Real-time State Estimation and Feedback Control for n-qubit Stochastic Quantum Systems
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摘要: 研究了$n $比特随机量子系统实时状态估计及其反馈控制的问题. 对于连续弱测量(Continuous weak measurement, CWM)过程存在高斯噪声的情况, 基于在线交替方向乘子法(Online alternating direction multiplier method, OADM)推导出一种适用于$n $比特随机量子系统的实时量子状态估计算法, 即QSE-OADM (Quantum state estimation based on OADM). 运用李雅普诺夫方法设计控制律, 实现基于实时量子状态估计的反馈控制, 并证明所提控制律的收敛性. 以2比特随机量子系统为例进行数值仿真实验, 通过与基于QST-OADM (Quantum state tomography based on OADM)算法和OPG-ADMM (Online proximal gradient-based alternating direction method of multipliers)算法的量子反馈控制方案的性能对比, 验证了所提控制方案的优越性.Abstract: The problem of real-time state estimation and feedback control of $n{\text{-}}{\rm{qubit}} $ stochastic quantum systems is investigated in this paper. Considering the presence of Gaussian noises in the continuous weak measurement (CWM) process, a real-time quantum state estimation (QSE) algorithm based on the online alternating direction multiplier method (OADM), short for QSE-OADM, is deduced for $ n{\text{-}}{\rm{qubit}} $ stochastic quantum systems. The control law is designed via the Lyapunov-based approach to realize real-time quantum state estimation-based feedback control, and the convergence of the proposed control law is proved. Numerical simulations are carried out for 2-qubit stochastic quantum systems, and the superiority of the proposed control scheme is verified in comparison with the performance of quantum feedback control schemes based on the QST-OADM (quantum state tomography based on the online alternating direction multiplier method) algorithm and the OPG-ADMM (online proximal gradient-based alternating direction method of multipliers) algorithm.
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图 1 不同外部控制场的作用下的实时状态估计性能
Fig. 1 Real-time state estimation performance under various external control fields
图 2 不同初始测量算符作用下的实时状态估计性能
Fig. 2 Real-time state estimation performance under various initial measurement operators
图 3 不同的相互作用强度作用下的实时状态估计性能
Fig. 3 Real-time state estimation performance under various interaction strengths
图 4 第30次采样时2比特量子系统估计状态与真实状态比较$( H' = {H_0} + {\;}1\cdot{\sigma _x} $, $ {M_1} = {\sigma _z} \otimes {\sigma _z} $, $ L' = {\;}0.7{\sigma _z}) $
Fig. 4 Comparison between the estimated state and the real state of a 2-qubit system at the 30th sampling time $( H' = {H_0} + {\;}1\cdot{\sigma _x} $, $ {M_1} = {\sigma _z} \otimes {\sigma _z} $, $ L' = {\;}0.7{\sigma _z} )$
图 5 基于实时状态估计的n比特随机量子系统反馈控制方案的框图
Fig. 5 Real-time state estimation-based feedback control scheme for n-qubit stochastic quantum systems
图 7 叠加态反馈控制的仿真结果
Fig. 7 Simulation results on feedback control of a superposition state
表 1 测量值序列的构造方法
Table 1 Construction approach of the measurement record sequence
$y_1$ $y_2$ $ \cdots $ $y_k$ $b_1$ ${\rm{tr}}(M_1^\dagger {\rho _1})$ — $ \cdots $ — $b_2$ ${\rm{tr}}(M_2^\dagger {\rho _2})$ ${\rm{tr}}(M_1^\dagger {\rho _2})$ $ \cdots $ — $\vdots$ $\vdots$ $\vdots$ $\ddots$ $\vdots$ $b_k$ ${\rm{tr}}(M_k^\dagger {\rho _k})$ ${\rm{tr}}(M_{k - 1}^\dagger {\rho _k})$ $\cdots$ ${\rm{tr}}(M_1^\dagger {\rho _k})$ 
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表 2 本征态反馈控制性能指标的对比
Table 2 Comparison of performance indicators of feedback control of an eigenstate
指标 方案 1 方案 2 方案 3 $k_{s1}$ 14 18 25 $k_{s2}$ 15 18 28 $Fidelity(30)$(%) 99.84 99.40 98.29 $V(30)$ $5.399 \times {10^{ - 4}}$ $1.484 \times {10^{ - 3}}$ $6.312 \times {10^{ - 3}}$ $J(30)$ 18.247 28.721 52.112
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表 3 叠加态反馈控制性能指标的对比
Table 3 Comparison of performance indicators of feedback control of a superposition state
指标 方案 1 方案 2 方案 3 $k_{s1}$ 16 22 29 $k_{s2}$ 16 25 33 $Fidelity(40)$(%) 99.90 99.40 98.30 $V(40)$ $1.651 \times {10^{ - 4}}$ $6.180 \times {10^{ - 4}}$ $2.854 \times {10^{ - 3}}$ $J(40)$ 133.179 133.880 136.223
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