发表机构
Chongqing Normal University; Universität Würzburg（重庆师范大学; 维尔茨堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Lindblad方程最优控制，提出保结构的全秩与低秩指数方法（FREM-SQH和LREM-SQH），基于Pontryagin原理和SQH求解，数值验证收敛性与低秩效率。
AI 中文摘要
由Lindblad主方程控制的开放量子系统的最优控制需要反复求解前向和伴随演化方程。对于大的希尔伯特空间，这些计算需要保结构、精确且计算高效，同时与不可微优化技术兼容的数值方法。本文提出了一个用于Lindblad方程最优控制的统一框架。连续最优性系统由Pontryagin最大值原理导出，并通过序列二次哈密顿（SQH）方法求解。其数值实现基于具有时间依赖哈密顿量的前向和伴随Lindblad方程的二阶指数中点传播子。开发了保持密度矩阵厄米和正半定结构的全秩格式，以及大幅降低存储需求和计算成本的低秩公式。为全秩和低秩的前向及伴随传播子建立了严格的误差估计。数值实验确认了预测的收敛阶，证明了低秩近似的有效性，并展示了所提出的FREM-SQH和LREM-SQH算法在具有光滑和不可微控制成本的最优控制问题中的性能。
英文摘要
Optimal control of open quantum systems governed by the Lindblad master equation requires the repeated solution of forward and adjoint evolution equations. For large Hilbert spaces, these computations demand numerical methods that are structure-preserving, accurate, and computationally efficient, while remaining compatible with nonsmooth optimization techniques. In this work, a unified framework for optimal control of Lindblad equations is presented. The continuous optimality system is derived from the Pontryagin maximum principle and solved by a sequential quadratic Hamiltonian (SQH) method. Its numerical realization is based on second-order exponential midpoint propagators for the forward and adjoint Lindblad equations with time-dependent Hamiltonians. Full-rank schemes preserving the Hermitian and positive-semidefinite structure of the density matrix are developed together with low-rank formulations that substantially reduce storage requirements and computational cost. Rigorous error estimates are established for the full- and low-rank forward and adjoint propagators. Numerical experiments confirm the predicted convergence rates, demonstrate the effectiveness of the low-rank approximations, and illustrate the performance of the proposed FREM-SQH and LREM-SQH algorithms for optimal control problems with smooth and nonsmooth control costs.
