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arXiv:2204.01055 · EPJ Quantum Technology

A stochastic evaluation of quantum Fisher information matrix with generic Hamiltonians

Le Bin Ho

Published Sep 19, 2023• 10 citations
arXivHTMLPDF

Abstract

Abstract Quantum Fisher information matrix (QFIM) is a fundamental quantity in quantum physics, which closely links to diverse fields such as quantum metrology, phase transitions, entanglement witness, and quantum speed limit. It is crucial in quantum parameter estimation, central to the ultimate Cramér-Rao bound. Recently, the evaluation of QFIM using quantum circuit algorithms has been proposed for systems with multiplicative parameters Hamiltonian. However, systems with generic Hamiltonians still lack these proposed schemes. This work introduces a quantum-circuit-based approach for evaluating QFIM with generic Hamiltonians. We present a time-dependent stochastic parameter-shift rule for the derivatives of evolved quantum states, whereby the QFIM can be obtained. The scheme can be executed in universal quantum computers under the family of parameterized gates. In magnetic field estimations, we demonstrate the consistency between the results obtained from the stochastic parameter-shift rule and the exact results, while the results obtained from a standard parameter-shift rule slightly deviate from the exact ones. Our work sheds new light on studying QFIM with generic Hamiltonians using quantum circuit algorithms.

DOI
10.1140/epjqt/s40507-023-00195-w

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Abstract Recently, quantum metrology with multiplicative Hamiltonians has been proposed in the variational quantum algorithms, from which the estimation precision can be adaptively optimized via the variational circuits. Systems with general Hamiltonians, however, still lack these variational schemes. This work introduces a quantum-circuit-based approach for studying quantum metrology with general Hamiltonians. We present a stochastic parameter-shift rule for the derivatives of the evolved quantum state under the parameterized gates in the circuit, whereby the quantum Fisher information can be obtained. Here the parameters are those we wish to estimate. We find that our scheme can be executed in universal quantum computers under the family of parameterized gates. Moreover, in the examples of the magnetic field estimation, we show the consistency between the results obtained from the stochastic parameter-shift rule and the exact results, while the results obtained from the standard parameter-shift rule slightly deviate from the exact ones. Our work sheds new light on studying quantum metrology with general Hamiltonians using quantum circuit algorithms.
Abstract Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér–Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.
Quantum Fisher Information Matrix (QFIM) is a fundamental concept in quantum-theoretical measurement that emphasizes the importance of quantum Cramér–Rao bound in estimating quantum parameters. Many relationships have been discovered between QFIM and many recent areas of research in quantum information. These relationships indicate that QFIM is not just a concept in quantum metrology but is a fundamental quantity in quantum information. The ability to estimate parameters and describe the optimal induced conditions for the thermal state of the two-axis twisting model was examined. QFIM is used as an estimator for multiple parameters. We show that simultaneous estimation reduces the estimate of parameter variation, so the accuracy of simultaneously estimating parameters is much better than individually. Each parameter can be used to control the criterion to reduce the minimum differences between other parameters.
Abstract Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér–Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A: Math. Theor. 52 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in terms of their first and second moments is given. We give the analytical formulas of right logarithmic derivative and symmetric logarithmic derivative operators. Then we derive the general expressions of the corresponding quantum Fisher information matrices. We also derive an explicit expression of the condition which ensures the saturation of the quantum Cramér–Rao bound in estimating several parameters. Finally, we examine some examples to clarify the use of our results.
Quantum Fisher information matrix (QFIM) is a cornerstone of modern quantum metrology and quantum information geometry. Apart from optimal estimation, it finds applications in description of quantum speed limits, quantum criticality, quantum phase transitions, coherence, entanglement, and irreversibility. We derive a surprisingly simple formula for this quantity, which, unlike previously known general expression, does not require diagonalization of the density matrix, and is provably at least as efficient. With a minor modification, this formula can be used to compute QFIM for any finite-dimensional density matrix. Because of its simplicity, it could also shed more light on the quantum information geometry in general.
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