Optimal ultrarobust quantum gates by inverse optimization

We propose a systematic method to construct robust optimal quantum control at high orders by inverse optimization and apply it for the generation of ultrarobust optimal quantum gates. We derive explicit integrals that characterize robustness up to seventh order against pulse amplitude for time or energy minimization. At fifth order, time optimization is achieved for a flat pulse and a detuning featuring a dual-frequency oscillation. We analyze its performance and compare it to composite pulse techniques: Ultrarobust inverse optimization is considerably faster (in the range 16%--40% depending on the considered gate) than the best composite pulses with similar performance.