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arXiv:1905.05323 · IEEE Transactions on Information Theory

Two-Stage Estimation for Quantum Detector Tomography: Error Analysis, Numerical and Experimental Results

Yuanlong Wang, Shota Yokoyama, Daoyi Dong, Ian R. Petersen, Elanor H. Huntington, Hidehiro Yonezawa

Published Feb 26, 2021• 21 citations
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Abstract

Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, a novel quantum detector tomography method is proposed. First, a series of different probe states are used to generate measurement data. Then, using constrained linear regression estimation, a stage-1 estimation of the detector is obtained. Finally, the positive semidefinite requirement is added to guarantee a physical stage-2 estimation. This Two-stage Estimation (TSE) method has computational complexity O(nd2M), where n is the number of d-dimensional detector matrices and M is the number of different probe states. An error upper bound is established, and optimization on the coherent probe states is investigated. We perform simulation and a quantum optical experiment to testify the effectiveness of the TSE method.

DOI
10.1109/tit.2021.3062596

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A tomography method for binary detectors is developed. In this method, different input states are employed and the measurement data are then collected. First a primary estimation of the detector is obtained through least squares estimation, without considering the restriction on the eigenvalues of the detector. Then this possibly nonphysical estimation is projected onto the physical subspace to obtain a final estimation. We analyze the computational complexity of this algorithm, and present a theoretical error upper bound. Numerical simulation on a two-qubit example validates the effectiveness of the algorithm.
Quantum process tomography is a critical task for characterizing the dynamics of quantum systems and achieving precise quantum control. In this paper, we propose a two-stage solution for both trace-preserving and non-trace-preserving quantum process tomography. Utilizing a tensor structure, our algorithm exhibits a computational complexity ofO(MLd2)O(MLd^{2})O(MLd2)where d is the dimension of the quantum system andM,L (M≥d2,L≥d2)M, L~(M\geq d^{2}, L\geq d^{2})M,L (M≥d2,L≥d2)represent the numbers of different input states and measurement operators, respectively. We establish an analytical error upper bound and then design the optimal input states and the optimal measurement operators, which are both based on minimizing the error upper bound and maximizing the robustness characterized by the condition number. Numerical examples and testing on IBM quantum devices are presented to demonstrate the performance and efficiency of our algorithm.
In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that ℓ2\ell_2ℓ2​-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.
Quantum Hamiltonian identification (QHI) is important for characterizing the dynamics of quantum systems, calibrating quantum devices, and achieving precise quantum control. In this paper, an effective two-step optimization (TSO) QHI algorithm is developed within the framework of quantum process tomography. In the identification method, different probe states are input into quantum systems and the output states are estimated using the quantum state tomography protocol via linear regression estimation. The time-independent system Hamiltonian is reconstructed based on the experimental data for the output states. The Hamiltonian identification method has computational complexity O(d6), where d is the dimension of the system Hamiltonian. An error upper bound O( d3/√N ) is also established, where N is the resource number for the tomography of each output state, and several numerical examples demonstrate the effectiveness of the proposed TSO Hamiltonian identification method.