  A general scheme for an adiabatic geometric phase gate is proposed which is
maximally robust against parameter fluctuations. While in systems with SU(2)
symmetry geometric phases are usually accompanied by dynamical phases and are
thus not robust, we show that in the more general case of a SU(2) x SU(2)
symmetry it is possible to obtain a non-vanishing geometric phase without
dynamical contributions. The scheme is illustrated for a phase gate using two
systems with dipole-dipole interactions in external laser fields which form an
effective four-level system.
