The optimization of flux trajectories for the adiabatic controlled-Z gate on split-tunable transmons

In a system of two tunable-frequency qubits, it is well-known that adiabatic tuning into strong coupling-interaction regions between the qubit subspace and the rest of the Hilbert space can be used to generate an effective controlled-Z rotation. We address the problem of determining a preferable adiabatic trajectory along which the qubit frequency is tuned and apply this to the flux-tunable transmon model. The especially minimal anharmonic nature of these quantum processors makes them good candidates for qubit control using non-computational states as long as higher-level leakage is properly addressed. While the statement of this method has occurred multiple times in the literature, there have been few discussions on which trajectories may be used. We present a generalized method for optimizing parameterized families of possible flux trajectories and provide examples of use on five test families of one and two parameters.