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分别对压缩传感理论、优化算法及其在系统状态重构中的应用3个方面进行了研究.在压缩传感理论方面,包括对所压缩信号的稀疏或低秩要求、编码测量以及与优化算法之间的关系进行了较为深入的研究,重点分析了原始信号的稀疏与低秩之间的关系、测量矩阵与压缩矩阵之间的关系、满足限制等距特性(RIP)的测量矩阵,以及由压缩传感理论提供的最少测量次数.在压缩信号重构过程中所需要采用的优化算法,着重讨论了核函数的凸优化问题描述,分别对常用的优化算法,包括最小二乘(LS)法、最大熵法、极大似然法和贝叶斯方法的求解过程中所用到的性能指标、优化目标和求解条件等进行了归纳与特性分析.对量子态估计中的交替方向乘子法(ADMM)以及作者最新提出的迭代阈值收缩法(IST)进行了专门的性能对比,并通过量子位分别5、6和7情况下纯态估计的应用为例,对不同测量比率对参数估计性能的影响,以及算法在不同量子位数下性能的表现,进行了不同层次上的对比和分析,完整地阐述基于压缩传感理论与优化的系统参数估计的研究过程.
The compression sensing theory, the optimization algorithm and its application in the system state reconstruction are respectively studied in the aspects of compressive sensing theory, including the sparse or low rank requirements of the compressed signal, the coding measurement and the optimization The relationship between the algorithm and the algorithm is studied in more depth. The relationship between the sparse and low rank of the original signal, the relationship between the measurement matrix and the compression matrix, the measurement matrix that satisfies the restriction of isometric characteristics (RIP), and The least number of measurements provided by the compression sensing theory.Optimized algorithm which is used in the process of reconstruction of compressed signal, the convex optimization problem description of kernel function is emphatically discussed, and the commonly used optimization algorithms including least-squares (LS) Method, maximum entropy method, maximum likelihood method and Bayesian method, the optimization target and the solution conditions are summarized and analyzed. The alternating direction multiplier method in quantum state estimation (ADMM) and the latest proposed iterative thresholding method (IST) for a specific performance comparison, and through the quantum bit 5,6 and 7 respectively, the case of pure state estimation as an example , The effect of different measurement rates on the performance of parameter estimation and the performance of the algorithm under different quantum bit numbers are compared and analyzed at different levels to fully describe the system parameter estimation based on the theory of compressive sensing and optimization process.