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Numerical approximation of the Koopman-von Neumann equation: Operator learning and quantum computing

Stefan Klus1, Feliks Nüske2, and Patrick Gelß3

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Phys. Rev. E 114, 045304 – Published 13 October, 2026

DOI: https://doi.org/10.1103/23fl-9hw8

Abstract

The Koopman-von Neumann equation describes the evolution of wave functions associated with autonomous ordinary differential equations and can be regarded as a quantum physics-inspired formulation of classical mechanics. The main advantage compared to conventional transfer operators such as Koopman and Perron-Frobenius operators is that the Koopman-von Neumann operator is unitary even if the dynamics are non-Hamiltonian. Projecting this operator onto a finite-dimensional subspace allows us to represent it by a unitary matrix, which in turn can be expressed as a quantum circuit. We will exploit relationships between the Koopman-von Neumann framework and classical transfer operators to derive numerical methods to approximate the Koopman-von Neumann operator and its eigenvalues and eigenfunctions from data. Furthermore, we will show that the choice of basis functions and domain are crucial to ensure that the operator is well-defined. We will illustrate the results with the aid of guiding examples, including simple undamped and damped oscillators and the Lotka-Volterra model.

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Phys. Rev. E 114, 045304– Published 13 October, 2026
  • Received 9 April 2026
  • Accepted 24 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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