Non-adiabatic holonomic quantum computation

We develop a non-adiabatic generalization of holonomic quantum computation in which high-speed universal quantum gates can be realized using non-Abelian geometric phases. We show how a set of non-adiabatic holonomic one-and two-qubit gates can be implemented by utilizing optical transitions in a generic three-level 3 configuration. Our scheme opens up the possibility of realizing universal holonomic quantum computation on qubits characterized by short coherence time.