  Nonadiabatic holonomic quantum computation (NHQC) has been developed to
shorten the construction times of geometric quantum gates. However, previous
NHQC gates require the driving Hamiltonian to satisfy a set of rather
restrictive conditions, reducing the robustness of the resulting geometric
gates against control errors. Here we show that nonadiabatic geometric gates
can be constructed in an extensible way, called NHQC{\th}, for maintaining both
flexibility and robustness against certain types of noises. Consequently, this
approach makes it possible to incorporate most of the existing optimal control
methods, such as dynamical decoupling, composite pulses, and a shortcut to
adiabaticity, into the construction of single-looped geometric gates.
Furthermore, this extensible approach of geometric quantum computation can be
applied to various physical platforms such as superconducting qubits and
nitrogen-vacancy centers. Specifically, we performed numerical simulation to
show how the noise robustness in recent experimental implementations [Phys.
Rev. Lett. 119, 140503 (2017); Nat. Photonics 11, 309 (2017)] can be
significantly improved by our NHQC{\th}.approach. These results cover a large
class of new techniques combing the noise robustness of both geometric phase
and optimal control theory.
