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IEEE International Conference on Quantum Computing and Engineering (QCE)

P T-Enhanced Bayesian Parameter Estimation

Yaroslav Balytskyi, Manohar Raavi, Sang‐Yoon Chang

Published Oct 1, 2021• 2 citations
Publisher

Abstract

The frequentist (local) quantum parameter estimation based on the Cramer-Rao bound is an asymptotic theory and therefore requires an infinite number of samples for the measurements. On the other hand, by introducing the prior probability distribution, Bayesian parameter estimation approach relaxes this requirement and can be successfully applied under realistic assumption of possessing a limited prior information about the parameter value under investigation. Meanwhile, recent advances in ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$-symmetric quantum mechanics, both theoretically and experimentally, provide an opportunity to significantly enhance the parameter estimation precision by manipulating the Hilbert space of the system. Taking advantage of these, we develop the Bayesian parameter estimation approach for the two-level system possessing ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$ symmetry. We consider both options available in ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$-symmetric quantum mechanics, namely the ${{{\mathcal{C}}}}{{{\mathcal{P}}}}{{{\mathfrak{T}}}}$ and Hermitian measurements, for encoding the probe state and manipulation of the quantum Fisher information of the system and take into account possible decoherence effects inherent for the practical implementations. Our approach should be particularly relevant for applications where the accuracy of quantum parameter estimation is crucial, such as quantum randomness sources and the quantum key distribution. Finally, to bridge the gap with the practical implementations, we consider the relation of our scheme with the current implementation of ${{{\mathcal{P}}}}{{{\mathcal{T}}}}$ symmetry on the IBM quantum processor and formulate our approach as Bayesian parameter estimation employing an ancilla qubit.

DOI
10.1109/qce52317.2021.00022

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The frequentist (local) quantum parameter estimation based on the Cramer-Rao bound is an asymptotic theory and therefore requires an infinite number of samples for the measurements. On the other hand, by introducing the prior probability distribution, Bayesian parameter estimation approach relaxes this requirement and can be successfully applied under realistic assumption of possessing a limited prior information about the parameter value under investigation. Meanwhile, recent advances in PT{{{\mathcal{P}}}}{{{\mathcal{T}}}}PT-symmetric quantum mechanics, both theoretically and experimentally, provide an opportunity to significantly enhance the parameter estimation precision by manipulating the Hilbert space of the system. Taking advantage of these, we develop the Bayesian parameter estimation approach for the two-level system possessing PT{{{\mathcal{P}}}}{{{\mathcal{T}}}}PT symmetry. We consider both options available in PT{{{\mathcal{P}}}}{{{\mathcal{T}}}}PT-symmetric quantum mechanics, namely the CPT{{{\mathcal{C}}}}{{{\mathcal{P}}}}{{{\mathfrak{T}}}}CPT and Hermitian measurements, for encoding the probe state and manipulation of the quantum Fisher information of the system and take into account possible decoherence effects inherent for the practical implementations. Our approach should be particularly relevant for applications where the accuracy of quantum parameter estimation is crucial, such as quantum randomness sources and the quantum key distribution. Finally, to bridge the gap with the practical implementations, we consider the relation of our scheme with the current implementation of PT{{{\mathcal{P}}}}{{{\mathcal{T}}}}PT symmetry on the IBM quantum processor and formulate our approach as Bayesian parameter estimation employing an ancilla qubit.
To realize the computational advantages of quantum mechanics, it is necessary to be able to effectively perform quantum state tomography, and Bayesian methods provide significant opportunities in this regard, being applicable in the uncertainty regimes, where the local approaches based on Cramer-Rao bound become ill-defined. PT{\mathcal{P}}{\mathcal{T}}PT -symmetric quantum mechanics provides a unique advantage for Bayesian parameter estimation having an additional degree of freedom, absent in the regular Hermitian case, which significantly enhances the quantum Fisher information and reduces variance of the obtained results. While previous studies have examined PT{\mathcal{P}}{\mathcal{T}}PT -symmetric Bayesian parameter estimation in theory, we conduct experimental studies using IBM Quantum Experience and compare them with theoretical calculations. In our approach, the evolution of a qubit controlled by a PT{\mathcal{P}}{\mathcal{T}}PT-symmetric Hamiltonian is realized by the dilation method using ancilla qubit. We show that our implementation is consistent with the theoretical model and, in particular, we observe a good correspondence between the theoretical and experimental likelihood distributions. Thus we bridge the gap between theory and practical implementations in PT{\mathcal{P}}{\mathcal{T}}PT-symmetric quantum mechanics and its practical applications in quantum information science.
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.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.
Abstract Chapter 24 provides a compact introduction to the topic of quantum metrology, focusing on Hamiltonian parameter estimation in the frequentist and in the Bayesian paradigms. We first discuss how estimates of non-directly measurable quantities such as phases are obtained from measurement statistics in the frequentist approach. We then discuss the Cramér-Rao bound and the Fisher information and study single-qubit phase estimation in the light of this result. We then turn to the multi-qubit setting and discuss the quantum Cramér-Rao bound, for which the quantum Fisher information is the central quantity of interest. In this context we discuss the Uhlmann fidelity in detail. We further contrast Heisenberg scaling with the standard quantum limit in the phase-estimation scenario. Finally, we analyse phase estimation in the Bayesian-estimation approach and derive the van Trees inequality as a Bayesian version of the Cramér-Rao bound
Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér-Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér-Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.
Abstract Bayesian analysis is a framework for parameter estimation that applies even in uncertainty regimes where the commonly used local (frequentist) analysis based on the Cramér–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.