  The ability to perform fast, high-fidelity entangling gates is an important
requirement for a viable quantum processor. In practice, achieving fast gates
often comes with the penalty of strong-drive effects that are not captured by
the rotating-wave approximation. These effects can be analyzed in simulations
of the gate protocol, but those are computationally costly and often hide the
physics at play. Here, we show how to efficiently extract gate parameters by
directly solving a Floquet eigenproblem using exact numerics and a perturbative
analytical approach. As an example application of this toolkit, we study the
space of parametric gates generated between two fixed-frequency transmon qubits
connected by a parametrically driven coupler. Our analytical treatment, based
on time-dependent Schrieffer-Wolff perturbation theory, yields closed-form
expressions for gate frequencies and spurious interactions, and is valid for
strong drives. From these calculations, we identify optimal regimes of
operation for different types of gates including $i$SWAP, controlled-Z, and
CNOT. These analytical results are supplemented by numerical Floquet
computations from which we directly extract drive-dependent gate parameters.
This approach has a considerable computational advantage over full simulations
of time evolutions. More generally, our combined analytical and numerical
strategy allows us to characterize two-qubit gates involving parametrically
driven interactions, and can be applied to gate optimization and cross-talk
mitigation such as the cancellation of unwanted ZZ interactions in multi-qubit
architectures.
