Quantum simulation of real-world nonlinear dynamics via Koopman method

Simulating nonlinear dynamics on quantum computers is fundamentally challenging because quantum evolution is inherently unitary and linear. While nonlinear phenomena like ocean circulation and chemical kinetics are ubiquitous, existing quantum algorithms often struggle with either excessive circuit depth or a lack of physical inductive bias. This paper addresses the challenge of embedding dissipative, nonlinear dynamics into the unitary framework of near-term quantum processors.

The authors propose the Quantum Koopman Method (QKM), which leverages Koopman operator theory to lift nonlinear dynamics into an infinite-dimensional linear space. This space is then projected onto a finite-dimensional subspace spanned by learned observables. The framework uses a classical neural network encoder to map physical initial conditions to circuit parameters, while the non-unitary propagator is decomposed into parallel spectral channels using a Linear Combination of Hamiltonian Simulation (LCHS) approach. This allows the simulation to run on shallow quantum circuits, specifically optimized for the connectivity of the superconducting processor "Yudu."

The QKM was validated across three progressively complex systems: a 3D Gray-Scott reaction-diffusion system, spherical shallow-water fluid dynamics, and satellite-derived Gulf Stream observations. The simulations successfully captured dominant multiscale patterns and statistical signatures. The study identifies a practical boundary for quantum utility: in weakly nonlinear systems, performance is limited by hardware noise, whereas in more complex systems, performance is limited by the finite-dimensional Koopman representation. The QKM achieved significant speedups over classical Koopman propagation by utilizing up to 32 parallel circuits of 10 qubits.

This work provides a hardware-validated, systematic route for simulating moderately nonlinear dynamics on near-term quantum hardware. By replacing heuristic circuit design with a physics-informed LCHS decomposition, the framework offers better interpretability and error traceability. It effectively bridges the gap between theoretical quantum algorithms and the practical constraints of noisy intermediate-scale quantum (NISQ) devices, expanding the scope of problems accessible to quantum simulation.