Quantum-Informed Machine Learning for Predicting Spatiotemporal Chaos with Practical Quantum Advantage

Abstract

We introduce a quantum-informed machine learning (QIML) framework for modelling the long-term behaviour of high-dimensional chaotic systems. QIML combines a one-time, offline-trained quantum generative model with a classical autoregressive predictor for spatiotemporal field generation. The quantum model learns a quantum prior (Q-Prior) that guides the representation of small-scale interactions and improves the modelling of fine-scale dynamics. We evaluate QIML on the Kuramoto-Sivashinsky equation, two-dimensional Kolmogorov flow, and the three-dimensional turbulent channel flow used as a realistic inflow condition. Across these systems, QIML improves predictive distribution accuracy by up to 17.25% and full-spectrum fidelity by up to 29.36% relative to classical baselines. For turbulent channel inflow, the Q-Prior is trained on a superconducting quantum processor and proves essential: without it, predictions become unstable, whereas QIML produces physically consistent long-term forecasts that outperform leading PDE solvers. Beyond accuracy, QIML offers a memory advantage by compressing multi-megabyte datasets into a kilobyte-scale Q-Prior, enabling scalable integration of quantum resources into scientific modelling.

Figures (19)

• Figure 1: Architecture of the QIML framework.A. High-dimensional dynamical flow fields are generated using high-resolution numerical solvers and used to construct training, validation and test datasets. B. A quantum circuit is trained on superconducting hardware to learn invariant statistical properties from the data (Q-Prior). C. The learned Q-Prior is integrated into a classical transformer machine learning, guiding it toward physically consistent predictions and improved long-term stability.
• Figure 2: Evaluation of the QIML framework on the KS system. A. This panel displays time-averaged mean velocity fields (left column) and corresponding relative error $E_r$ maps (right column) across the ground truth, the classical ML without Q-Prior, and the quantum-informed ML model with Q-Prior (from top to bottom). B. This panel presents the probability distribution of $u$. C. This panel shows the energy spectrum $\langle E(k) \rangle$ as a function of spatial wavenumber $k$, characterizing kinetic energy distribution across spectral modes. D. This panel visualizes the invariant density when dynamics are projected into the $(u_x, u_{xx})$ space, revealing the geometric support of the underlying invariant measure. E. This panel plots the temporal autocorrelation, denoted by $C(t^*)$, computed from the time series of the field $u$ averaged over all spatial points. The correlation is shown as a function of the dimensionless time lag $t^* = t/t_{Lyapunov}$.
• Figure 3: Early-time chaotic evolution of streamwise velocity fields. The streamwise velocity rollout of ground truth (top), the QIML with Q-Prior (middle), and the classical model without Q-Prior (bottom) given the same initial state at $\hat{t}$ = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 (from left to right), corresponding to Lyapunov times $t^{*} = 0.0, 0.04, 0.08, 0.12, 0.16,$ and $0.20$, respectively.
• Figure 4: Evaluation of the QIML framework on the 2D Kolmogorov flow. A. This panel displays time-averaged mean $u$ fields (left column) and corresponding TKE fields (right column) for the ground truth, the classical model without Q-Prior, and the quantum-informed model with Q-Prior. These maps are evaluated on the test set, highlighting long-term flow patterns and energy distribution. B. This panel presents the ensemble-averaged energy spectrum $\langle E(k) \rangle$ as a function of spatial wavenumber $k$. C. This panel shows the PDF of the field variable $u$, comparing the distribution from the raw test data with the PDFs predicted by the models with and without the Q-Prior. D. This panel plots the temporal autocorrelation $C(t^*)$ of the velocity field as a function of dimensionless time $t^*$. E. This panel reports the normalized relative error, $E_r$, with respect to the ground truth, calculated over time on the same validation trajectory for a 1000-step rollout.
• Figure 5: Generation of Synthetic Turbulent Inflow from Turbulent Channel Flow Data.A. Instantaneous flow field from a turbulent channel flow simulation, representing the raw data used for turbulent inflow generation. The red ellipse indicates the cross-section where velocity field data are extracted. B. The QIML generative model can be used as turbulent inflow conditions in 3D turbulent channel flow for large eddy simulations.