  Realization of fast fault-tolerant quantum gates on a single spin is the core
requirement for solid-state quantum-information processing. As polarized light
shows geometric interference, spin coherence is also geometrically controlled
with light via the spin-orbit interaction. Here, we show that a geometric spin
in a degenerate subspace of a spin-1 electronic system under a zero field in a
nitrogen vacancy center in diamond allows implementation of optical
non-adiabatic holonomic quantum gates. The geometric spin under quasi-resonant
light exposure undergoes a cyclic evolution in the spin-orbit space, and
acquires a geometric phase or holonomy that results in rotations about an
arbitrary axis by any angle defined by the light polarization and detuning.
This enables universal holonomic quantum gates with a single operation. We
demonstrate a complete set of Pauli quantum gates using the geometric spin
preparation and readout techniques. The new scheme opens a path to holonomic
quantum computers and repeaters.
