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IEEE Conference on Decision and Control (CDC)

Simultaneous quantum state and detector tomography through multiple Hamiltonians

Shuixin Xiao, Weichao Liang, Yuanlong Wang, Daoyi Dong, Ian R. Petersen, Valery Ugrinovskii

Published Dec 16, 2024• 1 citation
Publisher

Abstract

The estimation of all the parameters in an unknown quantum state/measurement device, commonly referred to as quantum state/detector tomography (QST/QDT), plays a pivotal role in characterization and control of quantum systems. In this paper, we introduce a novel method to concurrently identify a quantum state and a detector using multiple Hamiltonians. We then develop two solutions, i.e., a closed-form algorithm and the sum of squares (SOS) optimization. The numerical examples demonstrate that the closed-form solution offers computational efficiency, while the SOS optimization method ensures accuracy.

DOI
10.1109/cdc56724.2024.10885898

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Quantum state tomography (QST) is a technique for characterizing, benchmarking, and verifying quantum systems and devices. In this paper, we focus on collective QST using data from collective measurements performed on multiple copies of the state. We propose a closed-form solution and provide an analytical characterization of its computational complexity and mean squared error (MSE) scaling. Additionally, we reformulate the problem as a sum of squares (SOS) optimization problem with semialgebraic constraints, enabling the application of SOS tools for its solution. The effectiveness of the proposed methods is demonstrated through numerical simulations. Furthermore, we validate the algorithms using two-copy collective experimental data, where the entangled measurement provides information about the state purity. Compared to previous methods, our algorithms achieve lower MSEs and approach the collective MSE bound by leveraging this purity information more efficiently.
The estimation of all the parameters in an un-known quantum state or measurement device, known as quantum state tomography (QST) and quantum detector tomography (QDT), is fundamental to the characterization and control of quantum systems. This paper presents a unified state space framework for QST and QDT based on observable measurement results at different evolution moments, applicable to both closed and Markovian open quantum systems. We derive lower bounds on the number of sampling points required for a unique estimation and analyze the computational complexity and mean squared error (MSE) scaling. As the estimator may yield unphysical results, we introduce new correction techniques to enforce physicality, which may also improve the infidelity scaling. The effectiveness of the proposed methods is validated through numerical simulations.
Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design optimal probe states for detector estimation based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the minimum UMSE and the minimum condition number for quantum detectors and provide concrete examples that can achieve optimal detector tomography. In order to enhance estimation precision, we also propose a two-step adaptive detector tomography algorithm and investigate how this adaptive strategy can be used to achieve efficient estimation of quantum detectors. Moreover, the superposition of coherent states are used as probe states for quantum detector tomography and the estimation error is analyzed. Numerical results demonstrate the effectiveness of both the proposed optimal and adaptive quantum detector tomography methods.
Quantum state tomography (QST) is an important tool for estimating an unknown quantum state, which includes a measurement process and a reconstruction process. The state estimation error involves the measurement-induced (probability estimation) error in the measurement process and the calculation error in the reconstruction process. Via the mean-square error methods, we propose two fusion measurement schemes with multiple measurement devices (MMDs) to improve the efficiency and accuracy of quantum state measurement and QST by using information fusion theory. These two schemes are founded on the parallel synchronous measurements of MMDs and, therefore, can improve the efficiency of quantum state measurement and tomography. At the same time, by fusing measurement data from different measurement devices in optimal and suboptimal manners, the proposed multiple-measurement-device fusion measurement schemes achieve the improvement of quantum state measurement and tomography accuracy. Numerical simulations are presented to demonstrate the proposed method.