  The controls enacting logical operations on quantum systems are described by
time-dependent Hamiltonians that often include rapid oscillations. In order to
accurately capture the resulting time dynamics in numerical simulations, a very
small integration time step is required, which can severely impact the
simulation run-time. Here, we introduce a semi-analytic method based on the
Dyson expansion that allows us to time-evolve driven quantum systems much
faster than standard numerical integrators. This solver, which we name Dysolve,
efficiently captures the effect of the highly oscillatory terms in the system
Hamiltonian, significantly reducing the simulation's run time as well as its
sensitivity to the time-step size. Furthermore, this solver provides the exact
derivative of the time-evolution operator with respect to the drive amplitudes.
This key feature allows for optimal control in the limit of strong drives and
goes beyond common pulse-optimization approaches that rely on rotating-wave
approximations. As an illustration of our method, we show results of the
optimization of a two-qubit gate using transmon qubits in the circuit QED
architecture.
