  Hole-spins localized in semiconductor structures, such as quantum dots or
defects, serve to the realization of efficient gate-tunable solid-state quantum
bits. Here we study two electrically driven spin $3/2$ holes coupled to the
electromagnetic field of a microwave cavity. We show that the interplay between
the non-Abelian Berry phases generated by local time-dependent electrical
fields and the shared cavity photons allows for fast manipulation, detection,
and long-range entanglement of the hole-spin qubits in the absence of any
external magnetic field. Owing to its geometrical structure, such a scheme is
more robust against external noises than the conventional hole-spin qubit
implementations. These results suggest that hole-spins are favorable qubits for
scalable quantum computing by purely electrical means.
