Learning Quantum Operator Dynamics from Short-Time Data

Abstract

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 physics-driven designs to reconstruct long-time operator dynamics from short-time measurements. 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 data. Applied to the transverse-field Ising model, the method accurately extrapolates long-time behavior and resolves excitation spectra from noisy short-time signals. Our results demonstrate a scalable and data-efficient strategy for extracting dynamical and spectral information from practical quantum hardware.

Figures (5)

• Figure 1: Schematic illustration of the operator-learning framework. (a) Input: Short-time evolutions of Pauli coefficients $c_i(t)$ to be measured by the evolution of the initial state $\rho_i$ on noisy quantum hardware. (b) Operator learning: The reduced operator dynamics are modeled using a physics-driven Neural ODE, which enforces frequency-aware modules $F$ while learning smooth temporal trajectories. (c) Output: The trained network enables long-time predictions of operator dynamics. By Fourier transforming the predicted correlation functions, excitation spectra and energy gaps can be reliably extracted, showing improved frequency resolution compared with the input data.
• Figure 2: Long-time prediction of one-point functions $\langle O_X(t)\rangle$ for different initial states in $N=3$ system without physics truncation. The training set is in the left blue panel with 64 coefficient time-series. The expectation value is defined as $\langle O_X(t)\rangle_{XYZ} =\mathrm{Tr}[\hat{O}_X(t)X_1Y_2Z_3]$. Initial values at $t_0=3.0, 5.0, 7.0$ are input to the Neural ODE to generate long-time predictions from both training and test data (yellow, red, and green curves).
• Figure 3: Excitation spectrum of the domain-wall pair obtained from short-time data and Neural ODE prediction with qubits $N=3$ without physics truncation. The training set is in the left blue panel with 64 coefficient time-series. (Top) Two-point correlation function $C(t)$ and its long-time predictions with different initial points at $t_0=3.0$ (yellow), $5.0$ (red), and $7.0$ (green). (Bottom) Corresponding excitation spectra from the Fourier transform. The spectrum (blue) is computed using only training data ranging from 0 to 5, while the spectrum (black) uses the full evolution data.
• Figure 4: Spectrum distribution in the presence of both decoherence and stochastic noise for the $N=5$ system with the local Pauli truncation (trained on 52 time-series of the coefficients). We set dimensionless time $\Gamma=0.05$ for decoherence noise and a $1\%$ relative strength for Gaussian noise. The training and full datasets cover the time intervals $[0,5]$ and $[0,200]$, respectively, with the initial value fixed at $t_0=5$. The comparison shows that Gaussian noise has little effect on the spectrum, while decoherence mainly modifies the low-frequency modes below $0.2~\mathrm{Hz}$.
• Figure 5: Comparison between the baseline FCN predictions(upper panel) and the FAN-augmented predictions (lower panel).