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Computation of derivatives (gradient and Hessian) of a fidelity function isone of the most crucial steps in many optimization algorithms. Having access toaccurate methods to calculate these derivatives is even more desired where theoptimization process requires propagation of these calculations over manysteps, which is in particular important in optimal control of spin systems.Here we propose a novel numerical approach, ESCALADE (Efficient Spin Controlusing Analytical Lie Algebraic Derivatives) that offers the exact first andsecond derivatives of the fidelity function by taking advantage of theproperties of the Lie group of $2\times 2$ Hermitian matrices,$\mathrm{SU}(2)$, and its Lie algebra, the Lie algebra of skew-Hermitianmatrices, $\mathfrak{su}(2)$. A full mathematical treatment of the proposedmethod along with some numerical examples are presented.

Optimal Control of Spins by Analytical Lie Algebraic Derivatives