  Nonadiabatic holonomic quantum computation uses non-Abelian geometric phases
to implement a universal set of quantum gates that are robust against control
imperfections and decoherence. Until now, a number of three-level-based schemes
of nonadiabatic holonomic computation have been put forward, and several of
them have been experimentally realized. However, all these works are based on
the same class of nonadiabatic paths, which originates from the first
nonadiabatic holonomic proposal. Here, we propose a universal set of
nonadiabatic holonomic gates based on an extended class of nonadiabatic paths.
We find that nonadiabatic holonomic gates can be realized with paths shorter
than the known ones, which provides the possibility of realizing nonadiabatic
holonomic gates with less exposure to decoherence. Furthermore, inspired by the
form of this new type of paths, we find a way to eliminate decoherence from
nonadiabatic holonomic gates without resorting to redundancies.
