An efficient online estimation algorithm with measurement noise for time-varying quantum states
Introduction
Quantum state estimation (QSE), also known as quantum state tomography, is a fundamental problem in quantum state preparation, quantum computation and quantum state feedback control [1], [2], [3]. The aim of QSE is to reconstruct the quantum state from a number of measurement times [4]. The state of an -qubit system usually can be described by a () density matrix, which has the physical constraints of positive semidefinite and unit-trace Hermitian. QSE can be converted to an optimization problem with the constraints and solved by numerical methods [5], [6]. The commonly used methods of QSE to measure the quantum state are based on strong, destructive measurements [7]. As a consequence, people have to prepare a large number of identical copies of the original quantum state. Moreover, the measurement apparatus need to be reconfigured at each measurement. For real-time applications, however, both measuring process and computation need to be carried out in a very short time. Thus such strong measurement is not deemed suitable for online QSE [8].
Continuous weak measurement (CWM) provides a new approach to estimate quantum states continuously [7], [9]. In the measurement process, by using CWM it is possible to gain the measurement information regarding the estimated state without disturbing it substantially, and the quantum state can be recovered by computing the ensemble averaging [10]. Due to the non-complete destructive characteristic of CWM, the online estimation of time-varying quantum states becomes feasible.
Many optimization algorithms have been developed for offline QSE. For offline estimation of a fixed quantum state, there are a series of optimization algorithms based on the alternating direction multiplier method (ADMM) [11], which is an offline computing framework that effectively solves the optimization problem with separable objective functions. Li et al. adopted ADMM to solving QSE by ignoring the constraints of quantum density matrix [12]. Considering partial quantum state constraints, Zheng et al. and Zhang et al. proposed FP-ADMM based on fixed point equation [13] and IST-ADMM based on iterative shrinkage-thresholding [14], respectively. After that, Zhang et al. proposed three improved ADMM optimization algorithms for different types of interference in the quantum system [15], [16], [17]. All the above offline QSE algorithms need to pass the same set of measurement data through multiple iterations to estimate a fixed quantum state.
For online QSE, its estimated state is a more general state of dynamic evolution. In contrast, people usually used picture transformation to simplify the online problem of estimating dynamic states to a static problem of estimating the initial state [7], [10], [18]. Throughout the online QSE process in their researches, the estimation state is always the initial state, and then the dynamic state was obtained by the evolutionary model. Yang et al. designed an online estimation algorithm for dynamic quantum states [19]. They converted the state estimation problem at each sampling time into a constrained least square problem and solved it using the convex optimization toolbox (CVX) in MATLAB [20]. Since CVX is an offline optimization tool, the algorithm proposed by Yang (called CVX-LS) was essentially a double-loop algorithm, that is, changing the optimization problem online in the outer loop, and performing offline iterative optimization in the inner loop, which is very time-consuming. In terms of the online algorithm which focuses on processing data in sequence or incremental way, Youssry et al. proposed an online quantum state learning algorithm based on matrix exponential gradient method (MEG) [21]. MEG guarantees the positive semidefiniteness of the estimated matrix by adding the gradient update rule of logarithmic and exponential operations. More recently, Zhang et al. proposed an online QSE algorithm combined with the online proximal gradient-ADMM (OPG-ADMM) framework [22]. OPG-ADMM performs an online proximal gradient update by adding a proximal term about the density matrix. Essentially, both MEG and OPG-ADMM resort first-order stochastic gradient information to estimate the quantum state density matrix online. The main difference is that OPG-ADMM adopts an adaptive learning rate and exploits the ADMM framework, while the learning rate of MEG is a constant.
Considering the real-time application of QSE (such as quantum state feedback control), the currently sampled quantum state is expected to be reconstructed instantly. It motivates us to develop a real online QSE algorithm, which is able to efficiently complete dynamic state reconstructions at each sampling time. In other words, the goal of the online QSE algorithm is to track the time-varying quantum state in real time. The task of real-time reconstruction for the dynamic state is challenging. Because in each weak measurement sampling, only one noisy measurement value can be obtained. Meanwhile, the online algorithm only performs one iteration at each estimation, and the estimated density matrix has to meet the quantum state constraints.
In this paper, we propose a novel online estimation algorithm to achieve an accurate reconstruction of dynamic quantum state from noisy measurements. The online QSE problem is divided into two subproblems that can be solved by online alternating direction multiplier method (OADM) [23] which is an online variation framework of ADMM. One subproblem is the minimization of a quadratic problem subject to quantum state constraints for estimating the dynamic and the other is the minimization of an unconstrained problem for estimating the measurement noise . Since the involved quantum state constraints make the direct solution of the subproblem of cumbersome, we simplify the solution by two steps: first solving an unconstrained subproblem, then updating the solution subject to the constraints through projection. The proposed QSE-OADM algorithm processes the measurement values in a serialized manner, and exactly solves two sub-problems in every estimation. Our proposed algorithm is used to estimate the dynamic state density matrix of multi-qubit systems, and compares its performance with existing algorithms for online QSE.
The rest of this paper is organized as follows. Section 2 introduces the stochastic open quantum system evolution model in CWM. Section 3 describes the optimization problem of online QSE with measurement noise. The QSE-OADM algorithm is proposed in Section 4. Numerical experiments are carried out in Section 5. Finally, a conclusion is drawn in Section 6.
Section snippets
Discrete evolution model of -qubit system
The open quantum system can be described by the continuous stochastic master equation in Schrödinger picture as [10]:
where denotes the quantum state density matrix; is the Hamiltonian representing the total energy of the system; is a bounded operator pertaining to the Lindblad interaction; denotes the conjugate transposition of ; is the weak measurement time; is set to 1; is the measure
Problem statement of online QSE
In order to make full use of the measurement record sequence and alleviate the computational burden in the online processing, a sliding window containing the most recent measurements is adopted in consideration. Therefore, we rewrite the measurement record sequence as:where is the size of the sliding window and represents the starting index of the window.
When the number of obtained measurements is less than the size of the window is equal to the
Online QSE Algorithm with measurement noise
We introduce the online alternating direction multiplier method (OADM) [23] to develop the online QSE algorithm. For the constrained online convex optimization problem (11) with separable two-objective variables and the basic idea of OADM is to decompose it into two sub-problems and solve them alternately. The framework of OADM is to minimize the corresponding augmented Lagrangian functions of the two primal variables in turn, and finally update the Lagrangian multiplier by dual gradient
Numerical experiments and results analysis
In this section, numerical experiments are carried out to assess online properties of the proposed QSE-OADM algorithm in the reconstruction performance of time-varying states. In the experiments, the measurement record sequence is constructed by . The true quantum state of the estimated system is generated by (2). The corresponding sampling matrix is defined by (10). For the discrete evolution model the -qubit quantum system, the parameters are set as:
Conclusion
In this paper, a novel online algorithm QSE-OADM in a noisy CWM process was developed, which processed the measurement values in a serialized manner, and exactly solved two sub-problems in every estimation. Furthermore, for achieving high estimation accuracy and improving efficiency, the sliding window of measurements was adopted. The algorithm we proposed was efficient and fast to estimate the dynamic quantum state. Numerous experiments are supportive of the potential merits of the method as
Credit Author Statement
Kun Zhang developed the measurement record sequence, the method of solving online quantum state estimation optimization problem, performed the experiments and wrote the original draft. Shuang Cong conceived and supervised the project. Kezhi Li validated the experiments and analyzed the results. All authors contributed to review the manuscript.
Declaration of Competing Interest
The authors declare no competing interests.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grants No. 61973290 and 61720106009.
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