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.
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