  We demonstrate how non-Abelian geometric phases can be used to universally
process a spin qubit in heavy hole quantum dots in the absence of magnetic
fields. A time dependent electric quadrupole field is used to perform any
desired single qubit operation by virtue of non-Abelian holonomy. During the
proposed operations, the degeneracy of the time dependent two level system
representing the qubit is not split. Since time reversal symmetry is preserved
and hyperfine coupling is known to be weak in spin qubits based on heavy holes,
we expect very long coherence times in the proposed setup.
