# Data-driven quantum Koopman method for simulating nonlinear dynamics

arXiv: https://arxiv.org/abs/2507.21890 · HTML: https://mlsanity.com/paper/2507.21890/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 94 references that are in the map.
- First posted on arXiv 29 Jul 2025.

## Builds on (most cited first, 17 of 17)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [SegFormer: Simple and Efficient Design for Semantic Segmentation with Transformers](https://mlsanity.com/paper/2105.15203/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Improving Vision Transformers by Revisiting High-frequency Components](https://mlsanity.com/paper/2204.00993/) (2022)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes](https://mlsanity.com/paper/2304.11247/) (2023)
- [Deep Learning Models for Global Coordinate Transformations that Linearize PDEs](https://mlsanity.com/paper/1911.02710/) (2019)
- [KoopmanLab: machine learning for solving complex physics equations](https://mlsanity.com/paper/2301.01104/) (2023)
- [Disentangling Hype from Practicality: On Realistically Achieving Quantum Advantage](https://mlsanity.com/paper/2307.00523/) (2023)
- [H-DenseFormer: An Efficient Hybrid Densely Connected Transformer for Multimodal Tumor Segmentation](https://mlsanity.com/paper/2307.01486/) (2023)
- [Parametric Learning of Time-Advancement Operators for Unstable Flame Evolution](https://mlsanity.com/paper/2402.10238/) (2024)
- [Koopman Theory-Inspired Method for Learning Time Advancement Operators in Unstable Flame Front Evolution](https://mlsanity.com/paper/2412.08426/) (2024)

## Built on by (most cited first, 1 of 1)

- [Quantum simulation of real-world nonlinear dynamics via Koopman method](https://mlsanity.com/paper/2607.07338/) (2026)

## Abstract (arXiv)

Quantum computation offers potential exponential speedups for simulating certain physical systems, but its application to nonlinear dynamics is inherently constrained by the requirement of unitary evolution. We propose the quantum Koopman method (QKM), a data-driven framework that bridges this gap through transforming nonlinear dynamics into linear unitary evolution in higher-dimensional observable spaces. Leveraging the Koopman operator theory to achieve a global linearization, our approach maps system states into a hierarchy of Hilbert spaces using a deep autoencoder. Within the linearized embedding spaces, the state representation is decomposed into modulus and phase components, and the evolution is governed by a set of unitary Koopman operators that act exclusively on the phase. These operators are constructed from diagonal Hamiltonians with coefficients learned from data, a structure designed for efficient implementation on quantum hardware. This architecture enables direct multi-step prediction, and the operator's computational complexity scales logarithmically with the observable space dimension. The QKM is validated across diverse nonlinear systems. Its predictions maintain relative errors below 6% for reaction-diffusion systems and shear flows, and capture key statistics in 2D turbulence. This work establishes a practical pathway for quantum-accelerated simulation of nonlinear phenomena, exploring a framework built on the synergy between deep learning for global linearization and quantum algorithms for unitary dynamics evolution.