Electrical two-qubit gates within a pair of clock-qubit magnetic molecules

Enhanced coherence in HoW$_{10}$ molecular spin qubits has been demonstrated by use of Clock Transitions (CTs). More recently it was shown that, while operating at the CTs, it was possible to use an electrical field to selectively address HoW$_{10}$ molecules pointing in a given direction, within a crystal that contains two kinds of identical but inversion-related molecules. Herein we theoretically explore the possibility of employing the electric field to effect entangling two-qubit quantum gates among two neighbouring CT-protected HoW$_{10}$ qubits within a diluted crystal. We estimate the thermal evolution of $T_1$, $T_2$, find that CTs are also optimal operating points from the point of view of phonons, and lay out how to combine a sequence of microwave and electric field pulses to achieve coherent control within a 2-qubit operating space that is protected both from spin-bath and from phonon-bath decoherence. Finally, we found a highly protected 1-qubit subspace resulting from the interaction between two clock molecules.