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IEEE Conference on Decision and Control (CDC)

Control-Enhanced Quantum Metrology Guided by Pontryagin’s Minimum Principle

Shouliang Hu, Hailan Ma, Yunyan Lee, Daoyi Dong, Ian R. Petersen

Published Dec 9, 2025• 0 citations
Publisher

Abstract

Quantum metrology utilizes quantum resources to enhance the precision of parameter estimation. A central objective is to maximize the Quantum Fisher information (QFI), which determines the ultimate estimation precision based on the Cramér-Rao bound. In this work, we formulate QFI optimization as an optimal control problem and apply Pontryagin’s Minimum Principle (PMP) to derive the optimal control protocol. We show that when the control amplitude is constrained below a certain threshold, the optimal solution reduces to a constant control. This strategy enables the QFI to scale quadratically with the total evolution time T, thereby improving estimation precision with longer probe durations.

DOI
10.1109/cdc57313.2025.11312173

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Quantum metrology utilizes quantum resources to enhance the precision of parameter estimation. A central objective is to maximize the Quantum Fisher information (QFI), which determines the ultimate estimation precision based on the Cramér-Rao bound. In this work, we formulate QFI optimization as an optimal control problem and apply Pontryagin’s Minimum Principle (PMP) to derive the optimal control protocol. We show that when the control amplitude is constrained below a certain threshold, the optimal solution reduces to a constant control. This strategy enables the QFI to scale quadratically with the total evolution time T, thereby improving estimation precision with longer probe durations.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.
Optimal control theory, also known as Pontryagin's maximum principle, is applied to the quantum parameter estimation in the presence of decoherence. An efficient procedure is devised to compute the gradient of quantum Fisher information with respect to the control parameters and is used to construct the optimal control protocol. The proposed procedure keeps the control problem in the time-invariant form so that both first-order and second-order optimality conditions derived from Pontryagin's maximum principle apply; the second-order condition turns out to be crucial when the optimal control contains singular arcs. Concretely we look for the optimal control that maximizes quantum Fisher information for the ``twist and turn'' problem. We find that the optimal control is singular without dissipation but can become unbounded once the quantum decoherence is introduced. An amplitude constraint is needed to guarantee a bounded solution. With quantum decoherence, the maximum quantum Fisher information happens at a finite time due to the decoherence, and the asymptotic value depends on the specific decoherence channel and the control of consideration.
Estimating multiple unknown parameters simultaneously is essential for practical quantum sensing. However, it faces a fundamental challenge: the optimal strategy for estimating one parameter is often incompatible with that for another, making it impossible to simultaneously achieve the ultimate precision limits for all parameters. Here we develop a general and efficient computational framework that jointly optimizes probe states, control operations, and measurements across different strategy families, including parallel, sequential, and those with indefinite causal order. Our approach provides exact semidefinite-program formulations for several precision bounds, including the Holevo, Nagaoka-Hayashi, and quantum Cramér-Rao bounds. We demonstrate the capabilities of the framework in multiparameter magnetometry and frequency estimation, identifying optimal protocols within each class and revealing a strict hierarchy among the achievable performances of different classes in the multiparameter regime. The framework also directly incorporates resource constraints, such as energy budgets, enabling systematic investigation of experimentally realistic sensing scenarios. Furthermore, we develop a finite-memory optimization method for sequential strategies with restricted ancillary-memory dimension. By decomposing the protocol into initial probe preparation and intermediate control operations, this method provides a practical route to designing resource-constrained sequential sensing schemes. Our work establishes a versatile computational tool for determining fundamental precision limits and designing optimal quantum-sensing protocols in complex multiparameter settings.
Quantum metrology uses the principles of quantum mechanics to improve the accuracy of parameter estimation so that it can surpass the classical limit. However, noise and the challenge of preparing multipartite entangled states hinder practical applications. In this work, we use the Lipkin-Meshkov-Glick model as the experimental platform and the quantum parameter estimation package QuanEstimation as a tool to improve the quantum parameter estimation in many-body systems by using Hamiltonian control optimization. We apply auto-GRAPE, PSO, and DE algorithm to optimize the time-dependent control field. Our results show that the optimal control strategy can significantly enhance the quantum Fisher information and reduce the quantum Cramér-Rao bound even under environmental noise. These findings provide a way to achieve the parameter estimation limit in a noisy environment and promote the development of practical quantum metrology applications.
Reliable high-fidelity quantum state transformation has always been considered as an inseparable part of quantum information processing. In this regard, Pontryagin Minimum (or maximum) Principle (PMP) has proved to play an important role to achieve the maximum fidelity in an optimum time or energy. Motivated by this, in this work, we formulate a control constrained optimal control problem where we aim to minimize time and also energy subjected to a quantum system satisfying the bilinear Schrödinger equation. We derive the first order optimality conditions through the application of PMP resulting in a boundary value problem. Next, in order to obtain efficient numerical results, we exploit a particular family of physics-informed neural networks that are specifically designed to tackle the indirect method based on the PMP. We show that this method can significantly speed up the process by first obtaining a set of relations which finally let us compute the optimal control strategy to determine the time- and energy-optimal protocol driving a general initial state to a target state by a quantum Hamiltonian with bounded control. We make use of the so-called “qutip” package in python, and the newly developed “tfc” python package.
Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known, but is given by the solution of a non-trivial minimisation problem. We present a new expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.