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arXiv:1412.4282 · Physical Review A

Ubiquitous problem of learning system parameters for dissipative two-level quantum systems: Fourier analysis versus Bayesian estimation

S. G. Schirmer, Frank C. Langbein

Published Feb 27, 2015• 16 citations
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Abstract

We compare the accuracy, precision, and reliability of different methods for estimating key system parameters for two-level systems subject to Hamiltonian evolution and decoherence. It is demonstrated that the use of Bayesian modeling and maximum likelihood estimation is superior to common techniques based on Fourier analysis. Even for simple two-parameter estimation problems, the Bayesian approach yields higher accuracy and precision for the parameter estimates obtained. It requires less data, is more flexible in dealing with different model systems, can deal better with uncertainty in initial conditions and measurements, and enables adaptive refinement of the estimates. The comparison results show that this holds for measurements of large ensembles of spins and atoms limited by Gaussian noise as well as projection noise limited data from repeated single-shot measurements of a single quantum device.

DOI
10.1103/physreva.91.022125

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We compare the accuracy, precision, and reliability of different methods for estimating key system parameters for two-level systems subject to Hamiltonian evolution and decoherence. It is demonstrated that the use of Bayesian modeling and maximum likelihood estimation is superior to common techniques based on Fourier analysis. Even for simple two-parameter estimation problems, the Bayesian approach yields higher accuracy and precision for the parameter estimates obtained. It requires less data, is more flexible in dealing with different model systems, can deal better with uncertainty in initial conditions and measurements, and enables adaptive refinement of the estimates. The comparison results show that this holds for measurements of large ensembles of spins and atoms limited by Gaussian noise as well as projection noise limited data from repeated single-shot measurements of a single quantum device.
The deterioration of precision caused by quantum decoherence in dissipative environments is a longstanding problem in the development of quantum metrology. Therefore, it is significant to effectively analyze the evolution of quantum systems under dissipative environments and quantitatively estimate the parameters given a certain probe. Here, we propose a continuous-variable (CV) quantum metrology scheme within the Bayesian paradigm to estimate phase shift, loss coefficient and temperature of a dissipative environment. Through simulation analysis, we investigate the achievable precision of Bayesian quantum parameter estimation with Gaussian states and homodyne measurements, and the influence of different parameters on the estimation process. More interestingly, the scheme can be applied in more practical scenarios where not enough measurements are obtained and the frequentist analysis is not well defined.
As the size of quantum devices continues to grow, the development of scalable methods to characterise and diagnose noise is becoming an increasingly important problem. Recent methods have shown how to efficiently estimate Hamiltonians in principle, but they are poorly conditioned and can only characterize the system up to a scalar factor, making them difficult to use in practice. In this work we present a Bayesian methodology, called Bayesian Hamiltonian Learning (BHL), that addresses both of these issues by making use of any or all, of the following: well-characterised experimental control of Hamiltonian couplings, the preparation of multiple states, and the availability of any prior information for the Hamiltonian. Importantly, BHL can be used online as an adaptive measurement protocol, updating estimates and their corresponding uncertainties as experimental data become available. In addition, we show that multiple input states and control fields enable BHL to reconstruct Hamiltonians that are neither generic nor spatially local. We demonstrate the scalability and accuracy of our method with numerical simulations on up to 100 qubits. These practical results are complemented by several theoretical contributions. We prove that a kkk-body Hamiltonian HHH whose correlation matrix has a spectral gap ΔΔΔ can be estimated to precision ε\varepsilonε with only O~(n3k/(εΔ)3/2)\tilde{O}\bigl(n^{3k}/(\varepsilon Δ)^{3/2}\bigr)O~(n3k/(εΔ)3/2) measurements. We use two subroutines that may be of independent interest: First, an algorithm to approximate a steady state of HHH starting from an arbitrary input that converges factorially in the number of samples; and second, an algorithm to estimate the expectation values of mmm Pauli operators with weight ≤k\le k≤k to precision εεε using only O(ε−23klog⁡m)O(ε^{-2} 3^k \log m)O(ε−23klogm) measurements, which quadratically improves a recent result by Cotler and Wilczek.
Bayesian analysis is a framework for parameter estimation that applies even in uncertainty regimes where the commonly used local (frequentist) analysis based on the Cramer-Rao bound (CRB) is not well defined. In particular, it applies when no initial information about the parameter value is available, e.g., when few measurements are performed. Here, we consider three paradigmatic estimation schemes in continuous-variable (CV) quantum metrology (estimation of displacements, phases, and squeezing strengths) and analyse them from the Bayesian perspective. For each of these scenarios, we investigate the precision achievable with single-mode Gaussian states under homodyne and heterodyne detection. This allows us to identify Bayesian estimation strategies that combine good performance with the potential for straightforward experimental realization in terms of Gaussian states and measurements. Our results provide practical solutions for reaching uncertainties where local estimation techniques apply, thus bridging the gap to regimes where asymptotically optimal strategies can be employed.
Abstract The quest for precision in parameter estimation is a fundamental task in different scientific areas. The relevance of this problem thus provided the motivation to develop methods for the application of quantum resources to estimation protocols. Within this context, Bayesian estimation offers a complete framework for optimal quantum metrology techniques, such as adaptive protocols. However, the use of the Bayesian approach requires extensive computational resources, especially in the multiparameter estimations that represent the typical operational scenario for quantum sensors. Hence, the requirement to characterize protocols implementing Bayesian estimations can become a significant challenge. This work focuses on the crucial task of robustly benchmarking the performances of these protocols in both single and multiple-parameter scenarios. By comparing different figures of merits, evidence is provided in favor of using the median of the quadratic error in the estimations in order to mitigate spurious effects due to the numerical discretization of the parameter space, the presence of limited data, and numerical instabilities. These results, providing a robust and reliable characterization of Bayesian protocols, find natural applications to practical problems within the quantum estimation framework.
We consider the problem of estimating a dynamical parameter of a Markovian quantum open system (the atom maser), by performing continuous time measurements in the systemʼs output (outgoing atoms). Two estimation methods are investigated and compared. Firstly, the maximum likelihood estimator (MLE) takes into account the full measurement data and is asymptotically optimal in terms of its mean square error. Secondly, the 'likelihood-free' method of approximate Bayesian computation (ABC) produces an approximation of the posterior distribution for a given set of summary statistics, by sampling trajectories at different parameter values and comparing them with the measurement data via chosen statistics. Building on previous results which showed that atom counts are poor statistics for certain values of the Rabi angle, we apply MLE to the full measurement data and estimate its Fisher information. We then select several correlation statistics such as waiting times, distribution of successive identical detections, and use them as input of the ABC algorithm. The resulting posterior distribution follows closely the data likelihood, showing that the selected statistics capture 'most' statistical information about the Rabi angle.
We investigate schemes for Hamiltonian parameter estimation of a two-level system using repeated measurements in a fixed basis. The simplest (Fourier based) schemes yield an estimate with a mean-square error (MSE) that decreases at best as a power law $\ensuremath{\sim}{N}^{\ensuremath{-}2}$ in the number of measurements NNN. By contrast, we present numerical simulations indicating that an adaptive Bayesian algorithm, where the time between measurements can be adjusted based on prior measurement results, yields a MSE which appears to scale close to $\mathrm{exp}(\ensuremath{-}0.3N)$. That is, measurements in a single fixed basis are sufficient to achieve exponential scaling in NNN.