优化和量子控制的梯度流：基础和应用
Gradient Flows for Optimisation and Quantum Control: Foundations and Applications
摘要：
为了解决有限维量子系统的优化任务，我们给出了一个全面的基础上的梯度流黎曼流形，包括新的发展：我们延长了以前的结果从李群，如全酉群封闭子群，如分区因式分解成张量积，其中最好的分区由纯粹的本地酉操作。此外，公共框架保持足够的一般性，并允许在（子）流形，李群，商群和约化齐性空间上建立梯度流。讨论了相关的收敛条件，旨在为最近和新的成果提供依据，并为进一步的研究奠定基础。（a）在抽象层次上，梯度流可以建立相关质量函数的精确上界，即在状态空间动力学的基本流形内可达到的上界;（b）在涉及具体实验设置的第二阶段中，可以建立分段恒定控制幅度$\R^m$的流形上的梯度流，以提供（近似）在现实条件下的量子设备的最优控制。说明性的例子和新的应用有关的距离措施的纯态纠缠。我们建立了高阶张量的最佳秩1近似的对应关系。
英文摘要：
For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such as the full unitary group to closed subgroups like partitionings by factorisation into tensor products, where the finest partitioning consists of purely local unitary operations. Moreover, the common framework is kept sufficiently general and allows for setting up gradient flows on (sub-)manifolds, Lie (sub-)groups, quotient groups, and reductive homogeneous spaces. Relevant convergence conditions are discussed meant to serve as justification for recent and new achievements, and as foundation for further research. Exploiting the differential geometry of quantum dynamics under different scenarios helps to provide highly useful algorithms: (a) On an abstract level, gradient flows may establish the exact upper bounds of pertinent quality functions, i.e. upper bounds reachable within the underlying manifold of the state space dynamics; (b) in a second stage referring to a concrete experimental setting, gradient flows on the manifold of piecewise constant control amplitudes $\R^m$ may be set up to provide (approximations to) optimal control of quantum devices under realistic conditions. Illustrative examples and new applications are given relating to distance measures of pure-state entanglement. We establish the correspondence to best rank-1 approximations of higher-order tensors.
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