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International Conference on Intelligent Control and Information Processing

Efficient and Fast Optimization Algorithms for Quantum State Filtering and Estimation

Kun Zhang, Shuang Cong, Jiao Ding, Jiaojiao Zhang, Kezhi Li

Published Dec 1, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.1109/icicip47338.2019.9012097

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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.
Based on the theoretical lower bounds of the number of measurements given by Compressive Sensing (CS), the quantum state estimation task can be formulated as a convex optimization problem subject to positive semi-definite and unit-trace Hermitian constraints. This paper proposes an improved Alternating Direction Multiplier Method (AD-MM) for high-dimensional quantum state reconstruction. The proposed method not only greatly reduces the computational complexity and speeds up the convergence, but also has a strong robustness to the noise. For the density matrix reconstruction of 11-qubit quantum system, the proposed algorithm achieves 4.54e-2 reconstruction error using 0.6% of total measurement data and 99.26 seconds.
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.
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.
The possible state space dimension increases exponentially with respect to the number of qubits. This feature makes the quantum state tomography expensive and impractical for identifying the state of merely several qubits. The recent developed approach, compressed sensing, gives us an alternative to estimate the quantum state with fewer measurements. It is proved that the estimation then can be converted to a convex optimization problem with quantum mechanics constraints. In this paper we present an alternating augmented Lagrangian method for quantum convex optimization problem aiming for recovering pure or near pure quantum states corrupted by sparse noise given observables and the expectation values of the measurements. The proposed algorithm is much faster, robust to outlier noises (even very large for some entries) and can solve the reconstruction problem distributively. The simulations verify the superiority of the proposed algorithm and compare it to the conventional least square and compressive quantum tomography using Dantzig method.
A pure or nearly pure quantum state can be described as a low-rank density matrix, which is a positive semidefinite and unit-trace Hermitian. We consider the problem of recovering such a low-rank density matrix contaminated by sparse components, from a small set of linear measurements. This quantum state estimation task can be formulated as a robust principal component analysis (RPCA) problem subject to positive semidefinite and unit-trace Hermitian constraints. We propose an efficient and fast inexact alternating direction method of multipliers (I-ADMM), in which the subproblems are solved inexactly and hence have closed-form solutions. We prove global convergence of the proposed I-ADMM, and the theoretical result provides a guideline for parameter setting. Numerical experiments show that the proposed I-ADMM can recover state density matrices of 5 qubits on a laptop in 0.69 s, with 6 × 10-4accuracy (99.38% fidelity) using 30% compressive sensing measurements, which outperforms existing algorithms.
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.