Online Quantum State Estimation With Noise Or Disturbance

Online quantum state estimation is the foundation of quantum communication and quantum computing,and it is also an important prerequisite for information security and efficient computing.Using continuous weak measurement technique,one can reconstruct and track dynamically evolving quantum states in real time through solving a series of convex optimization problems.This dissertation mainly aims at the online quantum state estimation problem with noise or disturbance,proposes different online estimation algorithms respectively and compares the performances through simulation experiments to verify the effectiveness and superiority of our proposed algorithms.The main research content of this dissertation are:1.The Gaussian noise in the continuous weak measurement process of quantum system is studied,and an online quantum state estimation algorithm with adaptive learning rate matrix exponentiated gradient is proposed.For the online quantum state estimation problem with Gaussian noise in the measurement results,we convert it into a convex optimization problem about the density matrix to be measured with quantum state constraints.By introducing Von Neumann divergence(quantum relative entropy),the exponential quantum state iteration formula is derived under the first-order optimal condition,which ensures the positive semide finite and conjugate symmetry of the quantum state density matrix,and then by projecting the iteration result with one trace,the final quantum state estimate is obtained.Numerical simulation experiments of our proposed algorithm are carried out on 1,2,3 and 4 qubit systems respectively.Experiment results show that compared with the existing algorithms,our proposed algorithm has better rapid convergence and higher state estimation accuracy.2.The sparse disturbance of quantum system in the process of continuous weak measurement is studied,and an online quantum state estimation algorithm based on Fast Iterative Threshold Shrinkage Method is proposed.Sparse disturbance introduces outliers at some random positions in the density matrix.Mathematically,the online estimation problem which considers sparse disturbance can be converted into a bivariate convex optimization problem about the density matrix to be estimated and the disturbance matrix,which is decomposed by the Online Alternating Direction Multiplier Method.After decomposition,the subproblem about the state to be estimated is to minimize the ellipsoidal norm of the density matrix with quantum state constraints;subproblem about the sparse disturbance is to minimize the ellipsoidal norm and the l1 norm of the disturbance.Among them,the subproblem of sparse disturbance includes continuously differentiable ellipsoidal norm square term and discontinuous differentiable l1 norm,which can be transformed into the form of the optimization problem that can be solved by the soft threshold function.We use Fast Iterative Soft Threshold Algorithm(FISTA)to solve it.And the estimated system state is solved through the first-order optimal condition.Simulation experiments verify the superiority of our proposed algorithm.3.Study the online quantum state filtering problem with both Gaussian noise and sparse disturbance,an online quantum state filter(OQSF)is proposed in this dissertation for recovering the quantum density matrix from continuous weak measurement(CWM)in real time.There exists sparse disturbances on the quantum density matrix and Gaussian noise in the measurements.Mathematically,the problem is transformed into minimizing the ellipsoidal norm of the density matrix,the l2 norm of Gaussian noise and the l1 norm of sparse disturbance with quantum state constraints.Based on the Online Alternating Direction Multiplier Method,the quantum state estimation problem is decomposed into two subproblems,the sparse disturbance and density matrix estimation variables are configured into the same subproblem,which can be solved by first-order optimal condition and ISTA.In a complex measurement environment with disturbances and noise,higher estimation accuracy and operating efficiency of OQSF is shown,by using sliding window and processing data in stream.The superiority of OQSF is demonstrated in the numerical experiments compared with different state-of-the-art methods.