Berry-phase-based quantum gates assisted by transitionless quantum driving

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.