Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0411093

Optimal Experiment Design for Quantum State and Process Tomography and Hamiltonian Parameter Estimation

Robert L. Kosut, Ian A. Walmsley, HERSCHEL A. RABITZ

Published Nov 12, 2004• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0411093

Related papers

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

Matthijs P. A. Branderhorst, Ian A. Walmsley, Robert L. Kosut +1 more

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.

Journal of Physics B Atomic Molecular and Optical Physics•Mar 26, 2008•17 citations

Optimal experiment design for quantum state tomography: Fair, precise, and minimal tomography

J. Nunn, Brian J. Smith, Graciana Puentes +2 more

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.

arXiv:0911.4310•Physical Review A•Apr 22, 2010•39 citations

Convex Optimization-Based Parameter Estimation and Experiment Design for Pauli Channels

Gábor Balló, Katalin Mária Hangos, D�nes Petz

In this technical note, we study the problem of parameter estimation for quantum channels, which are (completely positive) maps acting on quantum states. We focus on an important subclass of channels called Pauli channels which are characterized by a certain set of vectors (directions) and a number of scalar parameters. For the case where the directions are known, a special parametrization turns the parameter estimation problem into a convex optimization problem. For the case of unknown directions we give a simple algorithm to estimate the directions for qubit Pauli channels. These results assume that the identification experiment configuration is given, namely, a set of quantum states are given on which the channel acts, and a set of positive measurement (operator)s are fixed from which information is gathered. In the second part of the technical note we consider the problem of determining the experiment design, namely, determining states and measurements for optimal parameter identification of the channels. We formalize this problem as a maximization problem for Fisher information and, assuming known channel directions, prove that this problem is convex. We also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal. For qubit Pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions. We illustrate the results of the technical note with numerical examples and simulations.

IEEE Transactions on Automatic Control•Jul 19, 2012•10 citations

Efficient convex optimization for bosonic state tomography

Shengyong Li, Yanjin Yue, Ying Hu +7 more

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.

arXiv:2509.06305•Physical Review Applied•Dec 4, 2025•0 citations

Optimal quantum multiparameter estimation and application to dipole- and exchange-coupled qubits

Kevin C Young, Mohan Sarovar, Robert L. Kosut +1 more

We consider the problem of quantum multiparameter estimation with experimental constraints and formulate the solution in terms of a convex optimization. Specifically, we outline an efficient method to identify the optimal strategy for estimating multiple unknown parameters of a quantum process and apply this method to a realistic example. The example is two electron-spin qubits coupled through the dipole and exchange interactions with unknown coupling parameters---explicitly, the position vector relating the two qubits and the magnitude of the exchange interaction are unknown. This coupling Hamiltonian generates a unitary evolution which, when combined with arbitrary single-qubit operations, produces a universal set of quantum gates. However, the unknown parameters must be known precisely to generate high-fidelity gates. We use the Cram\'er-Rao bound on the variance of a point estimator to construct the optimal series of experiments to estimate these free parameters and present a complete analysis of the optimal experimental configuration. Our method of transforming the constrained optimal parameter estimation problem into a convex optimization is powerful and widely applicable to other systems.

arXiv:0812.4635•Physical Review A•Jun 2, 2009•35 citations

Experiment design and parameter estimation of Pauli channels using convex optimization

Gábor Balló, Katalin Mária Hangos

A unified framework is proposed in this paper for parameter estimation using convex optimization and experiment design applying convex maximization for Pauli channels, that can be extended to generalized Pauli channels, too. In the case of known channel directions, an affine parametrization of the Choi matrix turns the LS parameter estimation into a convex optimization problem also for the generalized Pauli channels. A simple iterative algorithm for estimating the channel directions is also given for qubit Pauli channels. The experiment design was performed by maximizing the trace of the Fisher information matrix of the output quantum system to find optimal input state and measurement POVM for the channel estimation. For the known channel direction case it was found that the optimal input state should be pure and the optimal measurement POVM is extremal. It was also shown that both the input state and the POVM elements in the optimal configuration should be parallel to the channel directions in the qubit Pauli channel case. The proposed methods and algorithms are illustrated by simple numerical examples.

arXiv:1107.0890•Jul 5, 2011•3 citations
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.
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.
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.
In this technical note, we study the problem of parameter estimation for quantum channels, which are (completely positive) maps acting on quantum states. We focus on an important subclass of channels called Pauli channels which are characterized by a certain set of vectors (directions) and a number of scalar parameters. For the case where the directions are known, a special parametrization turns the parameter estimation problem into a convex optimization problem. For the case of unknown directions we give a simple algorithm to estimate the directions for qubit Pauli channels. These results assume that the identification experiment configuration is given, namely, a set of quantum states are given on which the channel acts, and a set of positive measurement (operator)s are fixed from which information is gathered. In the second part of the technical note we consider the problem of determining the experiment design, namely, determining states and measurements for optimal parameter identification of the channels. We formalize this problem as a maximization problem for Fisher information and, assuming known channel directions, prove that this problem is convex. We also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal. For qubit Pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions. We illustrate the results of the technical note with numerical examples and simulations.
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.
We consider the problem of quantum multiparameter estimation with experimental constraints and formulate the solution in terms of a convex optimization. Specifically, we outline an efficient method to identify the optimal strategy for estimating multiple unknown parameters of a quantum process and apply this method to a realistic example. The example is two electron-spin qubits coupled through the dipole and exchange interactions with unknown coupling parameters---explicitly, the position vector relating the two qubits and the magnitude of the exchange interaction are unknown. This coupling Hamiltonian generates a unitary evolution which, when combined with arbitrary single-qubit operations, produces a universal set of quantum gates. However, the unknown parameters must be known precisely to generate high-fidelity gates. We use the Cram\'er-Rao bound on the variance of a point estimator to construct the optimal series of experiments to estimate these free parameters and present a complete analysis of the optimal experimental configuration. Our method of transforming the constrained optimal parameter estimation problem into a convex optimization is powerful and widely applicable to other systems.
A unified framework is proposed in this paper for parameter estimation using convex optimization and experiment design applying convex maximization for Pauli channels, that can be extended to generalized Pauli channels, too. In the case of known channel directions, an affine parametrization of the Choi matrix turns the LS parameter estimation into a convex optimization problem also for the generalized Pauli channels. A simple iterative algorithm for estimating the channel directions is also given for qubit Pauli channels. The experiment design was performed by maximizing the trace of the Fisher information matrix of the output quantum system to find optimal input state and measurement POVM for the channel estimation. For the known channel direction case it was found that the optimal input state should be pure and the optimal measurement POVM is extremal. It was also shown that both the input state and the POVM elements in the optimal configuration should be parallel to the channel directions in the qubit Pauli channel case. The proposed methods and algorithms are illustrated by simple numerical examples.