自旋的解析李代数导数最优控制
Optimal Control of Spins by Analytical Lie Algebraic Derivatives
摘要：
在许多优化算法中，计算保真度函数的导数（梯度和Hessian）是最关键的步骤之一。在优化过程需要在许多步骤上传播这些计算的情况下，甚至更需要获得精确的方法来计算这些导数，这在尾旋系统的最佳控制中特别重要。在这里，我们提出了一种新的数值方法，ESCALADE（有效的自旋控制使用分析李代数导数），它提供了精确的第一和第二导数的保真度功能，利用的性质的李群的$2\times 2$厄米矩阵，$\mathrm{SU}(2)$，和它的李代数，李代数的斜厄米矩阵，$\mathfrak{su}(2)$。一个完整的数学处理所提出的方法以及一些数值例子。
英文摘要：
Computation of derivatives (gradient and Hessian) of a fidelity function is one of the most crucial steps in many optimization algorithms. Having access to accurate methods to calculate these derivatives is even more desired where the optimization process requires propagation of these calculations over many steps, which is in particular important in optimal control of spin systems. Here we propose a novel numerical approach, ESCALADE (Efficient Spin Control using Analytical Lie Algebraic Derivatives) that offers the exact first and second derivatives of the fidelity function by taking advantage of the properties of the Lie group of $2\times 2$ Hermitian matrices, $\mathrm{SU}(2)$, and its Lie algebra, the Lie algebra of skew-Hermitian matrices, $\mathfrak{su}(2)$. A full mathematical treatment of the proposed method along with some numerical examples are presented.
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