An online optimization algorithm for the real-time quantum state tomography

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.