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Moscow University Physics Bulletin

Reconstruction of a Time-Dependent Quantum Hamiltonian Using Neural Differential Equations

A. S. Naumov, V. Yu. Popov

Published Feb 1, 2026• 0 citations
Publisher

Abstract

Abstract The problem of reconstructing an unknown time-dependent Hamiltonian of a two-level quantum system from data on the system’s dynamics is considered. A method based on a neural network within the Neural ODE framework is proposed, in which the continuous evolution of the quantum state is modeled by integrating the Schrödinger equation with a parametric Hamiltonian model implemented by the network. As input data, measured time series of the expectation values of Pauli operators for a qubit are used. The network is trained by minimizing the difference between the experimentally observed and modeled trajectories of quantum observables. Using the example of simulating spin dynamics in a time-varying magnetic field, it is shown that the proposed approach enables high-accuracy reconstruction of the time-dependent Hamiltonian parameters without interrupting the quantum evolution. The results demonstrate that the Neural ODE model provides continuous tracking of the time parameter and reliable reconstruction of the system’s dynamics under conditions of limited data and noise.

DOI
10.3103/s0027134926700104

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Abstract The problem of reconstructing an unknown time-dependent Hamiltonian of a two-level quantum system from data on the system’s dynamics is considered. A method based on a neural network within the Neural ODE framework is proposed, in which the continuous evolution of the quantum state is modeled by integrating the Schrödinger equation with a parametric Hamiltonian model implemented by the network. As input data, measured time series of the expectation values of Pauli operators for a qubit are used. The network is trained by minimizing the difference between the experimentally observed and modeled trajectories of quantum observables. Using the example of simulating spin dynamics in a time-varying magnetic field, it is shown that the proposed approach enables high-accuracy reconstruction of the time-dependent Hamiltonian parameters without interrupting the quantum evolution. The results demonstrate that the Neural ODE model provides continuous tracking of the time parameter and reliable reconstruction of the system’s dynamics under conditions of limited data and noise.
The problem of reconstructing an unknown time-dependent Hamiltonian of a two-level quantum system from its observed dynamics is considered. A method based on a neural network in the Neural ODE framework is proposed, where the continuous evolution of the quantum state is modeled by integrating the Schrodinger equation with a parametric Hamiltonian model implemented by the network. The input data consist of measured time series of the expectation values of Pauli operators for a qubit. The network is trained by minimizing the difference between the experimentally observed and simulated trajectories of quantum observables. Using the example of spin dynamics in a time-varying magnetic field, it is shown that the proposed approach allows accurate reconstruction of the parameters of the time-dependent Hamiltonian without interrupting the quantum evolution. The results demonstrate that the Neural ODE model provides continuous handling of the time parameter and robust reconstruction of system dynamics under conditions of limited data and noise.
Real-time dynamics of quantum observables provide direct access to excitation spectra and correlation functions in quantum many-body systems, but currently available quantum devices are limited to short evolution times due to decoherence. We propose a neural ordinary differential equation (Neural ODE) framework with a physics-driven neural network to reconstruct long-time operator dynamics from short-time quantum data. By expanding observables in the Pauli basis and exploiting locality and symmetry constraints, the operator evolution is reduced to a tractable set of coefficients whose dynamics are learned from the measurement data. Applied to the Ising spin chain with mixed fields, the operator learning method accurately extrapolates its long-time correlation function and resolves its excitation spectrum. We validate the framework by collecting short-time data from IBM's superconducting quantum processor and successfully reconstruct the Fourier spectrum of an N=8N=8N=8 Ising spin chain with 5 times enhanced precision. Our results demonstrate a scalable and data-efficient strategy for extracting dynamical and spectral information from practical quantum hardware.
At its core, quantum mechanics is a theory developed to describe fundamental observations in the spectroscopy of solids and gases.Despite these practical roots, however, quantum theory is infamous for being highly counterintuitive, largely due to its intrinsically probabilistic nature.Neural networks have recently emerged as a powerful tool that can extract nontrivial correlations in vast datasets.These networks routinely outperform state-of-the-art techniques in language translation, medical diagnosis, and image recognition.It remains to be seen if neural networks can be trained to predict stochastic quantum evolution without a priori specifying the rules of quantum theory.Here, we demonstrate that a recurrent neural network can be trained in real time to infer the individual quantum trajectories associated with the evolution of a superconducting qubit under unitary evolution, decoherence, and continuous measurement from physical observations only.The network extracts the system Hamiltonian, measurement operators, and physical parameters.It is also able to perform tomography of an unknown initial state without any prior calibration.This method has the potential to greatly simplify and enhance tasks in quantum systems such as noise characterization, parameter estimation, feedback, and optimization of quantum control.
Quantum systems governed by time-dependent Hamiltonians pose significant challenges for the accurate computation of unitary time-evolution operators, which are essential for predicting quantum state dynamics. In this work, we introduce a physics-informed deep learning approach based on Physics-Informed Neural Networks to estimate these operators over the full time domain. By incorporating physical constraints such as unitarity and leveraging the second-order Magnus expansion on the evolution operator, the proposed framework enables the estimation of unitary matrices at different time intervals. The model is trained using simulated unitary operators and evaluated on quantum systems ranging from 2 to 6 qubits. For larger many-body systems, specifically those with 7 and 8 qubits, the same methodology is employed to reconstruct an effective time-dependent Hamiltonian, from which the corresponding time-evolution operator is computed over the entire temporal domain. The proposed framework achieves fidelities exceeding 0.92 using a limited number of unitary samples, indicating a potential reduction in measurement and data acquisition costs. These results highlight the effectiveness of the approach for data-driven simulation and identification of quantum dynamical systems, with direct relevance to quantum computing and quantum simulation applications.
In the Hamiltonian-based quantum dynamics, to estimate Hamiltonians from the measured data is a vital topic.In this work, we propose a recurrent neural network to learn the target Hamiltonians from the temporal records of single-qubit measurements, which does not require the ground states and only measures single-qubit observables.It is applicable on both time-independent and time-dependent Hamiltonians and can simultaneously capture the magnitude and sign of Hamiltonian parameters.Taking the Hamiltonians with the nearest-neighbor interactions as numerical examples, we trained our recurrent neural networks to learn different types of Hamiltonians with high accuracy.The study also shows that our method has good robustness against the measurement noise and decoherence effect.Therefore, it has widespread applications in estimating the parameters of quantum devices and characterizing the Hamiltonian-based quantum dynamics.
Capturing the dynamics of quantum many-body systems under time-dependent driving protocols is a central challenge for numerical simulations. Existing methods such as tensor networks and time-dependent neural quantum states, however, must be re-run for every protocol. In this work, we introduce the Neural Operator Quantum State (NOQS) as a foundation model for quantum dynamics. Rather than solving the Schrödinger equation for individual trajectories, our approach aims to \emph{learn the solution operator} that maps entire driving protocols to time-evolved quantum states. Once trained, the NOQS predicts time evolution under unseen protocols in a single forward pass, requiring no additional optimization. We validate NOQS on the two-dimensional Ising model with time-dependent longitudinal and transverse fields, demonstrating accurate prediction not only for unseen in-distribution protocols, but also for qualitatively different, out-of-distribution functional forms of driving. Further, a single NOQS model can be transferred between different temporal resolutions, and can be efficiently fine-tuned with sparse experimental measurements to improve predictions across all observables at negligible cost. Our work introduces a new paradigm for quantum dynamics simulation and provides a practical computational-experimental interface for driven quantum systems.