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IEEE Transactions on Automatic Control

Quantum State Filter With Disturbance and Noise

Jiaojiao Zhang, Shuang Cong, Qing Hua Ling, Kezhi Li, HERSCHEL A. RABITZ

Published Aug 12, 2019• 10 citations
Publisher

Abstract

A quantum state filter (QSF) is proposed in this paper to estimate a low-rank quantum density matrix from informationally incomplete and contaminated measurements. There exist sparse disturbances on the quantum density matrix and Gaussian noise in the measurements. A proximal Jacobian variant of the alternating direction method of multipliers (PJ-ADMM) is proposed to design the QSF. The closed-form solutions to three resulting subproblems are given and the iterative QSF is developed. The proposed QSF is proved to be convergent and its superiority is demonstrated in the numerical illustrations compared with different state-of-the-art methods.

DOI
10.1109/tac.2019.2934755

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A quantum state filter (QSF) is proposed in this paper to estimate a low-rank quantum density matrix from informationally incomplete and contaminated measurements. There exist sparse disturbances on the quantum density matrix and Gaussian noise in the measurements. A proximal Jacobian variant of the alternating direction method of multipliers (PJ-ADMM) is proposed to design the QSF. The closed-form solutions to three resulting subproblems are given and the iterative QSF is developed. The proposed QSF is proved to be convergent and its superiority is demonstrated in the numerical illustrations compared with different state-of-the-art methods.
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In this paper, based on Alternating Direction Multiplier Method (ADMM) and Compressed Sensing (CS), we develop three types of novel convex optimization algorithms for the quantum state estimation and filtering. Considering sparse state disturbance and measurement noise simultaneously, we propose a quantum state filtering algorithm. At the same time, the quantum state estimation algorithms for either sparse state disturbance or measurement noise are proposed, respectively. Contrast with other algorithms in literature, simulation experiments verify that all three algorithms have low computational complexity, fast convergence speed and high estimation accuracy at lower measurement rates.
An online quantum state filter (OQSF) optimization algorithm in the existence of the sparse disturbance and Gaussian noise is proposed in this paper. The optimization algorithm is based on the continuous weak measurements (CWM) and the online measurement record sequence constructed. The OQSF problem is decomposed into three subproblems, and three variables are estimated in real-time at the same time. The convergence OQSF algorithm proposed is strictly proved by means of two lemmas proof of the lower bound function of the distance between the optimal solution and original one having sufficient decreasing space. By comparing the performances with existing algorithms, the developed OQSF algorithm proposed demonstrates advantages in the numerical performance comparisons.
The dimension of the density matrix of a quantum system increases with the qubits of the quantum system, which makes the quantum state estimation time consuming and requires huge computation. In order to reduce the computational time in quantum state estimation, the problem of quantum state estimation based on compressive sensing is changed to the optimization problem with error constraint. In this paper, an improved Alternating Direction Method of Multipliers (ADMM) algorithm is proposed to design the optimization scheme of solving the pure state of quantum state estimation in the cases of with and without external noise. The experiments are implemented in the MATLAB. The comparison results between adaptive and fixed weight value indicate that the improved algorithm has better performances in both aspects of estimation accuracy and robustness to external disturbances. We also extend the quantum state estimation to the qubits of six and seven.
In this paper, we propose an online optimization algorithm for estimating state density in free evolution quantum systems from noisy continuous weak measurements. The problem is formulated via sparsity-promoting semidefinite programming, and an online quantum state estimation algorithm is developed based on online proximal gradient and alternating direction multiplier method. The proposed algorithm is computationally efficient and further features high robustness to measurement noise. The merits of the approach are illustrated by numerical experiments in 1-, 2-, 3-, and 4-qubit systems.
Reconstructing high-dimensional quantum state accurately with noise is a challenging problem, because of the probabilistic measurements of quantum states and the exponential growth of the computation in terms of the number of qubits. In this paper, an improved Alternating Direction Multiplier Method (ADMM) is proposed for the quantum state estimation based on the lower limit of the measurement rate inferred by Compressive Sensing (CS) theory. The proposed algorithm solves the density matrix correlation subproblems by introducing a proximal gradient step to avoid large-scale matrix inversion. Furthermore, it reduces the computational complexity by changing the order of operations. The algorithm we proposed can dramatically reduces the reconstruction time on the premise of achieving high reconstruction accuracy. When the number of qubits n = 14 and sampling rate η = 0.0819%, it takes about 3 hours to reconstruct the density matrix of a pure quantum state with the reconstruction error of 4.97e-2 and fidelity of 98.79%.