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 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.