Considering the presence of measurement noise in the continuous weak measurement process, the optimization problem of online quantum state tomography (QST) with corresponding constraints is formulated. Based on the online alternating direction multiplier method (OADM) and the continuous weak measurement (CWM), an online QST algorithm (QST-OADM) is designed and derived. Specifically, the online QST problem is divided into two subproblems about the quantum state and the measurement noise. The proposed algorithm adopts adaptive learning rate and reduces the computational complexity to \({\mathscr {O}}(d^3)\), which provides a more efficient mechanism for real-time quantum state tomography. Compared with most existing algorithms of online QST based on CWM which require time-consuming iterations in each estimation, the proposed QST-OADM can exactly solve two subproblems at each sampling. The merits of the proposed algorithm are demonstrated in the numerical experiments of online QST for 1-, 2-, 3-, and 4-qubit systems.

考虑到连续弱测量过程中存在测量噪声，提出了具有相应约束条件的在线量子状态层析成像（QST）的优化问题。基于在线交替方向乘数法（OADM）和连续弱测量（CWM），设计并推导了在线QST算法（QST-OADM）。具体而言，在线QST问题分为关于量子态和测量噪声的两个子问题。提出的算法采用自适应学习率，将计算复杂度降低到\（{\ mathscr {O}}（d ^ 3）\），它为实时量子状态层析成像提供了一种更有效的机制。与大多数现有的基于CWM的在线QST算法相比，在每次估计中都需要耗时的迭代，所提出的QST-OADM可以在每次采样时准确地解决两个子问题。在针对1、2、3和4量子位系统的在线QST的数值实验中证明了该算法的优点。