  In quantum control, geometrical operations could provide an extra layer of
robustness against control errors. However, the conventional non-adiabatic
holonomic quantum computation (NHQC) is limited by the fact that all of the
operations require exactly the same amount of evolution time, even for a
small-angle rotation. Furthermore, NHQC confines the driving part of the
Hamiltonian to strictly cover a fixed pulse area, making it sensitive to
control errors. Here we present an unconventional approach of NHQC, termed
brachistochronic NHQC, for bypassing these limitations. Specifically, with
B-NHQC, non-Abelian geometric gates can be time-optimized by following the
brachistochrone curve, minimizing the impact from the environmental
decoherence. Additionally, we demonstrate that B-NHQC is compatible with
composed pulses, which can further enhance the robustness against pulse errors.
For benchmarking, we provide a thorough analysis on the performance of B-NHQC
under experimental conditions; we found that the gate error can be reduced by
as much as 64% compared with NHQC.
