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arXiv:2506.10379 · Physical Review Research

Hamiltonian learning via inverse physics-informed neural networks

Jie Liu, Xin Wang

Published Oct 6, 2025• 1 citation
arXivHTMLPDF

Abstract

Hamiltonian learning (HL), enabling precise estimation of system parameters and underlying dynamics, plays a critical role in characterizing quantum systems. However, conventional HL methods face challenges in noise robustness and resource efficiency, especially under limited measurements. In this work, we present Inverse Physics-Informed Neural Networks for Hamiltonian Learning (iPINN-HL), an approach that incorporates the Schrödinger equation as a soft constraint via a loss function penalty into the ML procedure. This formulation allows the model to integrate both observational data and known physical laws to infer Hamiltonian parameters with greater accuracy and resource efficiency. We benchmark iPINN-HL against a deep-neural-network-based quantum state tomography method [denoted as DNN-HL (Deep Neural Network for Hamiltonian Learning)] and demonstrate its effectiveness across several different scenarios, including one-dimensional spin chains, cross-resonance gate calibration, crosstalk identification, and real-time compensation to parameter drift. Our results show that iPINN-HL can approach the Heisenberg limit and exhibits robustness to noises, while outperforming DNN-HL in accuracy and resource efficiency. Therefore, iPINN-HL is a powerful and flexible framework for quantum system characterization for practical tasks.

DOI
10.1103/nx97-zjdf

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Hamiltonian learning (HL), enabling precise estimation of system parameters and underlying dynamics, plays a critical role in characterizing quantum systems. However, conventional HL methods face challenges in noise robustness and resource efficiency, especially under limited measurements. In this work, we present Inverse Physics-Informed Neural Networks for Hamiltonian Learning (iPINN-HL), an approach that incorporates the Schrödinger equation as a soft constraint via a loss function penalty into the ML procedure. This formulation allows the model to integrate both observational data and known physical laws to infer Hamiltonian parameters with greater accuracy and resource efficiency. We benchmark iPINN-HL against a deep-neural-network-based quantum state tomography method [denoted as DNN-HL (Deep Neural Network for Hamiltonian Learning)] and demonstrate its effectiveness across several different scenarios, including one-dimensional spin chains, cross-resonance gate calibration, crosstalk identification, and real-time compensation to parameter drift. Our results show that iPINN-HL can approach the Heisenberg limit and exhibits robustness to noises, while outperforming DNN-HL in accuracy and resource efficiency. Therefore, iPINN-HL is a powerful and flexible framework for quantum system characterization for practical tasks.
Abstract Understanding and characterising quantum many-body dynamics remains a significant challenge due to both the exponential complexity required to represent quantum many-body Hamiltonians, and the need to accurately track states in time under the action of such Hamiltonians. This inherent complexity limits our ability to characterise quantum many-body systems, highlighting the need for innovative approaches to unlock their full potential. To address this challenge, we propose a novel method to solve the Hamiltonian learning (HL) problem -inferring quantum dynamics from many-body state trajectories. Our approach has the unprecedented advantage of distilling and filtering an estimator Hamiltonian from a Neural ODE representation of state trajectories. By distilling Hamiltonian information from Neural ODE onto an ansatz Hamiltonian, our method is reliably convergent, experimentally friendly, and interpretable. In addition to this, we propose a new quantitative benchmark based on power laws, which can objectively compare the reliability and generalisation capabilities of any two HL algorithms. Finally, we benchmark our method against state-of-the-art HL algorithms with a 1D spin- 1 / 2 chain proof of concept, giving a stable solution for HL on a set of Hamiltonians previously unlearnable in the literature. Code for our method is available and open source.
Characterizing the environmental interactions of quantum systems is a critical bottleneck in the development of robust quantum technologies. Traditional tomographic methods are often data-intensive and struggle with scalability. In this work, we introduce a novel framework for performing Lindblad tomography using Physics-Informed Neural Networks (PINNs). By embedding the Lindblad master equation directly into the neural network's loss function, our approach simultaneously learns the quantum state's evolution and infers the underlying dissipation parameters from sparse, time-series measurement data. Our results show that PINNs can reconstruct both the system dynamics and the functional form of unknown noise parameters, presenting a sample-efficient and scalable solution for quantum device characterization. Ultimately, our method produces a fully-differentiable digital twin of a noisy quantum system by learning its governing master equation.
The precision required to characterize a Hamiltonian is central to developing advantageous quantum computers, providing powerful advances in quantum sensing and crosstalk mitigation. Traditional methods to determine a Hamiltonian are difficult due to the intricacies of quantum systems, involving numbers of equations and parameters that grow exponentially with the number of qubits. To mitigate these shortcomings, in this paper, we introduce an innovative and effective procedure integrating a physics-informed neural network (PINN) with a freezing mechanism to learn the Hamiltonian parameters efficiently. Although PINN and experimental data alone would become impractical as NNN increases, the mechanism we introduce freezes the interactions of most of the qubits, leaving just a qubit subsystem to be analyzed by the PINN method. Determination of all physical parameters is accomplished by analyzing the system by parts until completion. We validated the efficacy of our method using simulation data obtained from the IBM quantum computer to obtain the training data and we found that a PINN can learn the two-qubit parameters with high accuracy, achieving a median error of less than 0.10.1%0.1 for systems of up to four qubits. We have successfully combined the PINN analysis of two qubits with the freezing mechanism in the case of a four-qubit system.
Determining the microscopic Hamiltonian of a quantum magnet from inelastic neutron scattering measurements remains a central challenge in condensed matter physics. While the dynamical spin structure factor contains, in principle, sufficient information about the underlying interactions, extracting Hamiltonian parameters from experimental spectra constitutes a highly nontrivial inverse problem due to the large parameter space and the presence of experimental noise. Here we demonstrate that neural networks can efficiently solve this inverse problem for a broad class of quantum magnets. As a benchmark, we consider an eight-parameter family of honeycomb lattice Heisenberg spin models in the fully polarized phase, where the dynamical spin structure factor can be computed exactly within linear spin-wave theory. Using a large synthetic dataset of neutron scattering spectra, we train neural networks to infer the underlying Hamiltonian parameters directly from the dynamical response. We compare the performance of three different architectures---fully connected neural networks (FCNNs), one-dimensional convolutional neural networks (CNN1Ds), and two-dimensional convolutional neural networks (CNN2Ds)---and find that all achieve high predictive accuracy. Remarkably, the trained models remain robust in the presence of substantial experimental uncertainty, reliably recovering the Hamiltonian parameters even when the input spectra are contaminated by random noise with amplitudes reaching 10%10\%10% of the signal intensity. Our results establish machine learning as a powerful framework for quantitative Hamiltonian reconstruction from neutron scattering data and provide a practical route toward automated characterization of quantum magnetic materials.
In this paper, authors present a physics-informed neural networks (PINN) approach for solving the optimal control problem (OCP) of the stochastic Schrödinger equation (SSE), with an emphasis on applications in quantum optics. The SSE models quantum systems under uncertainty, a scenario often encountered in open quantum systems where environmental interactions introduce randomness into the system dynamics. These stochastic dynamics are particularly relevant in quantum optics, where precise control over quantum states is crucial for applications such as quantum computing, secure communication, and measurement protocols. Traditional numerical methods for solving such problems face limitations due to high computational costs and the complexity associated with stochastic partial differential equations. The proposed approach utilises PINN, a type of deep learning model that integrates physical laws directly into the learning process by incorporating loss functions based on governing equations, initial and boundary conditions. The fidelity-based objective function is constructed to achieve the desired quantum state so as to develop an OCP solving algorithm. The proposed work is distinguished between deterministic drift-based training and post-training stochastic validation. Extensive numerical experiments are demonstrated near-unity deterministic fidelity. Further, a systematic architecture study validated that a moderately deep network yielded optimal performance with reduced computational cost. Fidelity dissipation analysis confirmed that the method maintained accuracy even under strong damping. These findings established the PINN as an efficient and physically consistent framework for quantum optimal control in open systems.
Physics-informed neural networks (PINNs) represent a methodology that integrates physical equations into neural networks. By incorporating the residuals of physical equations as part of the loss function, PINNs enable the network to learn data features while simultaneously satisfying the constraints imposed by physical laws. The underlying principle of PINNs involves using optimization algorithms to iteratively update the network parameters until the value of a specified physics-informed loss function decreases to an acceptable level, thereby driving the network towards the solution of the differential equations. This approach is not merely a tool but serves as a bridge integrating scientific computing with artificial intelligence, offering novel paradigms for modeling and predicting complex systems. In this paper, we propose a non-Hermitian shortcut to adiabaticity (STA) scheme based on physics-informed neural networks. We utilize the physics-informed neural networks to solve parameterized differential equations, employing the neural networks as an approximating function for the quantum adiabatic evolution process. This provides an interpretable and end-to-end framework (termed STAPINNs) for control-field optimization, distinct from traditional gradient-based approaches like gradient ascent pulse engineering (GRAPE), which require pre-defined pulse parameterization. The parameterized differential equations, along with various physical constraints, are incorporated into the network's loss function. Training the networks allows them to fit the quantum system's evolution process and obtain the driving control function for population inversion. Compared to conventional STA techniques, our approach introduces machine learning into non-Hermitian quantum systems, enabling highly robust and high-fidelity population transfer. Numerical simulations demonstrate that the proposed method significantly outperforms traditional STA protocols under dissipative conditions, achieving fidelities above 0.99 even in the presence of substantial decoherence. Moreover, the framework demonstrates strong generalization across wide parameter ranges and is inherently scalable to multi-level and many-body systems due to the mesh-free, high-dimensional handling capability of PINNs. Neural networks possess strong computational capabilities suitable for generating driving control functions in complex systems, and are equally applicable to non-Hermitian STA techniques. The STAPINNs framework not only enhances control flexibility but also provides a powerful tool for optimizing quantum operations in open and noisy systems.