  We propose a novel proposal for geometric quantum gates using three- or
two-level systems, in which a controllable variable, the detuning between the
driving frequency and the atomic energy spacing, is introduced to realize
geometric transformations. In particular, we can have two instantaneous
eigenstates with opposite eigenvalues constituting a closed loop in the
parameter space. The accumulated dynamical phase is then exactly cancelled when
the loop is completed, which is beyond the traditional parallel-transport
restriction. We apply the transitionless quantum driving, which renders
revisions in both amplitudes and phases of the driving fields, to enhance the
speed and the fidelity of geometric transformation in both universal
single-qubit gates and nontrivial double-qubit gates. Gate fidelity under
decoherence is also estimated.
