  Fastness and robustness are both critical in the implementation of
high-fidelity gates for quantum computation, but in practice, a trade-off has
to be made between them. In this paper, we investigate the underlying robust
time-optimal control problem so as to make the best balance. Based on the
Taylor expansion of the system's unitary propagator, we formulate the design
problem as the optimal control of an augmented finite-dimensional system at its
quantum speed limit (QSL), where the robustness is graded by the degree of
series truncation. The gradient-descent algorithm is then introduced to
sequentially seek QSLs corresponding to different orders of robustness.
Numerical simulations for single-qubit systems show that the obtained
time-optimal control pulses can effectively suppress gate errors (to the
prescribed robustness order) caused by qubit frequency and field amplitude
uncertainties. These results provide a practical guide for selecting pulse
lengths in the pulse-level compilation of quantum circuits.
