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Journal of Physics B Atomic Molecular and Optical Physics

Optimal experiment design for quantum state tomography of a molecular vibrational mode

Matthijs P. A. Branderhorst, Ian A. Walmsley, Robert L. Kosut, HERSCHEL A. RABITZ

Published Mar 26, 2008• 17 citations
Publisher

Abstract

Optimal experiment design is used to collect a data set for the complete reconstruction of the vibrational quantum state of diatomic potassium molecules with maximum accuracy and a minimum number of experiments. The optimal configuration of experimental settings is obtained by minimizing the Cramér–Rao lower bound to the accuracy, which is attained in our reconstruction. The quantum state is reconstructed from the data set using maximum-likelihood estimation. Both optimal experiment design and maximum-likelihood estimation are solved efficiently as convex optimization problems.

DOI
10.1088/0953-4075/41/7/074004

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Optimal experiment design is used to collect a data set for the complete reconstruction of the vibrational quantum state of diatomic potassium molecules with maximum accuracy and a minimum number of experiments. The optimal configuration of experimental settings is obtained by minimizing the Cramér–Rao lower bound to the accuracy, which is attained in our reconstruction. The quantum state is reconstructed from the data set using maximum-likelihood estimation. Both optimal experiment design and maximum-likelihood estimation are solved efficiently as convex optimization problems.
A number of problems in quantum state and system identification are addressed. Specifically, it is shown that the maximum likelihood estimation (MLE) approach, already known to apply to quantum state tomography, is also applicable to quantum process tomography (estimating the Kraus operator sum representation (OSR)), Hamiltonian parameter estimation, and the related problems of state and process (OSR) distribution estimation. Except for Hamiltonian parameter estimation, the other MLE problems are formally of the same type of convex optimization problem and therefore can be solved very efficiently to within any desired accuracy. Associated with each of these estimation problems, and the focus of the paper, is an optimal experiment design (OED) problem invoked by the Cramer-Rao Inequality: find the number of experiments to be performed in a particular system configuration to maximize estimation accuracy; a configuration being any number of combinations of sample times, hardware settings, prepared initial states, etc. We show that in all of the estimation problems, including Hamiltonian parameter estimation, the optimal experiment design can be obtained by solving a convex optimization problem. Software to solve the MLE and OED convex optimization problems is available upon request from the first author.
Given an experimental setup and a fixed number of measurements, how should one take data to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [R. Kosut, I. Walmsley, and H. Rabitz, e-print arXiv:quant-ph/0411093 (2004)]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of `minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias. Monte Carlo simulations confirm the utility of our results for qubits. Finally, we adapt our approach to deal with constrained techniques such as maximum-likelihood estimation. We find that these are less amenable to optimization than cruder reconstruction methods, such as linear inversion.
Quantum states encoded in electromagnetic fields, also known as bosonic states, have been widely applied in quantum sensing, quantum communication, and quantum error correction. Accurate characterization is therefore essential yet difficult when states cannot be reconstructed with sparse Pauli measurements. Tomography must work with dense measurement bases, high-dimensional Hilbert spaces, and often sample-based data. However, existing convex optimization-based techniques are not efficient enough and scale poorly when extended to large and multimode systems. In this work, we explore convex optimization as an effective framework to address problems in bosonic state tomography, introducing three techniques to enhance efficiency and scalability: efficient displacement operator computation, Hilbert-space truncation, and stochastic convex optimization, which mitigate common limitations of existing approaches. Then we propose a sample-based, convex maximum-likelihood estimation method specifically designed for flying-mode tomography. Numerical simulations of flying four-mode and nine-mode problems demonstrate the accuracy and practicality of our methods. This method provides practical tools for reliable bosonic mode quantum state reconstruction in high-dimensional and multimode systems.
A fundamental difficulty in demonstrating quantum state tomography is that the required resources grow exponentially with the system size. For pure states and nearly pure states, the task of tomography can be more efficient. We proposed two methods for state reconstruction, by (1) minimizing entropy and (2) maximizing likelihood. The algorithm of compressed sampling is employed to solve the optimization problem. Experiments are demonstrated considering 4-qubit photonic states. The results show that (1) much fewer measurements than the standard tomography are sufficient to obtain high fidelity, and (2) the method of maximizing likelihood is more accurate and noise robust than the original reconstruction method of compressed sampling. Furthermore, the physical meaning of the methods of minimizing entropy and maximizing likelihood is clear.
Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried out in a number of different settings on a fixed experimental setup. The collected data are often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how one should optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standard integer-programming algorithms allow us to find the minimum set of readout operations that can realize a target tomographic task. We apply the method to certain basic and practical state-tomographic problems in nuclear-magnetic-resonance experimental systems. The results show that considerably fewer readout operations can be found using our technique than by using the previous greedy search strategy. Therefore, our method could be helpful for simplifying measurement schemes to minimize the experimental effort.