunability of the geometric phase via the
detuning frequency is demonstrated through the high fidelity (about 80%) of
3-qubit off-resonant holonomic gates over the on-resonant ones. The transition
between the dark states shows the alignment of the gate dark state with the
qubits initial state hence decoherence of the multi-qubit system is
well-controlled through a 0.33pi rotation. The electron return probability can
exhibit spin-orbit coupling-like behavior as observed in topological materials
based on the extra geometric phase.
