Hamiltonian Learning via Inverse Physics-Informed Neural Networks

TL;DR

This paper addresses the challenge of robust, data-efficient Hamiltonian learning under noise by introducing Inverse Physics-Informed Neural Networks for HL (iPINN-HL), which embeds the Schrödinger dynamics as a physics-informed loss within neural learning using Neural Network Quantum States. The method reformulates HL as an inverse problem and couples data loss with a Schrödinger-equation-based physics loss and initial-condition constraints to infer Hamiltonian parameters. Across one-dimensional spin chains, cross-resonance gates, crosstalk identification, and drift compensation, iPINN-HL outperforms a deep-learning HL baseline in accuracy and data efficiency, approaching near-Heisenberg scaling under ideal conditions and demonstrating robustness to realistic noise. The work provides a flexible and practical framework for quantum system characterization and calibration, with avenues for integration with active learning and extension to open quantum systems.

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 \textit{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) 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.

Figures (11)

• Figure 1: Circuit illustration of the quantum query model. After the a query $x=(U,t,M)$ is input to the system, the output is an $n$-bit string.
• Figure 2: (a) Visualization of Physics-Informed Neural Networks (PINNs) for HL. The solid curve shows the true solution $\ket{\Psi(t; \bm{\theta})}$, while the dashed curve represents the estimated solution $\ket{\Psi(t; \bm{\hat{\theta}})}$. Red stars mark data points, green dots indicate prediction errors, and blue dots enforce physical laws by ensuring consistency with the Schrödinger equation. (b) Neural Network Quantum States (NNQS) representation and its tabular form at different time points. The output nodes $\alpha$ and $\beta$ of the neural network represent the real and imaginary parts, respectively, of the complex amplitude of the quantum state $\braket{m| \Psi(t)}$ at time $t$. The neural network efficiently captures the complex amplitudes of quantum states across various configurations, demonstrating high expressive capacity for representing many-body quantum systems. The table on the right shows the amplitudes of different configurations at discrete time steps $t_1, t_2, \dots, t_n$. Automatic differentiation of NNQS enables efficient computation of time derivatives of the quantum state, facilitating the enforcement of dynamical constraints of Schrödinger equation.
• Figure 3: Flowchart illustrating the use of Physics-Informed Neural Networks (PINNs) for HL. Experimental data is generated by evolving an initial quantum state $\ket{0}^{\otimes n}$ under a unitary transformation $U$ and time evolution $e^{-iH(\bm{\theta})t}$, followed by measurement $M$. The neural network takes configurations and time as inputs and outputs the amplitude of the quantum state, with its weights $w$ and estimation $\bm{\hat{\theta}}$ optimized using the Adam optimizer. The loss function combines a physical loss $\mathcal{L}_{\text{physics}}$, which ensures consistency with the Schrödinger equation, and a data loss $\mathcal{L}_{\text{data}}$, which minimizes discrepancies with experimental measurements. Automatic differentiation efficiently computes the time derivative $\partial \Psi_w(m, t)/\partial t$, facilitating the enforcement of physical constraints during training.
• Figure 4: The scaling plot of MSE of iPINN and DNN-HL with respect to the number of entries in the dataset $D$ in (a) and (b) respectively when the number of spins in one-dimensional spin chain is 4, 7 and 10. Here we set $J=1$, $\omega=0.5$ and the number of collocation points$P=50$.
• Figure 5: Schematic representation of a one-dimensional spin chain system with translational symmetry parameters $s\in\{1, 2, 4\}$, showing the corresponding patterns of interaction strengths and local external field distributions.