Quantum machine learning for efficient reduced order modelling of turbulent flows

TL;DR

This work introduces a hybrid quantum-classical NIROM for turbulence by coupling quantum POD (QPOD) with quantum deep kernel learning (QDKL). Variational quantum algorithms (VQAs) accelerate POD basis generation and enrich kernel expressivity through quantum feature maps, enabling faster training and reduced parameter counts while preserving or improving predictive accuracy for turbulent flows. Demonstrations on canonical 2D flows at Re=4000 show the quantum framework matching or surpassing classical NIROM performance with significantly fewer POD modes and epochs, highlighting potential for real-time quantum fluid simulations as hardware advances. The study outlines a scalable blueprint for integrating quantum components into nonlinear Navier–Stokes ROMs, while noting current NISQ limitations and pointing to hardware-aware future directions.

Abstract

Accurately predicting turbulent flows remains a central challenge in fluid dynamics due to their high dimensionality and intrinsic nonlinearity. Recent developments in quantum algorithms and machine learning offer new opportunities for overcoming the computational barriers inherent in turbulence modeling. Here we present a new hybrid quantum-classical framework that enables faster-than-real-time turbulence prediction by integrating machine learning, quantum computation, and fluid dynamics modeling, in particular, the reduced-order modeling. The novel framework combines quantum proper orthogonal decomposition (QPOD) with a quantum-enhanced deep kernel learning (QDKL) approach. QPOD employs quantum circuits to perform efficient eigenvalue decomposition for low-rank flow reconstruction, while QDKL exploits quantum entanglement and nonlinear mappings to enhance kernel expressivity and dynamic prediction accuracy. The new method is demonstrated on three benchmark turbulent flows, our architecture achieves significantly improved predictive accuracy at reduced model ranks, with training speeds up to 10 times faster and parameter counts reduced by a factor of 1/N compared to classical counterparts, where N is the input dimensionality. Although constrained by current noisy intermediate-scale quantum (NISQ) hardware, our results demonstrate the potential of quantum machine learning to transform turbulence simulation and lay a solid foundation for scalable, real-time quantum fluid modeling in future quantum computers.

Figures (17)

• Figure 1: The structure of the deep kernel learning.
• Figure 2: The structure of the hybrid quantum-classical deep kernel learning.
• Figure 3: The structure of the NIROM.
• Figure 4: The error on the test set using classical deep kernel learning model and quantum-enhanced deep kernel learning model.
• Figure 5: Lid-driven cavity flow at $\boldsymbol{Re=4000}$. The figure shows some of the first 8 basis functions of the problem.