  Nonadiabatic holonomic quantum computing is a novel and promising framework
for the implementation and efficient and robust execution of quantum gates
based on purely geometric principles. However, the parallel transport condition
that is central to nonadiabatic holonomic quantum computing has shortcomings.
In this paper, we address some of these shortcomings and show that a
projectivization of the standard gauge theory of nonadiabatic holonomic quantum
computation eliminates them. In addition, we extend the isoholonomic inequality
to projective gates and establish a minimum execution time-a quantum speed
limit-for projective holonomic quantum gates.
