  A practical quantum computer must be capable of performing high fidelity
quantum gates on a set of quantum bits (qubits). In the presence of noise, the
realization of such gates poses daunting challenges. Geometric phases, which
possess intrinsic noise-tolerant features, hold the promise for performing
robust quantum computation. In particular, quantum holonomies, i.e.,
non-Abelian geometric phases, naturally lead to universal quantum computation
due to their non-commutativity. Although quantum gates based on adiabatic
holonomies have already been proposed, the slow evolution eventually
compromises qubit coherence and computational power. Here, we propose a general
approach to speed up an implementation of adiabatic holonomic gates by using
transitionless driving techniques and show how such a universal set of fast
geometric quantum gates in a superconducting circuit architecture can be
obtained in an all-geometric approach. Compared with standard non-adiabatic
holonomic quantum computation, the holonomies obtained in our approach tends
asymptotically to those of the adiabatic approach in the long run-time limit
and thus might open up a new horizon for realizing a practical quantum
computer.
