Optimization Algorithm And Its Convergence Of Quantum State Estimation And Filtering Via Compressed Sensing

Quantum state estimation,also known as quantum state tomography,is the foun-dation of quantum information.Combining with compressive sensing theory,one can reconstruct a quantum state density matrix with fewer measurements.This disserta-tion studies the problem of quantum state estimation and filtering based on compres-sive sensing under different disturbance conditions,propose efficient optimization al-gorithms and strictly prove the global convergence.The main research contents consist of the following three aspects:1.Study the problem of quantum state estimation with sparse disturbance,propose an inexact ADMM and prove its convergence.The disturbance of the state itself intro-duces sparse outliers in elements of certain locations in the density matrix.The task of the quantum state estimation is to reconstruct the density matrix from the measure-ments affected by the sparse disturbance.Mathematically,this task can be transformed into an optimization problem of robust principal component analysis with quantum state constraints.Using the alternating direction method of multipliers,we decompose the original problem into two subproblems.One subproblem minimizes the nuclear norm of the density matrix with quantum state constraints,and the other subproblem minimizes l1 for sparse disturbance.Since there are no closed-form solutions for the two subprob-lems,the exact solutions cost largely.We propose an efficient and fast inexact alternat-ing direction method of multipliers,I-ADMM for short,which solves the subproblems inexactly to obtain closed-form solutions and reduces computational complexity.In addition,we update Lagrangian multipliers with adjustable step size to achieve faster convergence speed.We strictly prove the global convergence of the proposed algo-rithm and give the rules for selecting parameters.Simulations verify the superiority of the proposed I-ADMM.2.Study the problem of high-dimensional quantum state estimation with Gaussian noise,propose an improved ADMM to estimate 11-bit quantum state.When Gaussian noise is considered in the measurements,we transform the quantum state estimation into a convex optimization problem with quantum state constraints.That is,the estimated density matrix must be positive semidefinite,symmetric,and unite trace.In order to es-timate the high-dimensional quantum state,we set the number of measurements to the lower bounds according to the compressed sensing theory.We propose an improved alternating direction method of multipliers,which decompose the original problem into two subproblems,minimizing the nuclear norm of density matrix with quantum state constraints and minimizing the l2 norm of Gaussian noise.The proposed algorithm in-exactly solve the subproblem in regard to density matrix,thereby avoiding large-scale matrix inversion operations and reducing computational complexity.Also,by changing the operation sequence,the amount of calculations can be further reduced.In addition,we update Lagrangian multipliers with adjustable step size to achieve faster conver-gence speeds.Simulation experiments verify the superiority of the proposed algorithm for estimating high-dimensional quantum states.3.Study the problem of quantum state filtering with state disturbance and mea-surement noise,propose a quantum state filter based on proximal Jacobian ADMM and prove its convergence.Considering state disturbance and measurement noise simulta-neously,we design an efficient and convergent quantum state filter which filters state disturbance and measurement noise while estimating the quantum state.Mathemati-cally,we transform the quantum state filtering problem into minimizing the nuclear norm of the density matrix,the l1 norm of sparse disturbance,and the l2 norm of Gaus-sian noise with linear measurement constraints and quantum state constraints.We in-troduce proximal Jacobian ADMM to solve the problem of quantum state filtering.The algorithm decomposes the original problem into three sub-problems.An proximal term is added to each sub-problem to correct error timely.We give a selection method for the parameters of the proximal terms to simplify the solving of the sub-problems.In addition,we update Lagrangian multipliers with adjustable step size to speed up con-vergence.We strictly prove the convergence of the proposed filter and provide rulers of selecting parameters.Simulation experiments verify the superiority of the proposed filter.