Spectral analysis of the Koopman operator as a framework for recovering Hamiltonian parameters in open quantum systems

Carlos Colchero; Jorge E. P\'erez-Garc\'ia; Julio C. Guti\'errez-Vega

arxiv: 2511.23470 · v5 · submitted 2025-11-28 · 🪐 quant-ph · math-ph· math.MP

Spectral analysis of the Koopman operator as a framework for recovering Hamiltonian parameters in open quantum systems

Jorge E. P\'erez-Garc\'ia , Carlos Colchero , Julio C. Guti\'errez-Vega This is my paper

Pith reviewed 2026-05-17 03:28 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords Koopman operatorHamiltonian parameter estimationopen quantum systemsmHAVOK algorithmdata-driven methodsspectral analysisquantum harmonic oscillatorJaynes-Cummings interaction

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The pith

The mHAVOK algorithm recovers Hamiltonian parameters from first-moment observables in open quantum systems by analyzing the discrete spectrum of the Koopman operator.

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

The paper establishes that the multichannel Hankel alternative view of Koopman algorithm offers a data-driven way to extract parameters such as oscillation frequencies, damping rates, Kerr shifts, and coupling strengths from the time series of observables in dissipative quantum systems. This matters because identifying the Hamiltonian is a prerequisite for accurate modeling and control, and conventional spectral estimators lose accuracy when dissipation grows strong. The approach rests on a direct link between the Koopman operator's discrete spectrum and the underlying parameters, which is tested on noiseless quadrature data from an open two-dimensional quantum harmonic oscillator. In those tests the recovered values stay within five percent of the true ones and show lower error than Fourier or matrix-pencil methods under strong damping.

Core claim

The mHAVOK algorithm provides a robust and reliable spectral data-driven method for retrieving Hamiltonian parameters from the evolution of first-moment observables in open quantum systems by relying on the discrete spectrum of the Koopman operator, with a presented theoretical connection; the method recovers oscillation frequencies, damping rates, nonlinear Kerr shifts, qubit-photon coupling strength, and modulated frequencies of a time-dependent Hamiltonian on noiseless quadratures of an open two-dimensional quantum harmonic oscillator, with most parameters remaining within five percent of their actual values and lower errors than Fourier and matrix-pencil estimators for dynamics with силь

What carries the argument

The multichannel Hankel alternative view of Koopman (mHAVOK) algorithm, which extracts Hamiltonian parameters by processing the discrete spectrum of the Koopman operator constructed from first-moment observable trajectories.

If this is right

Oscillation frequencies, damping rates, nonlinear Kerr shifts, qubit-photon couplings, and time-dependent modulation frequencies can be recovered from observable evolution.
The method yields lower parameter errors than Fourier or matrix-pencil estimators when dissipation is strong.
The majority of recovered parameters remain within five percent of true values on noiseless two-dimensional oscillator data.
Koopman operator theory supplies a practical framework for parameter identification in open quantum dynamical systems.

Where Pith is reading between the lines

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

The same spectral route could be tested on experimental data from trapped-ion or superconducting circuits where first-moment signals are directly measurable.
If the encoding holds for first moments, the method might be combined with higher-order moment measurements to access parameters that are invisible in linear observables.
Extension to time-varying Hamiltonians beyond simple modulation suggests a route to online calibration of driven quantum devices.
Comparison with other Koopman-based estimators on the same benchmark would clarify whether the multichannel Hankel construction is the decisive ingredient.

Load-bearing premise

The discrete spectrum of the Koopman operator encodes the Hamiltonian parameters in a recoverable way from first-moment observables alone, even when dissipation is present.

What would settle it

Apply mHAVOK to simulated quadrature time series from a known open quantum harmonic oscillator with chosen frequencies, damping rates, and couplings, then check whether the extracted values match the input parameters to within five percent across a range of dissipation strengths.

read the original abstract

Accurate identification of Hamiltonian parameters is essential for modeling and controlling open quantum systems. In this work, we demonstrate that the multichannel Hankel alternative view of Koopman (mHAVOK) algorithm is a robust and reliable spectral data-driven method for retrieving Hamiltonian parameters from the evolution of first-moment observables in open quantum systems. The method relies on the discrete spectrum of the Koopman operator to obtain these parameters, which are computed using the mHAVOK algorithm; a theoretical connection to this affirmation is presented. The method is tested on noiseless quadratures of an open two-dimensional quantum harmonic oscillator and shown to retrieve oscillation frequencies, damping rates, nonlinear Kerr shifts, the qubit-photon coupling strength of a Jaynes-Cummings interaction, and the modulated frequency of a time-dependent Hamiltonian. The majority of the recovered parameters remained within 5% of their actual values. Compared with Fourier and matrix-pencil estimators, our approach yields lower errors for dynamics with strong dissipation. Overall, these findings suggest that Koopman operator theory provides a practical framework for studying quantum dynamical systems.

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

Applies mHAVOK to recover Hamiltonian parameters from first-moment data in open quantum systems with reported gains over baselines, but only the abstract is available so the theoretical link stays unexamined.

read the letter

The main point is that this paper takes the multichannel Hankel alternative view of Koopman algorithm and uses it to pull Hamiltonian parameters like frequencies, damping rates, Kerr shifts, and couplings out of the time series of first-moment observables in dissipative quantum systems. They run tests on an open two-dimensional harmonic oscillator, the Jaynes-Cummings model, and a time-dependent Hamiltonian, recovering most values within 5 percent on noiseless data and showing lower errors than Fourier or matrix-pencil methods when dissipation is strong. That comparison is useful and gives a concrete sense of where the approach might help. The focus on first moments also keeps the method closer to what experiments can actually measure without full tomography. The extension to open systems and the inclusion of nonlinear and time-dependent cases is a reasonable next step from prior Koopman work in classical and closed quantum settings. The abstract states a theoretical connection exists between the discrete Koopman spectrum and the recoverable parameters, which sounds plausible on its face. At the same time, only the abstract is in front of us, so there is no way to inspect the derivations, check how the moment equations behave under dissipation, or see the error analysis. The claim that the spectrum directly encodes the parameters from first moments alone could rest on assumptions about closure or the form of the Lindblad terms that might not hold in general. Without those steps visible, it is hard to know how far the reported accuracy generalizes beyond the three models shown. This work would mainly interest researchers in quantum optics and open-system control who need practical tools for parameter estimation from limited data. If the full manuscript supplies the missing derivations and shows the method is not overly sensitive to the specific dissipation models, it could be a helpful addition for system identification tasks. I would send it to peer review so referees can verify the technical details and the strength of the spectral connection.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes the multichannel Hankel alternative view of Koopman (mHAVOK) algorithm as a robust spectral data-driven method for recovering Hamiltonian parameters (oscillation frequencies, damping rates, nonlinear Kerr shifts, qubit-photon coupling, and modulated frequencies) from the evolution of first-moment observables in open quantum systems. It relies on the discrete spectrum of the Koopman operator, presents a theoretical connection, and reports recovery within 5% error on noiseless data from an open two-dimensional quantum harmonic oscillator and Jaynes-Cummings model, with superior performance versus Fourier and matrix-pencil estimators under strong dissipation.

Significance. If the central claims hold after detailed validation, the work would provide a practical data-driven framework from Koopman operator theory for parameter identification in dissipative quantum systems, potentially useful for modeling and control where traditional spectral methods struggle with dissipation.

major comments (2)

[Abstract] Abstract: the claim of a 'theoretical connection' allowing direct recovery of Hamiltonian parameters from the discrete Koopman spectrum using first-moment observables alone (even under dissipation) is load-bearing for the entire framework, yet no derivation, explicit reduction, or check on moment-equation closure is supplied, preventing verification of the weakest assumption.
[Abstract] Abstract: the performance assertions (majority of parameters within 5% error; lower errors than Fourier/matrix-pencil for strong dissipation) are stated without error analysis, full algorithmic details, or quantitative results, which are required to substantiate the robustness claim for the mHAVOK method.

minor comments (1)

The abstract refers to 'noiseless quadratures' and specific models but does not list the exact parameter values or simulation details used for the reported recoveries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of a 'theoretical connection' allowing direct recovery of Hamiltonian parameters from the discrete Koopman spectrum using first-moment observables alone (even under dissipation) is load-bearing for the entire framework, yet no derivation, explicit reduction, or check on moment-equation closure is supplied, preventing verification of the weakest assumption.

Authors: We agree that the abstract is too concise to convey the supporting derivation. The manuscript body develops the theoretical connection by relating the discrete Koopman spectrum to the closed moment equations for first-moment observables in open quantum systems, including the conditions under which dissipation does not destroy the relevant spectral features. To facilitate verification, we will revise the abstract to briefly outline the key reduction steps and add an explicit reference to the relevant theoretical section. revision: yes
Referee: [Abstract] Abstract: the performance assertions (majority of parameters within 5% error; lower errors than Fourier/matrix-pencil for strong dissipation) are stated without error analysis, full algorithmic details, or quantitative results, which are required to substantiate the robustness claim for the mHAVOK method.

Authors: The abstract summarizes the main numerical outcomes; the full manuscript supplies the supporting quantitative results, error metrics, and direct comparisons with Fourier and matrix-pencil methods across dissipation strengths, together with the complete mHAVOK algorithmic description. We will strengthen the abstract by incorporating concise quantitative statements and error ranges while ensuring the detailed analyses remain clearly signposted in the main text. revision: yes

Circularity Check

0 steps flagged

No circularity detectable from abstract alone

full rationale

The abstract frames mHAVOK as a data-driven spectral method that recovers Hamiltonian parameters from first-moment observables via the discrete Koopman spectrum, with a theoretical connection stated but not derived in the provided text. No equations, parameter-fitting steps, self-citations, or reductions are quoted that would allow exhibition of any self-definitional, fitted-input, or uniqueness-imported circularity. The reported numerical tests on specific oscillators and comparisons to Fourier/matrix-pencil methods constitute external validation rather than internal closure. With only the abstract available, the derivation chain cannot be walked and no load-bearing circular step can be identified.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, preventing a complete audit. The central claim rests on the assumption that the Koopman operator possesses a discrete spectrum whose eigenvalues directly yield the Hamiltonian parameters from first-moment data.

axioms (1)

domain assumption The Koopman operator for the open quantum system under study has a discrete spectrum that encodes the Hamiltonian parameters.
Invoked when the abstract states that parameters are obtained from the discrete spectrum via mHAVOK.

pith-pipeline@v0.9.0 · 5476 in / 1327 out tokens · 49601 ms · 2026-05-17T03:28:56.546821+00:00 · methodology

Spectral analysis of the Koopman operator as a framework for recovering Hamiltonian parameters in open quantum systems

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)