Quantum Koopman Method Overview

• Quantum Koopman Method is a framework that linearizes nonlinear, open, and hybrid quantum dynamics by mapping them onto linear evolution in observable or latent spaces.
• It encompasses various formulations including data-driven spectral analysis, hybrid quantum-classical coupling, and variational or quantum computing implementations with high precision.
• Applications demonstrate efficient system identification, accurate spectral recovery, and robust quantum-optical surrogate modeling, impacting band structure, topology, and quantum optimization.

Searching arXiv for papers on Quantum Koopman Method and related Koopman-based quantum dynamics. Searching arXiv for foundational Koopman-von Neumann and hybrid quantum-classical Koopman work relevant to QKM. Searching arXiv for data-driven Koopman applications to quantum systems, band structure, and variational dynamics. Quantum Koopman Method (QKM) denotes, across current arXiv literature, a family of operator-theoretic and data-driven constructions that use Koopman ideas to recast nonlinear, hybrid, or reduced quantum dynamics as linear evolution on observables, wavefunctions, or lifted latent variables. In this usage, QKM is not a single canonical formalism. Some works explicitly introduce the term for quantum-compatible latent linearization and unitary simulation, while others are better understood as partial bridges: open-system spectral identification from measured expectation values, Koopman–von Neumann or Koopman–van Hove embeddings of classical and hybrid dynamics, Koopman-DMD reconstruction of quantum bands and topology, and Koopman analysis of variational-parameter dynamics in quantum optimization and imaginary-time evolution (Zhang et al., 29 Jul 2025, Pérez-García et al., 28 Nov 2025, Pan et al., 7 May 2026, Bauer et al., 2023, Okuma, 25 Mar 2026).

1. Scope and principal variants

The literature does not present a single universally accepted definition of QKM. Instead, several strands recur. Some papers are explicitly quantum-physical: they infer Hamiltonian, dissipation, band, or topological information from quantum data. Others are hybrid quantum-classical: they represent the classical sector by Koopman wavefunctions and couple it to a quantum subsystem. A third group applies Koopman analysis to reduced quantum representations such as variational parameters or latent embeddings learned from trajectories. This suggests that QKM is best treated as an umbrella term for Koopman-based linearization strategies in settings where the target dynamics, observables, or computational architecture are quantum, quantum-classical, or quantum-computing-oriented (Pérez-García et al., 28 Nov 2025, Pan et al., 7 May 2026, Luo et al., 2022).

Form of QKM	Representative papers	Core object
Data-driven quantum spectral analysis	(Pérez-García et al., 28 Nov 2025, Pan et al., 7 May 2026, Hunstig et al., 2023)	Expectation values, wavefields, or controlled response signals
Hybrid Koopman formulations	(Bouthelier-Madre et al., 2023, Gay-Balmaz et al., 2021, Bauer et al., 2023)	Classical Koopman wavefunctions coupled to quantum states
Variational and algorithmic QKM	(Okuma, 25 Mar 2026, Luo et al., 2022, Zhang et al., 29 Jul 2025)	Parameter flows or latent unitary surrogates

A common misconception is to equate QKM with a fully quantum operator theory on Hilbert-space amplitudes or density matrices alone. Several papers explicitly work instead with expectation trajectories, reduced observables, or semiclassical models. In the open-system identification setting, for example, the Koopman operator acts on functions of an effective dynamical state built from measured moments rather than directly on density operators (Pérez-García et al., 28 Nov 2025). In the band-structure setting, the cleanest energy interpretation requires full wavefunction data, whereas observable-only data generally yields transition frequencies rather than absolute energies (Pan et al., 7 May 2026).

2. Operator-theoretic and hybrid foundations

One foundational route to QKM starts from Koopman–von Neumann mechanics. A general classical first-order system $\dot x^j=v^j(x,t)$ can be rewritten as a Schrödinger-like equation for a phase-space wavefunction $\psi$ with $f=\psi^\dagger\psi$ and Hermitian generator

$$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$

so that $i\hbar \partial_t\psi=\hat H\psi$. In this formulation, even nonlinear, non-Hamiltonian, and dissipative classical dynamics become unitary in an enlarged Hilbert space, which is why the construction is attractive for quantum simulation (Joseph, 2020). A closely related numerical program approximates the Koopman–von Neumann generator

$$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$

by Galerkin or data-driven projection, then whitens the projected matrix so that it becomes skew-symmetric and therefore exponentiates to a unitary propagator suitable for quantum-circuit realization (Klus et al., 9 Apr 2026).

A second foundational route concerns hybrid quantum-classical systems. In the hybrid $C^*$-algebraic formalism, observables live in $\mathcal A_H=\mathcal A_C\otimes \mathcal A_Q$, states are represented as density matrices on the GNS Hilbert space, and dynamics is defined as an automorphism of the observable algebra, with the dual evolution furnishing a hybrid master equation. The classified unitary generators have a Koopman form in the classical sector, but the admissible unitary family is structurally limited and does not include full back-reaction of the quantum subsystem on the classical drift (Bouthelier-Madre et al., 2023). Related variational formulations use hybrid wavefunctions $\Upsilon(q,p,x)$, exact factorization $\Upsilon=\chi\psi$, and a phase-space Berry connection to produce gauge-invariant hybrid wave equations that preserve both the positivity of the quantum density matrix and the positivity of the classical Liouville density. Their special cases include mean-field and Ehrenfest models (Gay-Balmaz et al., 2021, Gay-Balmaz et al., 2021).

The mixed quantum-classical “koopmon” method pushes this line further by regularizing a nonlinear hybrid continuum model and closing it with singular particles carrying both classical phase-space coordinates and local quantum density matrices. In Tully-type nonadiabatic benchmarks, these koopmons reproduce features of fully quantum dynamics beyond standard Ehrenfest mean field, while remaining fundamentally an MQC approximation rather than an exact many-body quantum scheme (Bauer et al., 2023).

3. Open quantum-system identification from measured observables

A concrete data-driven QKM for open quantum systems is provided by the spectral identification framework developed for Lindblad dynamics of an open two-dimensional quantum harmonic oscillator and its Kerr, Jaynes–Cummings, and time-modulated extensions (Pérez-García et al., 28 Nov 2025). The measured signals are first moments of experimentally accessible observables, such as quadratures,

$\psi$0

with Lindblad evolution

$\psi$1

For the linear open oscillator, the first moments close exactly, so the quadrature vector satisfies $\psi$2. In this regime, QKM is not merely a surrogate reconstruction of trajectories; it is a spectral identification method in which the learned matrix $\psi$3 approximates the discrete spectrum of the Koopman generator on a finite observable subspace.

The computational engine is multichannel Hankel Alternative View of Koopman (mHAVOK). Multichannel time series are delay embedded into block Hankel matrices, reduced by thin SVD, partitioned into resolved and forcing coordinates through per-column coefficients of determination, and finally regressed into a forced linear model

$\psi$4

The paper’s central QKM claim is that the spectrum of $\psi$5 approximates the discrete spectrum of the Koopman generator restricted to the identified finite-dimensional subspace. This gives explicit maps from Koopman eigenvalues to physical parameters: $\psi$6 for damping, $\psi$7 for oscillator frequencies, eigenvalue ladder spacings for Kerr nonlinearities, dressed-frequency splittings for Jaynes–Cummings coupling, and sideband spacing for modulation frequency.

The reported identification performance is strong in regimes where the relevant spectrum is well separated. For the linear open 2D oscillator with $\psi$8, the average frequency error is $\psi$9 and the damping-rate error is $f=\psi^\dagger\psi$0. With stronger damping $f=\psi^\dagger\psi$1, the frequency error rises to $f=\psi^\dagger\psi$2, but the damping-rate error improves to $f=\psi^\dagger\psi$3, and the method outperforms FFT and matrix-pencil estimators under strong dissipation. In nonlinear cases, Kerr parameters are recovered with errors as low as $f=\psi^\dagger\psi$4 and remain below $f=\psi^\dagger\psi$5 across the reported sweep; Jaynes–Cummings coupling is recovered best in weak-to-moderate coupling away from resonance; modulation frequencies are recoverable when sidebands remain spectrally resolvable. The main caveats are that the demonstrations use noiseless quadrature trajectories, identifiability deteriorates when higher moments dominate first-moment dynamics, and strong resonance or dense sideband structure destabilizes spectral interpretation (Pérez-García et al., 28 Nov 2025).

4. Controlled quantum-optical surrogate modeling

A separate QKM line treats driven quantum-optical response dynamics as a controlled Koopman problem. In the photon-echo setting, the system is an ensemble of inhomogeneously broadened two-level systems governed by the optical Bloch equations in the rotating-wave approximation, with microscopic polarization $f=\psi^\dagger\psi$6 and occupation $f=\psi^\dagger\psi$7 driven by the Rabi frequency $f=\psi^\dagger\psi$8 and detuning $f=\psi^\dagger\psi$9 (Hunstig et al., 2023). The observable of interest is the macroscopic polarization

$$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$0

which produces the photon echo.

The learned model is a bilinear Koopman surrogate constructed by EDMD. In the simplest BilinearEDMDc (BE) variant, the control is decomposed into pulse amplitude and detuning directions, and the Koopman step is interpolated as

$$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$1

Because finite-time interpolation errors accumulate, this naive bilinear approximation is unstable for realistic detuning ranges. The key stabilization device is BERG, a refined detuning grid with local interpolation between neighboring trained detunings. This is the main methodological finding of the paper: the location and density of control points directly affect long-horizon stability.

For the main photon-echo experiment with $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$2 two-level systems, BERG with $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$3 trained detunings achieves relative peak error $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$4 and $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$5 error $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$6; with $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$7, the relative peak error improves to $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$8 and the $$\hat H=\frac{1}{2}\left(\hat P\cdot v+v\cdot \hat P\right)+\hat W,$$9 error to $i\hbar \partial_t\psi=\hat H\psi$0. In a large-ensemble convergence study with $i\hbar \partial_t\psi=\hat H\psi$1, acceptable accuracy is obtained with training on only $i\hbar \partial_t\psi=\hat H\psi$2 or $i\hbar \partial_t\psi=\hat H\psi$3 detunings, which the paper describes as a reduction in the number of expensive simulations by a factor of $i\hbar \partial_t\psi=\hat H\psi$4. This strand of QKM is semiclassical rather than a full quantum-state Koopman theory: the state variables encode coherence and population, the driving field is classical, and the online benefit comes from replacing repeated ODE solves with low-dimensional matrix propagation (Hunstig et al., 2023).

5. Band structure, localization, and quantum geometry from Koopman-DMD

A more recent QKM formulation establishes a correspondence between Hamiltonian Floquet-Bloch decomposition and Koopman-DMD for quantum and wave systems (Pan et al., 7 May 2026). Here the inputs may be full wavefunction snapshots $i\hbar \partial_t\psi=\hat H\psi$5 or projected observables such as densities, currents, and correlation functions. When full wavefunction data are available, DMD directly recovers single-particle eigenenergies or quasienergies and their spatial modes. When only observables are available, the recovered frequencies are generally transition frequencies $i\hbar \partial_t\psi=\hat H\psi$6 rather than absolute energies. This distinction is central to the paper’s quantum interpretation.

The DMD representation

$i\hbar \partial_t\psi=\hat H\psi$7

is used to extract frequencies, growth or decay rates, and real-space mode profiles. Fourier analysis of extended modes yields dominant momenta, from which one reconstructs

$i\hbar \partial_t\psi=\hat H\psi$8

The same mode set furnishes DMD analogues of spectral density, local density of states, inverse participation ratio, Berry curvature, Zak phase, quantum metric, and Chern number. The topology and geometry are computed through discrete gauge-invariant overlap formulas rather than explicit derivatives, because DMD modes at neighboring $i\hbar \partial_t\psi=\hat H\psi$9 points carry arbitrary $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$0 phases.

The scope of the demonstrations is broad: Anderson localization; static, Floquet, and non-Hermitian SSH chains; 2D SSH higher-order topology; graphene; and the Haldane model. Reported results include data-driven recovery of topological zero modes and Floquet $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$1- and $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$2-modes, discrimination between robust-topological, NHSE-dominant, and Anderson-dominant regimes in non-Hermitian SSH, winding numbers quantized at $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$3 in 2D SSH, vanishing total Berry curvature and enhanced quantum metric near graphene Dirac points, and a quantized $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$4 in the Haldane model. The paper explicitly describes itself as a strong partial bridge rather than a complete foundational quantum Koopman theory. Its main limitations are dependence on high-quality spatiotemporal data, the need for full state access for the cleanest energy interpretation, and unresolved issues involving degeneracies, band tracking, non-Abelian geometry, and realistic noise (Pan et al., 7 May 2026).

6. Variational, optimization-space, and quantum-computing implementations

One active QKM direction treats reduced quantum dynamics themselves as nonlinear systems on parameter space. In variational imaginary-time evolution, the parameter flow $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$5 induced by TDVP or least-squares projection defines a Koopman generator

$$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$6

For overlap observables $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$7, exact closure implies

$$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$8

so physical energies appear as Koopman-generator eigenvalues. A data-driven EDMD implementation on the four-site transverse-field Ising model estimates the ground-state energy from the leading eigenvalue of the learned generator, even though the true ground state lies outside the variational manifold. The sample set is restricted to parameter points where the discrepancy between true and projected imaginary-time dynamics is small, and the same formalism is extended to uniform matrix product states on an infinite chain using TDVP error estimators (Okuma, 25 Mar 2026).

A related but distinct application is Koopman learning in variational quantum optimization. QuACK treats the parameter trajectory of a VQA as a nonlinear discrete-time dynamical system, learns a Koopman surrogate from short segments of exact gradient-based optimization, predicts future iterates, and then restarts from the predicted point with minimal loss. The paper reports more than $$\mathcal Q\psi=-b\cdot\nabla\psi-\frac{1}{2}\operatorname{div}(b)\psi$$9 speedup in the overparameterized regime, more than $C^*$0 speedup in the smooth regime, and about $C^*$1 speedup in the non-smooth regime, with a reported $C^*$2 speedup for a quantum natural-gradient Ising example. The method is guarded by a controlled alternating scheme, so the chosen predicted iterate is never worse than the exact gradient trajectory at the end of a block (Luo et al., 2022).

The most literal use of the name “Quantum Koopman Method” appears in a data-driven nonlinear-simulation framework that learns a hierarchy of latent Hilbert spaces with a deep autoencoder and evolves only latent phases by unitary Koopman operators of the form

$C^*$3

The latent observable is factorized as $C^*$4, the modulus $C^*$5 is held fixed, and the phase is advanced by block-diagonal unitary operators. This design makes the latent evolution explicitly quantum-hardware-friendly. The reported predictions maintain relative errors below $C^*$6 for reaction-diffusion systems and shear flows, and capture key statistics in 2D turbulence, although pointwise turbulence errors can become large. The claimed quantum advantage is conditional: it applies to latent evolution complexity under assumptions of efficient state preparation, measurement, and sufficiently compact observable dimension (Zhang et al., 29 Jul 2025).

7. Limitations, misconceptions, and research directions

A recurrent limitation across QKM is observable closure. Open-system first moments can close exactly for quadratic Lindbladians, but Kerr, cross-Kerr, resonance, and modulation sidebands quickly move the problem into approximately invariant rather than invariant observable subspaces, so parameter recovery becomes indirect or unstable (Pérez-García et al., 28 Nov 2025). In Koopman-DMD band reconstruction, clean energy interpretation depends on full-state access; with observable-only data one generally recovers transition frequencies, and topology or geometry reconstruction requires a smooth mode family across parameter space (Pan et al., 7 May 2026). In variational and optimization-space formulations, non-normality of the learned finite-dimensional generator can make eigenvalues unstable, even when regression error is small (Okuma, 25 Mar 2026, Luo et al., 2022).

Another misconception is that unitarity alone resolves the quantum-computing bottleneck. Koopman–von Neumann and latent-unitary QKM constructions do yield unitary propagators, but end-to-end computational advantage still depends on state preparation, readout, truncation, and basis or dictionary design. Both the Koopman–von Neumann simulation literature and the latent deep-learning QKM explicitly acknowledge that state preparation and measurement are critical unresolved costs (Joseph, 2020, Zhang et al., 29 Jul 2025, Klus et al., 9 Apr 2026). Hybrid formalisms face a different structural limit: the hybrid $C^*$7-algebraic unitary family excludes full back-reaction, while mixed quantum-classical particle methods lose fidelity when the nominally classical sector develops persistent Wigner negativity (Bouthelier-Madre et al., 2023, Bauer et al., 2023).

A broader interpretation of QKM is now emerging in closure and data assimilation. Quantum Mechanical Closure replaces unresolved variables by a density operator on a finite-dimensional Hilbert space of observables, evolves that density operator by a projected Koopman action, and updates it by a quantum Bayes rule implemented through operator-valued feature maps. This suggests that QKM is expanding beyond spectral estimation and surrogate propagation toward positivity-preserving latent-state closure models for partially observed dynamics (Freeman et al., 2022).

Taken together, these developments indicate that QKM is best understood not as a single mature theory, but as a rapidly differentiating research program. Its durable themes are linearization in observable space, spectral interpretation of lifted dynamics, and the use of Hilbert-space or operator-algebraic structure to make nonlinear, open, or hybrid dynamics tractable. Its central open problems remain equally clear: rigorous identifiability for Lindblad systems with partial observations, robust noisy-data theory, scalable handling of continuous spectrum and non-Abelian geometry, principled observable selection, and end-to-end quantum implementations in which state preparation and readout do not erase the advantage promised by unitary Koopman evolution (Pérez-García et al., 28 Nov 2025, Pan et al., 7 May 2026, Zhang et al., 29 Jul 2025).