Exact Gradients and Hessians for Quantum Optimal Control and Applications in Many-Body Matrix Product States

The demand for precise control in high-fidelity regimes places increasing emphasis on the role of optimization methodologies and their performance capacities. We discuss the importance of accurate derivatives for efficiently locally traversing and converging in optimization landscapes. We find the feasibility of meeting this central requirement critically depends on the choice of propagation scheme by deriving analytically exact control derivatives. Even when exact propagation is sufficiently cheap, we find, perhaps surprisingly, that it is always more efficient to optimize the approximate propagators: approximate dynamics is traded off for complexity reductions in the exact derivative calculations. We quantitatively verify these claims for a concrete problem and find that the best schemes (Trotterization with a certain problem structure) obtain unit fidelity to machine precision, but the results from exact propagator schemes are separated consistently at least by an order of magnitude in computation time and in worst case 10 orders of magnitude in achievable fidelity. We then discuss a many-body Bose-Hubbard model in a form compliant with the best scheme in the matrix product state ansatz. With the established concepts we optimize the superfluid-Mott transition to fidelities above 0.99 and find quantum speed limit estimates beyond exact diagonalization approaches. Compared to earlier gradient-free work on similar problems, we show the simpler optimization objectives employed there are not consistent with high-fidelity requirements, and find substantial qualitative differences (bang structures) and quantitative performance improvements (orders of magnitude) and transformation times (factor three) depending on the comparative measure. We attribute the success and efficiency of our methodology to the exact derivatives, which are immediately applicable to all pure state-transfers.