【作者】 郑凯；

【导师】 丛爽；

【作者基本信息】 中国科学技术大学 ， 控制科学与工程， 2017， 硕士

【摘要】 在量子系统中,状态是携带信息的载体。量子状态估计,也被称为量子层析,是获取量子信息的一种重要手段。基于压缩传感的量子态估计结合了压缩传感理论以及量子态估计,试图通过少量的测量值重构出量子系统状态的全部信息,该信息可用密度矩阵表示。本论文研究了基于压缩传感的量子态估计的测量矩阵分析方法,快速且高精度的重构算法,以及最少测量数目。主要研究内容为以下4部分:1.测量矩阵的研究。结合已有理论研究成果,系统地总结了基于压缩传感的量子态估计中,测量矩阵的研究准则和方法;并用这些方法对五种常用的测量矩阵进行了分析,得出测量配置下界的理论值;并运用仿真实验的方法,对五种测量矩阵重构性能进行比较,由此获得具有实际运用指导意义的最优测量值数目及对应的重构误差。2.快速重构算法研究。为了加快密度矩阵的重构,结合不动点方程和ADMM算法,提出一种专门用于基于压缩传感的量子状态估计的改进算法,称之为基于不动点方程的交替方向乘子法(FP_ADMM)。FP_ADMM通过近邻算子求解基于压缩传感的量子态估计优化问题的最优解满足的不动点方程,避免了大规模矩阵伪逆运算,从而大幅度地减少在密度矩阵重构的计算时间;另外,通过不动点方程求的解为优化问题的全局最优解,从而使得FP_ADMM算法具有更高的重构精度以及良好的鲁棒性:并且本文通过仿真实验结果验证了 FP_ADMM算法的优越性。3.具有低秩和稀疏结构的密度矩阵的重构。一些实际的物理实验中,量子系统的密度矩阵不但低秩,并且稀疏。本论文提出结合低秩与稀疏特性的密度矩阵重构模型,并从理论上证明,当测量数目满足一定条件时,该模型的解唯一且等于系统密度矩阵。另外本文通过扩展软阈值算子到复数域,提出了求解该模型的算法,利用该算法以及核磁共振装置所测量到的数据进行量子态估计,实验表明,在相同的测量比率下,该算法重构出的密度矩阵比只运用低秩特性的传统模型的算法具有更高的保真度。4.基于泡利测量的本征态估计的最优测量配置集构造方法。当待估计的量子态为本征态时,基于密度矩阵特有的先验空间结构,采用泡利矩阵直积作为测量矩阵,本论文证明此时存在一组或者多组测量配置数目为O(n)的最优测量配置集,可以精确重构出系统密度矩阵,n为比特位。相对于常规的量子层析方法,以及基于压缩传感的量子态估计方法,此时的所需要的测量数目又具有极大的降低。并且本文提出一种自下而上的最优测量配置集的构造方法。更多还原

【Abstract】 In quantum system,the state is the carrier carrying information.Quantum State Estimation,which is also known as Quantum State Tomography(QST),is an import means of obtaining quantum information.The Compressive Quantum State Estimation is a combination of the Compressive Sensing(CS)and the Quantum State Estimation.It aims to reconstruct the whole information of a quantum system through a small amount of measurements and the information can be expressed by the density matrix.This dissertation mainly studies the analysis method of measurement matrix,fast and high precision reconstruction algorithm and the minimum number of measurements in compressive quantum state estimation.The contents of researches can be divided into following four aspects:1.Study on measurement matrix.Based on the existing theoretical research,the dissertation systematically summarizes the methods of analyzing the measurement matrix in compressive quantum state estimation.By using these methods,five commonly used measurement matrices are analyzed and the lower bound of measurement settings is obtained.The theoretical optimal measurement settings obtained by comparing the performance of five measurement matrices with simulation experiment provide theoretical instructions for practical experiments.2.Study on fast reconstruction algorithm.An improved algorithm is specifically developed for compressive quantum state estimation to further accelerate the process of reconstructdensity matrix.By leveraging the fixed point equation approach to avoid the matrix inverse operation,a fixed-point alternating direction method of multipliers algorithm(FP ADMM)is proposed for compressive quantum state estimation that can handle both normal errors and large outliers in the density matrix(for which LS and MSL can be easily failed).Comparisons with other quantum estimation approaches in numerical experiment show the advantage of the proposed method.3.In recent physical experiments,It is found that many unknown density matrices are low-rank as well as sparse.Bearing this information in mind,a reconstruction algorithm is proposed that combines the low-rank and sparsity model of density matrix.It is proved that the solution of the optimization function can be and only be the density matrix satisfying the model with overwhelming probability as long as the necessary number of measurements is allowed.The solver is developed by utilizing an extended soft-threshold operator that copes with complex values.Numerical experiments of density matrix estimation for real nuclear magnetic resonance(NMR)device reveal that the proposed method achieves a better accuracy comparing to the some existing methods.4.When the quantum state to be estimated are eigenstates,based on the peculiar prior structure of the density matrix and the using of Pauli measurement,it’s proved that there are one or more measurement sets of size O(n)called the optimal measurement sets which can reconstruct the density matrix accurately.Comparing with the conventional quantum state estimation and the compressive quantum state estimation,the required minimum number of measurement settings can be greatly reduced,which can further accelerate the speed and efficiency of quantum state estimation.In this paper,a bottom-up method of constructing the optimal measurement sets is proposed,which can effectively construct the optimal measurement sets of any quantum system with any qubit.更多还原