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New Journal of Physics

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

Subhadeep Mondal, Amit Kumar Dutta

Published Nov 20, 2023• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1367-2630/ad0e49

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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.
A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. An asymptotic mean squared error (MSE) upper bound for all possible states to be estimated is given analytically, which depends explicitly upon the involved measurement bases. This analytical MSE upper bound can guide one to choose optimal measurement sets. The computational complexity of LRE is O(d(4)) where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.
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.
Quantum state tomography is one of the critical tasks in the development of quantum technology. This paper further investigates the adaptive linear regression estimation (ALRE) method that was developed recently for state reconstruction of qubit systems. In this method, adaptive measurements are used to improve the performance of state reconstruction where current measurement basis is adaptively optimized based on previous knowledge through minimizing an index related to the mean squared error (MSE). The numerical results on single-qubit and two-qubit systems demonstrate the effectiveness of this proposed adaptive estimation method over static one for quantum state reconstruction.
Quantum state tomography (QST) is a crucial technique for characterizing quantum states, forming the bedrock of quantum information processing and quantum metrology. However, traditional QST relies on a large number of measurements, making it computationally expensive and experimentally demanding. This work addresses this challenge by proposing a novel algorithm that integrates compressed sensing (CS) and Bayesian inference to achieve accurate quantum state reconstruction from limited measurement data. The core idea is to exploit the inherent sparsity of quantum states and leverage CS principles to significantly reduce the required measurement count. The algorithm utilizes a sparse dictionary learned from prior knowledge of the quantum state and employs Bayesian inference to estimate the state parameters, providing a robust and efficient reconstruction process. The results demonstrate the feasibility and effectiveness of this approach, offering a pathway towards more practical and accessible QST methods. Key mathematical formulations are presented, including the Bayesian inference framework and the CS reconstruction process.
Quantum state tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the dimension and sample size, just like stochastic gradient descent. In this paper, we propose a stochastic first-order algorithm that computes an ε\varepsilonε-approximate ML estimate in O((Dlog⁡D)/ε2)O( ( D \log D ) / \varepsilon ^ 2 )O((DlogD)/ε2) iterations with O(D3)O( D^3 )O(D3) per-iteration time complexity, where DDD denotes the dimension of the unknown quantum state and ε\varepsilonε denotes the optimization error. Our algorithm is an extension of Soft-Bayes to the quantum setup.
Quantum state tomography (QST) is a crucial technique for characterizing quantum states, vital for verifying quantum device performance and implementing quantum information processing. However, traditional QST methods are notoriously computationally demanding, requiring extensive experimental data acquisition and subsequent processing, often leading to high resource consumption. This work presents a novel framework for QST utilizing probabilistic programming. The core idea is to model the quantum state as a probability distribution and employ Bayesian inference, coupled with variational methods, to efficiently infer the density matrix from limited measurement data. We demonstrate the efficacy of this approach, showcasing its potential to significantly reduce experimental costs and improve the accuracy of QST. The framework's automation and optimization capabilities represent a substantial advancement in the field, paving the way for more accessible and efficient quantum state characterization.