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arXiv:1610.08841 · IEEE Transactions on Automatic Control

A Quantum Hamiltonian Identification Algorithm: Computational Complexity and Error Analysis

Yuanlong Wang, Daoyi Dong, Bo Qi, Jun Zhang, Ian R. Petersen, Hidehiro Yonezawa

Published Aug 30, 2017• 96 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tac.2017.2747507

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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.
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Identification of the Hamiltonian is vital for characterizing the dynamical evolution of a quantum system. The dimension of a multi-qubit system increases exponentially with the qubit number, which usually leads to daunting computational complexity for general Hamiltonian identification algorithms. In this paper, we design an efficient quantum Hamiltonian identification method based on periodical sampling. The computational complexity is O(M2 + MN2), where M is the number of unknown parameters to be identified in the Hamiltonian and N is the length of the sampling data. Numerical results with different data lengths demonstrate the effectiveness of the proposed identification algorithm.