Robust control of single-qubit gates at the quantum speed limit

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 robust time-optimal control problem that aims at the best balance. Based on the Taylor expansion of the system in terms of uncertainty parameters, 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 order of series truncation. The gradient-descent algorithm is then introduced to sequentially seek QSLs corresponding to different orders of robustness. Numerical simulations are carried out for single-qubit systems with frequency and field amplitude uncertainties, and the obtained time-optimal control pulses can effectively suppress gate errors to the prescribed robustness order. These results provide a practical guide for selecting pulse lengths in the pulse-level compilation of quantum circuits.