Approximate Dynamics Lead to More Optimal Control: Efficient Exact Derivatives

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.