Simultaneous gates in frequency-crowded multi-level systems using fast, robust analytic control shapes

We present a few-parameter ansatz for pulses to implement arbitrary simultaneous single-qubit rotations in frequency-crowded multi-level systems. Specifically, we consider a system of two qutrits whose working and leakage transitions suffer from spectral crowding (detuned by $\delta$). In order to achieve precise controllability, we make use of two driving fields (each having two quadratures) at two different tones to implement arbitrary simultaneous rotations. Expanding the waveforms in terms of Hanning windows, we show how analytic pulses containing smooth and composite-pulse features can easily achieve gate errors $<10^{-4}$ and considerably outperform known adiabatic techniques. Moreover, we find a generalization of the WahWah method by Schutjens et al. [Phys. Rev. A 88, 052330 (2013)] that allows precise separate single-qubit rotations for all gate times beyond a quantum speed limit. We find in all cases a quantum speed limit $\sim 2\pi/\delta$ for the gate time and show that our pulses are robust against variations in system parameters and filtering due to transfer functions, making them suitable for experimental implementations.