Robust optimal control of two-level quantum systems

We investigate the time- and energy-minimum optimal solutions for the robust control of two-level quantum systems against offset or control-field uncertainties. Using the Pontryagin maximum principle, we derive the global optimal pulses for the first robustness orders. We show that the dimension of the control landscape is lower than or equal to $2N$ for a field robust to the $N\mathrm{th}$ order, which leads to an estimate of its complexity.