An online optimization algorithm for the real-time quantum state tomography
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An online optimization algorithm for the real-time quantum state tomography Kun Zhang1 · Shuang Cong1
· Kezhi Li2 · Tao Wang1
Received: 11 June 2020 / Accepted: 9 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 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. Keywords Online quantum state tomography · Optimization algorithm · Online alternating direction multiplier method · Continuous weak measurement
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61720106009 and 61973290.
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Shuang Cong [email protected] Kezhi Li [email protected]
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Department of Automation, University of Science and Technology of China, Hefei 230027, China
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University College London, 222 Euston Rd, London NW1 2DA, UK 0123456789().: V,-vol
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K. Zhang et al.
1 Introduction Quantum tomography reconstructs the state of a system by repeatedly measuring all identical quantum duplicates that form a set of quantum ensemble [1–3]. Traditional quantum state tomography (QST) or estimation algorithms are offline, and the entire measurement dataset is required to estimate a static fixed quantum state through multiple iterations [4,5]. The goal of the online QST is to obtain the dynamic quantum state in real time which can be used in the quantum state feedback control system. So, the online QST algorithm focuses on processing only a fraction of the available measurement data in each sampling and calculation. Such method has been used for online learning [6,7]. The state of an n-qubit system can usually be described by a density matrix ρ ∈ Cd×d (d = 2n ), which satisfies the physical constraints of positive semidefinite and unit-trace Hermitian [8]. Different from the commonly used methods of QST based on projective and destructive measurements [9,10], the continuous weak measurement (CWM) proposed by Silberfarb provides a new approach to estimate quantum states [11]. Based on CWM, it is possible to gain the measurement information regarding the state to be estimated without being disturbed s
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