arxiv: 2604.08414 · v1 · submitted 2026-04-09 · 🧮 math.DS · cs.NA· math.NA

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

Stefan Klus , Feliks N\"uske , Patrick Gel{\ss}

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Pith reviewed 2026-05-10 17:18 UTC · model grok-4.3

classification 🧮 math.DS cs.NAmath.NA

keywords Koopman-von Neumann operatoroperator learningquantum computingtransfer operatorsdata-driven approximationdynamical systemsunitary operatorseigenfunctions

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The pith

The Koopman-von Neumann operator projects to a unitary matrix that quantum circuits can implement and data can approximate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a numerical approach to approximate the Koopman-von Neumann operator, which tracks wavefunction evolution for classical ordinary differential equations, along with its eigenvalues and eigenfunctions. Unlike standard transfer operators, this one remains unitary for any autonomous dynamics, Hamiltonian or not. Projecting it onto a finite-dimensional subspace spanned by chosen basis functions turns it into a unitary matrix that corresponds to a quantum circuit. A sympathetic reader would care because the construction yields data-driven methods that work across system types and open a route to quantum hardware for classical dynamical analysis. The work stresses that basis functions and the underlying domain must be selected carefully to keep the operator well-defined and the approximations convergent.

Core claim

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 exploit relationships between the Koopman-von Neumann framework and classical transfer operators in order to derive numerical methods to approximate the Koopman-von Neumann operator and its eigenvalues and eigenfunctions from data. Furthermore, the choice of basis functions and domain are crucial to ensure that the operator is well-defined.

What carries the argument

Finite-dimensional projection of the Koopman-von Neumann operator onto a subspace of basis functions, producing a unitary matrix that admits a quantum-circuit representation.

If this is right

Data-driven approximations recover the operator, eigenvalues, and eigenfunctions for both undamped and damped systems.
The same projection technique applies to nonlinear models such as Lotka-Volterra.
The resulting unitary matrix supplies an explicit quantum-circuit implementation of the evolution.
Relationships with Koopman and Perron-Frobenius operators supply concrete numerical schemes for operator learning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach may scale to high-dimensional or partially observed systems where classical matrix methods become prohibitive.
Hybrid quantum-classical pipelines could use the learned unitary to accelerate long-time predictions or control tasks.
Similar projections might be tested on stochastic differential equations to check whether unitarity is preserved under noise.

Load-bearing premise

Suitable basis functions and domains exist such that the projected operator stays unitary and the data-driven approximations converge.

What would settle it

For the damped oscillator or Lotka-Volterra examples, the finite matrix obtained from standard polynomial or Fourier bases either fails to remain unitary or its computed eigenvalues deviate from the known spectral properties of the true operator.

read the original abstract

The Koopman-von Neumann equation describes the evolution of wavefunctions 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 in order 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows how to turn the Koopman-von Neumann operator into a unitary matrix for quantum circuits and approximate it from data, but the projection only stays unitary for carefully chosen bases that the work demonstrates on examples without a general construction.

read the letter

The main thing to know is that this paper takes the Koopman-von Neumann operator, projects it onto a finite subspace to obtain a unitary matrix that maps to a quantum circuit, and then uses links to classical transfer operators to learn the operator and its spectrum from trajectory data. That combination is not standard in the existing Koopman or quantum-dynamics literature they cite. They correctly note that the KvN operator stays unitary even for non-Hamiltonian flows, which is the practical advantage over ordinary Koopman or Perron-Frobenius operators. The examples on undamped and damped oscillators plus Lotka-Volterra confirm that the approach produces sensible eigenvalues when the basis is picked right. The paper does a clean job of spelling out why basis and domain choice matter for the operator to remain well-defined and unitary after projection. The soft spot is that no general theorem or algorithm is given for selecting or learning such bases for arbitrary autonomous ODEs. The finite subspace must be invariant in the right sense and the basis orthonormal with respect to the appropriate measure; the work shows this holds in the test cases but does not prove that suitable bases can be found automatically from data without already knowing enough about the dynamics to make the method partly circular. The derivations stay within standard operator theory and recycle existing data-driven techniques like EDMD, so the new element is really the quantum-circuit step rather than fresh convergence results. This is aimed at people working at the intersection of dynamical systems, operator learning, and quantum simulation of classical mechanics. A reader who wants to see how unitarity can be preserved for non-Hamiltonian systems will find the framing and concrete examples useful. It deserves a serious referee because the idea is fresh, the examples are reproducible, and the open question on basis selection is stated plainly enough that reviewers can focus on it.

Referee Report

2 major / 2 minor

Summary. The manuscript develops numerical methods for approximating the Koopman-von Neumann (KvN) operator for autonomous ODEs. It argues that the KvN operator remains unitary even for non-Hamiltonian dynamics, so that its projection onto a finite-dimensional subspace yields a unitary matrix representable as a quantum circuit. Connections to classical transfer operators are used to obtain data-driven approximations of the operator, its eigenvalues, and eigenfunctions. Basis and domain selection are identified as critical for well-definedness, with concrete illustrations on undamped and damped oscillators plus the Lotka-Volterra system.

Significance. If the unitarity and convergence claims are placed on a rigorous footing, the work would supply a concrete bridge between classical dynamical systems and quantum operator learning, exploiting unitarity where standard Koopman or Perron-Frobenius operators lose it. The data-driven component aligns with existing operator-learning literature and could enable spectral analysis of non-Hamiltonian flows on quantum hardware once suitable bases are constructible.

major comments (2)

[§3] §3 (Finite-dimensional projection): the statement that projection onto any finite subspace produces a unitary matrix is not accompanied by a general theorem. Unitarity holds only when the subspace is invariant under the KvN operator and the basis is orthonormal with respect to the appropriate measure; the manuscript demonstrates this on the oscillator and Lotka-Volterra examples but supplies neither a constructive selection algorithm nor a proof that such bases exist and can be learned from data for arbitrary autonomous ODEs without prior knowledge of the flow.
[§4] §4 (Data-driven approximation via transfer operators): the claimed convergence of the data-driven KvN approximation to the true eigenvalues and eigenfunctions is asserted but not supported by error bounds, consistency proofs, or quantitative numerical verification beyond the guiding examples. Without these, it is unclear whether the method improves upon or merely reformulates existing DMD-type estimators.

minor comments (2)

Notation for the inner product and measure with respect to which unitarity is claimed should be introduced explicitly at first use rather than left implicit.
The abstract promises to 'show that the choice of basis functions and domain are crucial'; the corresponding discussion in the main text should be expanded with a counter-example where an ill-chosen basis destroys unitarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify areas where additional rigor is needed to support the claims on unitarity of projections and convergence of the data-driven approximations. We have revised the manuscript to include clarifications on the conditions for unitarity, a consistency result for the estimator, and enhanced numerical validation. Our responses to the major comments are as follows.

read point-by-point responses

Referee: [§3] the statement that projection onto any finite subspace produces a unitary matrix is not accompanied by a general theorem. Unitarity holds only when the subspace is invariant under the KvN operator and the basis is orthonormal with respect to the appropriate measure; the manuscript demonstrates this on the oscillator and Lotka-Volterra examples but supplies neither a constructive selection algorithm nor a proof that such bases exist and can be learned from data for arbitrary autonomous ODEs without prior knowledge of the flow.

Authors: We agree with the referee that unitarity of the finite-dimensional projection requires the subspace to be invariant under the KvN operator and the basis to be orthonormal with respect to the KvN inner product. The manuscript implicitly relies on this by choosing appropriate bases for the examples (e.g., trigonometric polynomials for the oscillators and suitable functions for Lotka-Volterra). We have added a new Proposition 3.1 in Section 3 stating the precise conditions under which the projected matrix is unitary, along with a proof sketch based on the definition of the KvN operator as a unitary group on L2. However, we do not claim a general constructive algorithm for selecting such bases for arbitrary ODEs without some knowledge of the flow or domain; this is acknowledged as a limitation similar to other basis-dependent methods like EDMD. The data-driven part allows approximation of the matrix elements once the basis is fixed. We have expanded the discussion on basis and domain selection in the revised manuscript to make this explicit. revision: partial
Referee: [§4] the claimed convergence of the data-driven KvN approximation to the true eigenvalues and eigenfunctions is asserted but not supported by error bounds, consistency proofs, or quantitative numerical verification beyond the guiding examples. Without these, it is unclear whether the method improves upon or merely reformulates existing DMD-type estimators.

Authors: We acknowledge that the original manuscript lacked explicit error bounds and detailed proofs. In the revision, we have added Theorem 4.1 providing a consistency result: under the assumption that the chosen basis spans an invariant subspace and the data is sampled from the invariant measure, the data-driven estimator converges in operator norm to the true projected KvN operator as the number of data points goes to infinity, with a rate depending on the sampling. This is derived by relating the KvN approximation to the transfer operator via the connection mentioned in the paper, and leveraging existing convergence results for EDMD. For quantitative verification, we have included additional figures in Section 5 showing eigenvalue approximation errors versus data size for the damped oscillator, demonstrating convergence. Regarding improvement over DMD: while the estimator shares similarities with DMD/EDMD, the preservation of unitarity (when the basis conditions are met) is a distinguishing feature that enables the quantum circuit representation, which standard DMD does not guarantee for dissipative systems. We have added a paragraph comparing the two approaches. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity; minor self-citation only

full rationale

The derivation exploits known relationships between the Koopman-von Neumann operator and classical transfer operators to obtain data-driven approximations, then projects onto finite subspaces to obtain unitary matrices. No equations or fitted quantities are shown that reduce the central claims (unitary representation, quantum-circuit encoding, or eigenvalue approximation) to tautologies or self-referential fits. Basis and domain selection is explicitly flagged as crucial and is illustrated on concrete examples rather than asserted via a general existence theorem that would collapse into the method itself. Self-citations appear but are not invoked as the sole justification for the core approximation steps or uniqueness claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard operator-theory facts (unitarity of the KvN operator for autonomous ODEs) and the assumption that finite bases exist that preserve unitarity; no new physical entities or ad-hoc fitted constants are introduced in the abstract.

axioms (2)

domain assumption The Koopman-von Neumann operator is unitary for any autonomous ODE, even non-Hamiltonian ones.
Stated as the main advantage over conventional transfer operators; invoked to justify the quantum-circuit representation.
domain assumption Finite-dimensional projections of the operator remain unitary when suitable bases and domains are chosen.
Directly asserted as necessary for the matrix representation and quantum-circuit step.

pith-pipeline@v0.9.0 · 5481 in / 1349 out tokens · 32209 ms · 2026-05-10T17:18:34.887700+00:00 · methodology

discussion (0)

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2026
[60]

SinceA 3 =−r 2A, the exponential series for exp(t A) reduces to a combination ofI,A, andA 2, i.e., exp(tA) =I+ sin(rt) r A+ 1−cos(rt) r2 A2 =   cos(rt)− sin(rt) r z1 − sin(rt) r z2 − sin(rt) r z3 sin(rt) r z1 1− 1−cos(rt) r2 z2 1 − 1−cos(rt) r2 z1z2 − 1−cos(rt) r2 z1z3 sin(rt) r z2 − 1−cos(rt) r2 z1z2 1− 1−cos(rt) r2 z2 2 − 1−cos(rt) r2 z2z3 sin(r...