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arXiv:1812.04815 · Results in Physics

Optimal control for multi-parameter quantum estimation with time-dependent Hamiltonians

Dong Xie, Chunling Xu

Published Sep 3, 2019• 4 citations
arXivHTMLPDF

Abstract

We investigate simultaneous estimation of multi-parameter with time-dependent Hamiltonians. We analytically obtain the maximum quantum Fisher information matrix for two-parameter encoded in time-dependent three-level systems. The optimal coherent control scheme is proposed to increase the simultaneous multi-parameter estimation precisions. In an example of a spin-1 particle in a uniformly rotating magnetic field, the optimal coherent Hamiltonians for different parameters can be chosen to be completely same. However, in general, the optimal coherent Hamiltonians for different parameters are incompatible. In this situation, we suggest a variance method to obtain the optimal coherent Hamiltonian for simultaneously estimating multiple parameters, and obtain the optimal simultaneous estimation precision of two-parameter in a three-level Landau-Zener Hamiltonian.

DOI
10.1016/j.rinp.2019.102620

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We investigate simultaneous estimation of multi-parameter with time-dependent Hamiltonians. We analytically obtain the maximum quantum Fisher information matrix for two-parameter encoded in time-dependent three-level systems. The optimal coherent control scheme is proposed to increase the simultaneous multi-parameter estimation precisions. In an example of a spin-1 particle in a uniformly rotating magnetic field, the optimal coherent Hamiltonians for different parameters can be chosen to be completely same. However, in general, the optimal coherent Hamiltonians for different parameters are incompatible. In this situation, we suggest a variance method to obtain the optimal coherent Hamiltonian for simultaneously estimating multiple parameters, and obtain the optimal simultaneous estimation precision of two-parameter in a three-level Landau-Zener Hamiltonian.
Abstract Multi-parameter quantum estimation has attracted considerable attention due to its broad applications. Due to the complexity of quantum dynamics, existing research places significant emphasis on estimating parameters in time-independent Hamiltonians. Here, our work makes an effort to explore multi-parameter estimation with time-dependent Hamiltonians. In particular, we focus on the discrimination of two close frequencies of a magnetic field by using a single qubit. We optimize the quantum controls by employing both traditional optimization methods and reinforcement learning to improve the precision for estimating the frequencies of the two magnetic fields. In addition to the estimation precision, we also evaluate the robustness of the optimization schemes against the shift of the control parameters. The results demonstrate that the hybrid reinforcement learning approach achieves the highest estimation precision, and exhibits superior robustness. Moreover, a fundamental challenge in multi-parameter quantum estimation stems from the incompatibility of the optimal control strategies for different parameters. We demonstrate that the hybrid control strategies derived through numerical optimization remain effective in enhancing the precision of multi-parameter estimation in spite of the incompatibilities, thereby mitigating incompatibilities between control strategies on the estimation precision. Finally, we investigate the trade-offs in estimation precision among different parameters for different scenarios, revealing the inherent challenges in balancing the optimization of multiple parameters simultaneously and providing insights into the fundamental distinction between quantum single-parameter estimation and multi-parameter estimation.
Abstract Quantum multi-parameter estimation has recently attracted increased attention due to its wide applications, with a primary goal of designing high-precision measurement schemes for unknown parameters. While existing research has predominantly concentrated on time-independent Hamiltonians, little has been known about quantum multi-parameter estimation for time-dependent Hamiltonians due to the complexity of quantum dynamics. This work bridges the gap by investigating the precision limit of multi-parameter quantum estimation for a qubit in an oscillating magnetic field model with multiple unknown frequencies. As the well-known quantum Cramér–Rao bound is generally unattainable due to the potential incompatibility between the optimal measurements for different parameters, we use the most informative bound instead which is always attainable and equivalent to the Holevo bound in the asymptotic limit. Moreover, we apply additional Hamiltonian to the system to engineer the dynamics of the qubit. By utilizing the quasi-Newton method, we explore the optimal schemes to attain the highest precision for the unknown frequencies of the magnetic field, including the simultaneous optimization of initial state preparation, the control Hamiltonian and the final measurement. The results indicate that the optimization can yield much higher precisions for the field frequencies than those without the optimizations. Finally, we study the robustness of the optimal control scheme with respect to the fluctuation of the interested frequencies, and the optimized scheme exhibits superior robustness to the scenario without any optimization.
Abstract Quantum metrology has been studied for a wide range of systems with time-independent Hamiltonians. For systems with time-dependent Hamiltonians, however, due to the complexity of dynamics, little has been known about quantum metrology. Here we investigate quantum metrology with time-dependent Hamiltonians to bridge this gap. We obtain the optimal quantum Fisher information for parameters in time-dependent Hamiltonians, and show proper Hamiltonian control is generally necessary to optimize the Fisher information. We derive the optimal Hamiltonian control, which is generally adaptive, and the measurement scheme to attain the optimal Fisher information. In a minimal example of a qubit in a rotating magnetic field, we find a surprising result that the fundamental limit of T2 time scaling of quantum Fisher information can be broken with time-dependent Hamiltonians, which reaches T4 in estimating the rotation frequency of the field. We conclude by considering level crossings in the derivatives of the Hamiltonians, and point out additional control is necessary for that case.
We investigate the fundamental limits in precision allowed by quantum mechanics from Landau-Zener transitions, concerning Hamiltonian parameters. While the Landau-Zener transition probabilities depend sensitively on the system parameters, much more precision may be obtained using the acquired phase, quantified by the quantum Fisher information. This information scales with a power of the elapsed time for the quantum case, whereas it is time independent if the transition probabilities alone are used. We add coherent control to the system and increase the permitted maximum precision in this time-dependent quantum system. The case of multiple passes before measurement, Landau-Zener-Stueckelberg interferometry, is considered, and we demonstrate that proper quantum control can cause the quantum Fisher information about the oscillation frequency to scale as T4{T}^{4}T4, where TTT is the elapsed time. These results are foundational for frequency standards and quantum clocks.
Simultaneously estimating multiple parameters at the ultimate limit is a central challenge in quantum metrology, often hindered by inherent incompatibilities in optimal estimation strategies. At its most extreme, this incompatibility culminates in a fundamental impossibility when the quantum Fisher information matrix (QFIM) becomes singular, rendering joint estimation unattainable. This is the case for a canonical problem: estimating the amplitude and frequency of an AC magnetic field, where the generators are parallel to each other. Here, we introduce a quantum control protocol that resolves this singularity. Our control protocol strategically engineers the sensor's time evolution so the generators for the two parameters become orthogonal. It not only removes the singularity but also restores the optimal scaling of precision with interrogation time for both parameters simultaneously. We experimentally validate this protocol using a nitrogen-vacancy center in diamond at room temperature, demonstrating the concurrent achievement of the optimal scaling for both parameters under realistic conditions.
Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-12\frac{1}{2}21​ particle and compare the results for estimating the amplitude and the direction of the magnetic field.