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

Journal of Applied Statistics

Bayesian inference for quantum state tomography

Douglas Soares Gonçalves, Caio Azevedo, Carlile Lavor, Márcia A. Gomes-Ruggiero

Published Nov 15, 2017• 9 citations
Publisher

Abstract

We present a Bayesian approach to the problem of estimating density matrices in quantum state tomography. A general framework is presented based on a suitable mathematical formulation, where a study of the convergence of the Monte Carlo Markov Chain algorithm is given, including a comparison with other estimation methods, such as maximum likelihood estimation and linear inversion. This analysis indicates that our approach not only recovers the underlying parameters quite properly, but also produces physically acceptable punctual and interval estimates. A prior sensitive study was conducted indicating that when useful prior information is available and incorporated, more accurate results are obtained. This general framework, which is based on a reparameterization of the model, allows an easier choice of the prior and proposal distributions for the Metropolis–Hastings algorithm.

DOI
10.1080/02664763.2017.1401049

Related papers

A Bayesian quantum state tomography along with adaptive frameworks based on linear minimum mean square error criterion

Subhadeep Mondal, Amit Kumar Dutta

Abstract Quantum state tomography (QST) is essential for characterizing unknown quantum states. Several methods of estimating quantum states already exist and can be classified mainly into three broad classes. They are based on the criteria like maximum likelihood, linear inversion, and Bayesian framework. The Bayesian framework for QST gives a better reconstruction performance. However, the existing methods of the Bayesian frameworks are computationally extensive and, most of the time require knowledge about the prior distribution of the quantum state. In this paper, we propose a Bayesian method of QST based on the linear minimum mean square error criterion, where the prior statistics are estimated and the computational complexity is comparable to that of the linear inversion based QST method. We also propose an adaptive version based on the block estimation of parameters. Extensive numerical simulations are conducted to demonstrate its efficacy over the linear inversion-based QST regarding trace distance error metric.

New Journal of Physics•Nov 20, 2023•3 citations

Markov chain Monte Carlo estimation of quantum states

James DiGuglielmo, Christopher Messenger, Jaromı́r Fiurášek +4 more

We apply a Bayesian data analysis scheme known as the Markov chain Monte Carlo to the tomographic reconstruction of quantum states. This method yields a vector, known as the Markov chain, which contains the full statistical information concerning all reconstruction parameters including their statistical correlations with no a priori assumptions as to the form of the distribution from which it has been obtained. From this vector we can derive, e.g., the marginal distributions and uncertainties of all model parameters, and also of other quantities such as the purity of the reconstructed state. We demonstrate the utility of this scheme by reconstructing the Wigner function of phase-diffused squeezed states. These states possess non-Gaussian statistics and therefore represent a nontrivial case of tomographic reconstruction. We compare our results to those obtained through pure maximum-likelihood and Fisher information approaches.

arXiv:0902.4603•Physical Review A•Mar 31, 2009•8 citations

Quantum state tomography of a single qubit: comparison of methods

Roman Schmied

The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography errors. As a measure of the accuracy of these methods, we average the trace distance between a given density matrix and the tomographic density matrices it can give rise to through experimental measurements. We find that the BME provides the most accurate estimate of the density matrix, and suggest using either the pure-state prior, if the system is known to be in a rather pure state, or the Bures prior if any state is possible. The MLE is found to be slightly less accurate. We comment on the extrapolation of these results to larger systems.

arXiv:1407.4759•Journal of Modern Optics•Feb 7, 2016•59 citations

Quantum State Vector Reconstruction via Bayesian Inference

Jincheng Zhang

Quantum state vector reconstruction is a fundamental problem in quantum information theory, crucial for tasks like quantum state tomography and quantum error correction. Traditional methods often struggle with noisy measurements and lack a principled framework for uncertainty quantification. This paper introduces a novel algorithm, Quantum State Vector Reconstruction via Bayesian Inference, that leverages Bayesian inference to reconstruct a quantum state vector from a set of measurements, enhancing accuracy and offering a more robust approach compared to existing methods. We detail the core mechanism, emphasizing the integration of Bayesian estimation for uncertainty quantification and refinement of the reconstruction process. The algorithm incorporates a probabilistic approach to estimate the state vector distribution, improving reconstruction by incorporating uncertainty estimates directly into the reconstruction process. This results in a more reliable and accurate state vector reconstruction, particularly when dealing with noisy measurements. This work addresses the limitations of conventional methods by providing a principled Bayesian framework for state vector reconstruction, offering a significant advancement in this field.

•Sep 3, 2026•0 citations

Bayesian Quantum State Tomography with Python's PyMC

Daniel J. Lum, Yaakov S. Weinstein

Quantum state tomography (QST) is typically performed from a frequentist viewpoint using maximum likelihood estimation (MLE) which seeks to find the best plausible state consistent with the data by maximizing a likelihood function / distribution. The likelihood function holds an implicit assumption that there is suitable data to infer frequency. In data-starved experiments, this may or may not be a feasible assumption. Moreover, MLE returns no error estimates on the final solution and users are forced to rely on alternative approaches involving either additional measurements or simulated data. Alternatively, Bayesian methods can return a solution with error estimates consistent with the data's uncertainty, but at the expense of a difficult integration over the likelihood distribution. The integration usually requires computational methods with appropriately chosen step sizes in a somewhat complicated problem formulation. This additional complexity serves as a strong deterrent from using Bayesian methods despite the advantages. Probabilistic programming is becoming a common alternative with growing computational power and the development of robust automated integration techniques such as Markov-Chain Monte Carlo (MCMC). Here, we show how to use Python-3's open source PyMC probabilistic programming package to transform an otherwise complicated QST optimization problem into a simple form that can be quickly optimized with efficient under-the-hood MCMC samplers.

arXiv:2212.10655•Dec 20, 2022•0 citations

An efficient adaptive MCMC algorithm for Pseudo-Bayesian quantum tomography

Tien Mai

Abstract We revisit the Pseudo-Bayesian approach to the problem of estimating density matrix in quantum state tomography in this paper. Pseudo-Bayesian inference has been shown to offer a powerful paradigm for quantum tomography with attractive theoretical and empirical results. However, the computation of (Pseudo-)Bayesian estimators, due to sampling from complex and high-dimensional distribution, pose significant challenges that hamper their usages in practical settings. To overcome this problem, we present an efficient adaptive MCMC sampling method for the Pseudo-Bayesian estimator by exploring an adaptive proposal scheme together with subsampling method. We show in simulations that our approach is substantially computationally faster than the previous implementation by at least two orders of magnitude which is significant for practical quantum tomography.

arXiv:2106.00577•Computational Statistics•Jul 23, 2022•3 citations
We present a Bayesian approach to the problem of estimating density matrices in quantum state tomography. A general framework is presented based on a suitable mathematical formulation, where a study of the convergence of the Monte Carlo Markov Chain algorithm is given, including a comparison with other estimation methods, such as maximum likelihood estimation and linear inversion. This analysis indicates that our approach not only recovers the underlying parameters quite properly, but also produces physically acceptable punctual and interval estimates. A prior sensitive study was conducted indicating that when useful prior information is available and incorporated, more accurate results are obtained. This general framework, which is based on a reparameterization of the model, allows an easier choice of the prior and proposal distributions for the Metropolis–Hastings algorithm.
Abstract Quantum state tomography (QST) is essential for characterizing unknown quantum states. Several methods of estimating quantum states already exist and can be classified mainly into three broad classes. They are based on the criteria like maximum likelihood, linear inversion, and Bayesian framework. The Bayesian framework for QST gives a better reconstruction performance. However, the existing methods of the Bayesian frameworks are computationally extensive and, most of the time require knowledge about the prior distribution of the quantum state. In this paper, we propose a Bayesian method of QST based on the linear minimum mean square error criterion, where the prior statistics are estimated and the computational complexity is comparable to that of the linear inversion based QST method. We also propose an adaptive version based on the block estimation of parameters. Extensive numerical simulations are conducted to demonstrate its efficacy over the linear inversion-based QST regarding trace distance error metric.
We apply a Bayesian data analysis scheme known as the Markov chain Monte Carlo to the tomographic reconstruction of quantum states. This method yields a vector, known as the Markov chain, which contains the full statistical information concerning all reconstruction parameters including their statistical correlations with no a priori assumptions as to the form of the distribution from which it has been obtained. From this vector we can derive, e.g., the marginal distributions and uncertainties of all model parameters, and also of other quantities such as the purity of the reconstructed state. We demonstrate the utility of this scheme by reconstructing the Wigner function of phase-diffused squeezed states. These states possess non-Gaussian statistics and therefore represent a nontrivial case of tomographic reconstruction. We compare our results to those obtained through pure maximum-likelihood and Fisher information approaches.
The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography errors. As a measure of the accuracy of these methods, we average the trace distance between a given density matrix and the tomographic density matrices it can give rise to through experimental measurements. We find that the BME provides the most accurate estimate of the density matrix, and suggest using either the pure-state prior, if the system is known to be in a rather pure state, or the Bures prior if any state is possible. The MLE is found to be slightly less accurate. We comment on the extrapolation of these results to larger systems.
Quantum state vector reconstruction is a fundamental problem in quantum information theory, crucial for tasks like quantum state tomography and quantum error correction. Traditional methods often struggle with noisy measurements and lack a principled framework for uncertainty quantification. This paper introduces a novel algorithm, Quantum State Vector Reconstruction via Bayesian Inference, that leverages Bayesian inference to reconstruct a quantum state vector from a set of measurements, enhancing accuracy and offering a more robust approach compared to existing methods. We detail the core mechanism, emphasizing the integration of Bayesian estimation for uncertainty quantification and refinement of the reconstruction process. The algorithm incorporates a probabilistic approach to estimate the state vector distribution, improving reconstruction by incorporating uncertainty estimates directly into the reconstruction process. This results in a more reliable and accurate state vector reconstruction, particularly when dealing with noisy measurements. This work addresses the limitations of conventional methods by providing a principled Bayesian framework for state vector reconstruction, offering a significant advancement in this field.
Quantum state tomography (QST) is typically performed from a frequentist viewpoint using maximum likelihood estimation (MLE) which seeks to find the best plausible state consistent with the data by maximizing a likelihood function / distribution. The likelihood function holds an implicit assumption that there is suitable data to infer frequency. In data-starved experiments, this may or may not be a feasible assumption. Moreover, MLE returns no error estimates on the final solution and users are forced to rely on alternative approaches involving either additional measurements or simulated data. Alternatively, Bayesian methods can return a solution with error estimates consistent with the data's uncertainty, but at the expense of a difficult integration over the likelihood distribution. The integration usually requires computational methods with appropriately chosen step sizes in a somewhat complicated problem formulation. This additional complexity serves as a strong deterrent from using Bayesian methods despite the advantages. Probabilistic programming is becoming a common alternative with growing computational power and the development of robust automated integration techniques such as Markov-Chain Monte Carlo (MCMC). Here, we show how to use Python-3's open source PyMC probabilistic programming package to transform an otherwise complicated QST optimization problem into a simple form that can be quickly optimized with efficient under-the-hood MCMC samplers.
Abstract We revisit the Pseudo-Bayesian approach to the problem of estimating density matrix in quantum state tomography in this paper. Pseudo-Bayesian inference has been shown to offer a powerful paradigm for quantum tomography with attractive theoretical and empirical results. However, the computation of (Pseudo-)Bayesian estimators, due to sampling from complex and high-dimensional distribution, pose significant challenges that hamper their usages in practical settings. To overcome this problem, we present an efficient adaptive MCMC sampling method for the Pseudo-Bayesian estimator by exploring an adaptive proposal scheme together with subsampling method. We show in simulations that our approach is substantially computationally faster than the previous implementation by at least two orders of magnitude which is significant for practical quantum tomography.