| Summary: | A Hessian-based state-to-state optimal control method in Liouville space is presented to mitigate previously undesirable polynomial scaling of Hessian computation time. This method, an improvement to the state-of-the-art Newton-Raphson gradient ascent pulse engineering (GRAPE) method, is derived with respect to two exact time-propagator derivative calculation techniques, auxiliary matrix and efficient spin control using analytical Lie algebraic derivatives (ESCALADE) methods. We observed that compared to the best current implementation of the Newton-Raphson GRAPE method, for an ensemble of two-level systems, with realistic conditions, our auxiliary matrix and ESCALADE Hessians can be 4–200 and 70–600 times faster, respectively. Additionally, the Newton-Raphson GRAPE method using ESCALADE is presented in a Liouville space for higher-level systems and with the derivation of x-, y-, and z-control propagator derivatives, also extending the application of ESCALADE and the recent quantum optimal control by adaptive low-cost algorithm (QOALA) method for coupled systems.
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