  Accurate derivatives are important for efficiently locally traversing and
converging in quantum optimization landscapes. By deriving analytically exact
control derivatives (gradient and Hessian) for unitary control tasks, we show
here that the computational feasibility of meeting this accuracy requirement
depends on the choice of propagation scheme and problem representation. Even
when exact propagation is sufficiently cheap it is, perhaps surprisingly, much
more efficient to optimize the (appropriately) approximate propagators:
approximations in the dynamics are traded off for significant complexity
reductions in the exact derivative calculations. Importantly, past the initial
analytical considerations, only standard numerical techniques are explicitly
required with straightforward application to realistic systems. These results
are numerically verified for two concrete problems of increasing Hilbert space
dimensionality. The best schemes obtain unit fidelity to machine precision
whereas the results for other schemes are separated consistently by orders of
magnitude in computation time and in worst case 10 orders of magnitude in
achievable fidelity. Since these gaps continually increase with system size and
complexity, this methodology allows numerically efficient optimization of very
high-dimensional dynamics, e.g. in many-body contexts, operating in the
high-fidelity regime which will be published separately.
