Studies On The Theory And Optimization Algorithm Of Compressive Quantum States Estimation

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.