Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator

In qubits made from a weakly anharmonic oscillator the leading source of error at short gate times is leakage of population out of the two dimensional Hilbert space that forms the qubit. In this article we develop a general scheme based on an adiabatic expansion to find pulse shapes that correct this type of error. We find a family of solutions that allows tailoring to what is practical to implement for a specific application. Our result contains and improves the previously developed derivative removal by adiabatic gate technique [F. Motzoi et al., Phys. Rev. Lett. 103, 110501 (2009)] and allows a generalization to other nonlinear oscillators with more than one leakage transition.