Inverse Physics-informed neural networks procedure for detecting noise in open quantum systems

Gubio G. de Lima; Iann Cunha; Leonardo Kleber Castelano

REVIEW 4 major objections 6 minor 22 references

Inverse Physics-informed neural networks procedure for detecting noise in open quantum systems

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that an inverse physics-informed neural network can learn Hamiltonian parameters and decay rates of open quantum systems from noisy expectation-value data, reaching about 1% error on two-qubit simulations.

desk verdict A sensible Lindblad extension of PINNverse with convincing synthetic results, but the experimental comparison is unfair and the identifiability question is left open. read the letter →

arxiv 2507.12552 v1 pith:V7GKBPQP submitted 2025-07-16 quant-ph physics.comp-ph

Gubio G. de Lima , Iann Cunha , Leonardo Kleber Castelano This is my paper · ORCID

classification quant-phphysics.comp-ph

keywords quantumsystemidentificationinversephysics-informedneuralnetworksLindbladmasterequationHamiltonianlearningnoisecharacterizationcrosstalkdetectionopensystemsparameterestimation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the inverse physics-informed neural network (PINNverse) approach from closed to open quantum systems by encoding the Lindblad master equation into the training loss. It claims that from time traces of fifteen two-qubit observables the network can simultaneously recover all Hamiltonian coupling coefficients and the four amplitude-damping and dephasing decay rates, reaching mean absolute percentage errors around 1% when more than fifteen collocation points are used. The same procedure identifies crosstalk coupling terms in noisy simulated data and, on a real single-qubit experiment, fits the observed expectation values with lower mean absolute error than the analytical model it is compared against. The broader claim is that this offers a scalable, noise-resilient, unsupervised route to quantum system identification that needs only expectation-value data.

What carries the argument

The central object is the inverse physics-informed neural network trained on the Heisenberg-picture Lindblad equations for the fifteen independent two-qubit observables $\langle S_{\mu,\nu}\rangle(t)$. The composite loss $L = L_m + L_d$ couples the residual of the differential equations ($L_m$) with the mismatch to measured expectation values ($L_d$); minimizing it with respect to both the network weights and the unknown physical parameters $J_{\mu,\nu}$ and $\gamma_k$ forces the network to simultaneously solve the dynamics and match the data. The dissipator $D[S_{\mu,\nu}](t)$ carries the assumed noise channels, amplitude damping ($\sigma_-$ on each qubit) and dephasing ($\sigma_3$ on each qubit), and it is this term that allows the decay rates to be learned alongside the coherent couplings.

What would settle it

Generate synthetic data with a known non-Markovian or correlated-noise master equation, or with an extra unmodeled channel such as depolarizing noise, and check whether the inferred Hamiltonian and decay rates remain close to the true values; if they shift substantially or the model loss grows, the central claim fails. A cheaper check is to construct two different parameter sets that produce the same fifteen observable trajectories and show that PINNverse chooses one arbitrarily, which would break the uniqueness assumption.

Watch

Extended reading notes

Core claim

On its own terms, the paper shows that PINNverse, trained with a composite loss made of the Heisenberg-picture Lindblad equations (the model loss) and deviations from measured expectation values (the data loss), can identify both coherent and dissipative parameters of an open quantum system without labeled parameter examples. In two-qubit synthetic experiments with randomly sampled couplings and decay rates, the MAPE drops to about $10^{-2}$ once the number of collocation points exceeds fifteen; with fifty collocation points and Gaussian measurement noise of standard deviation 0.02, the reconstructed crosstalk trajectories reach a MAPE of 0.23%, individual decay rates stay below 1%, and coupling coefficients stay below 1.3%. Applied to a published single-qubit dataset, the inferred parameters are close to the reference values (for instance, J2 = -1.52 MHz versus -1.57 MHz and gamma1 = 1.26e-1 MHz versus 1.28e-1 MHz), and the model yields lower mean absolute errors on two of the three measured observables.

Load-bearing premise

The load-bearing premise is that the device is exactly described by a Markovian Lindblad master equation whose only noise channels are the assumed local amplitude-damping and dephasing operators, and that the fifteen measured trajectories uniquely determine all fitted coefficients.

Editorial extensions

If this is right

With roughly fifteen to fifty collocation points, Hamiltonian coefficients and decay rates are recoverable at around 1% MAPE, so a modest amount of expectation-value data may replace costly full process tomography for simple open systems.
The same network can flag unwanted qubit-qubit couplings (crosstalk) in the presence of dissipation, because it separates the coherent $J_{\mu,\nu}$ terms from the decay rates $\gamma_k$.
On real single-qubit data, PINNverse fits the whole set of observables with a mean absolute error of $4.8 \times 10^{-3}$ for $\langle\sigma_1\rangle$, below the analytical benchmark's $1.1 \times 10^{-2}$, supporting the claim that the approach transfers from simulation to experiment.
Because training is unsupervised and requires only observable traces, the procedure can be rerun for many parameter sets without building a labeled training database.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's direct claims, the method's scalability to more than two qubits is not demonstrated by the reported experiments; the freezing mechanism cited from the earlier work would need to be tested in the open-system setting to support the scalability claim.
The identifiability question is left open: the paper fits nineteen parameters (fifteen $J_{\mu,\nu}$ and four $\gamma_k$) from fifteen observable trajectories, and no uniqueness analysis is given, so a natural extension is to search for distinct parameter sets that produce the same observable traces.
Editorial inference: if the assumed channel set (local amplitude damping and dephasing) is wrong, the inferred parameters are likely to absorb misspecification errors, so a practical extension would be to compare models with different Lindblad operator sets using the same data and PINNverse loss.
Editorial inference: the real-data benchmark only covers a single qubit, so a direct next step is to apply the same procedure to two-qubit experimental tomography data, where the crosstalk detection claim could be tested against independently characterized couplings.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the useful part: this paper does a clean, natural extension of the authors' PINNverse framework to open systems governed by Lindblad master equations. The synthetic two-qubit benchmarks show that with more than ~15 collocation points the method recovers random Hamiltonian and decay parameters to about 1% MAPE, and the crosstalk scenario (Fig. 4) keeps errors below ~1.3%. That is a genuine capability gain over the closed-system work in Ref. [13], and the unsupervised, physics-constrained approach is measurement-efficient. Credit where due: the core idea is not novel—it is the same inverse-PINN scheme—but applying it to Lindblad dynamics and demonstrating simultaneous estimation of coherent and dissipative parameters is a worthwhile step.

The soft spots are real but not fatal. The identifiability concern is the most important. The paper fits 19 parameters (15 J's plus 4 gammas) from 15 observable trajectories with a fixed initial state, and offers no identifiability or sensitivity analysis. The MAPE on random synthetic instances is reassuring but does not rule out degeneracies where different parameters produce nearly identical trajectories. The single-qubit experimental fit makes this concrete: PINNverse finds J1 and J3 ~1e-2 MHz where the reference model sets them to zero, and without knowing whether those are identifiable from the data, they might just be overfitting.

The experimental comparison is also unfair. PINNverse is given extra free parameters (J1, J3) that the analytical model does not have, so its lower MAE on ⟨σ1⟩ and ⟨σ3⟩ is expected; it does not demonstrate superiority. The noise-resilience claim is overstated when Fig. 2 shows Jmean MAPE hitting 23% at σ=0.02. And there is no code or hyperparameter listing, which makes the results hard to reproduce.

The Markovianity assumption (fixed local amplitude-damping and dephasing channels) is stated plainly in Sec. II, so it's not hidden, but the paper never tests it or discusses what happens if the true noise is non-Markovian.

Who is this for? Researchers working on ML-based quantum system identification and NISQ calibration. They will get a useful benchmark and a plausible method, but they should treat the experimental comparison as suggestive, not decisive.

My recommendation: send it to peer review. The synthetic evidence is solid enough that a serious referee should look at it, but the paper needs substantial revision—identifiability analysis or at least a discussion of local minima, a fair comparison with equal model complexity, and released code—before it supports the stronger claims made in the abstract.

Referee Report

4 major / 6 minor

Summary. This manuscript extends the inverse physics-informed neural network (PINNverse) method of Ref. [13] to open quantum systems described by Lindblad master equations. The network is trained to minimize a composite loss (Eqs. (5)-(7)) combining the residual of the Heisenberg-picture Lindblad equation with a term that matches experimentally measured expectation values, thereby learning both Hamiltonian coefficients and decay rates. The method is tested on synthetic two-qubit systems with amplitude-damping and dephasing channels, where it reports MAPE values near 1% for the parameters, on a crosstalk scenario, and on real single-qubit data from Ref. [22], where it achieves a lower mean absolute error than the analytical reference model. The paper concludes that PINNverse is a scalable and noise-resilient framework for quantum system identification.

Significance. If the missing identifiability analysis can be supplied, this would be a useful contribution to quantum system identification in the NISQ era, because it offers an unsupervised alternative to full process tomography and works directly from observable expectation-value trajectories. The paper's strengths are its standard formulation of the Lindblad loss, the synthetic self-consistency benchmarks that recover planted parameters with low error, the explicit crosstalk test case, and the demonstration on real data with a comparison to an analytical model. The principal weaknesses are the absence of any identifiability or uncertainty analysis, the untested Markovian and fixed-channel assumption, and the lack of reproducibility details; these weaknesses currently prevent the stronger claims of 'identification' and 'scalability' from being fully supported.

major comments (4)

[III.A, Eqs. (5)-(7)] The method is presented as identifying 19 parameters (15 Hamiltonian coefficients J_mu_nu with J_00=0 plus four decay rates gamma_k) from 15 observable trajectories, but the manuscript provides no identifiability, sensitivity, or Fisher-information analysis. The low MAPE on random synthetic instances demonstrates only that, for those particular parameter draws, the optimizer finds the planted values; it does not establish that the map from parameters to the observable trajectories is injective. If the map is non-injective, different parameter vectors can produce nearly identical data and the recovered parameters are not identified, even with zero loss. Please add a concrete identifiability analysis (for example, a numerical perturbation study, the Fisher information matrix of the observable map, or a search for alternative parameter vectors that reproduce the same trajectories within noise), and report uncertainties for all recovered parameters.
[III.C] The experimental single-qubit fit returns small nonzero values J1=2.4e-2 MHz and J3=-1.08e-2 MHz that were set to zero in the reference model, along with gamma3 about 4.4e-5 MHz. No error bars, confidence intervals, cross-validation, or model-comparison statistic are given. Since PINNverse has extra free parameters relative to the analytical model, the lower MAE in Figs. 5-7 is partly an expected consequence of increased model flexibility and does not by itself establish that these extra terms are physical. Please provide uncertainty quantification (e.g., bootstrap over the training runs or over measurement noise) and an out-of-sample or complexity-penalized comparison to support the claim that the method detects real features rather than overfitting.
[Section II, paragraph before Eq. (3)] The framework assumes a Markovian Lindblad master equation with a fixed set of local amplitude-damping and dephasing channels. This assumption is stated but never tested against model mismatch. If the true device noise is non-Markovian, time-correlated, or produced by different channels, the recovered 'Hamiltonian parameters' and 'decay rates' will be systematically biased, and the experimental demonstration in Section III.C cannot distinguish such bias from genuine parameter values. Please add a model-mismatch test (for example, generate synthetic data from a non-Markovian master equation or from a different channel set and show the resulting parameter bias and residuals), or justify the Markovian assumption for the specific device of Ref. [22].
[III (general)] The paper does not report the neural network architecture (number of layers and neurons, activation function), the optimizer, learning rate or scheduling, number of training iterations, initialization strategy, or computational cost, and no code or data are made available. These details are essential for reproducing Figs. 1-7 and for assessing the reliability of a machine-learning-based inference method. Please include a complete hyperparameter table and consider releasing the training code and synthetic data-generation scripts.

minor comments (6)

[Eq. (8)] The MAPE in Eq. (8) divides by |P_i_exact|; this is undefined for zero-valued parameters. Please specify the handling of zero or near-zero parameters, which is relevant for the crosstalk setting where some couplings may be small or zero.
[Fig. 2 text] In the text describing Fig. 2, 'gamma_mean = P_k gamma4/4' appears to be a typo; it should read gamma_mean = (1/4) sum_k gamma_k.
[Figs. 1, 2, 4] The error bars are described as the maximum and minimum values across runs, but the number of runs and the random seeds are not reported, making the displayed dispersion unverifiable.
[III.B] The text reports a MAPE of 0.23% for the reconstructed observables, while Fig. 4 reports MAPE for individual parameters; these are different quantities and should be labeled consistently to avoid confusion.
[Abstract and Conclusions] The term 'scalable' is used in the abstract and conclusions, but the numerical experiments cover only one- and two-qubit systems and the freezing mechanism of Ref. [13] is not implemented here; please soften this claim or provide a scaling study.
[III.C] The Pauli operators sigma1, sigma2, sigma3 in the single-qubit section are used without an explicit definition; please state the convention (e.g., sigma1 = sigma_x, sigma2 = sigma_y, sigma3 = sigma_z).

Circularity Check

1 steps flagged · score 2.0 of 10

Experimental MAE comparison is in-sample fit of a more flexible model, but the central synthetic identification is self-contained.

fitted input called prediction [Section III.C, 'Experimental data for one qubit', around Eq. (10) and Figs. 5-7]
"The MAE for PINNverse related to Fig. 5(b) reaches the value 4 .8 × 10−3, while the MAE for the analytical model is 1 .1 × 10−2. This result shows that PINNverse is more precise on average to predict the experimental data."

The PINNverse parameters are obtained by minimizing L = Lm + Ld, with Ld being the sum of squared differences between model trajectories and the experimental data at the same collocation points. The MAE reported here is computed on those same experimental data via Eq. (10), i.e., on the training set. The analytical reference model is a constrained subset (J1 = J3 = 0), so a strictly more flexible model fitted to the same data is expected to match the training data at least as well. Thus the claim of being 'more precise... to predict the experimental data' is not an independent prediction but a restatement of the fitting objective: the extra parameters J1 and J3 are chosen precisely to reduce this same loss. This is the fitted-input-called-prediction pattern.

full rationale

The paper's central derivation chain is not circular. For the synthetic two-qubit experiments, data are generated by numerically integrating Eq. (3) with the same Hamiltonian and Lindblad structure used in the PINNverse loss, and the reported MAPE measures recovery of those known parameters. This is a standard self-consistency check, not a circular prediction. The method builds on the authors' prior PINNverse work [13] as a legitimate algorithmic dependency, and no uniqueness theorem or ansatz is smuggled in via self-citation: the Markovian Lindblad assumption with dephasing and amplitude-damping channels is stated explicitly as a modeling choice with a standard reference. Two genuine concerns are the lack of identifiability analysis for the 19 fitted two-qubit parameters from 15 observable trajectories and the unverified Markovian assumption, but these are correctness risks, not circularity. The only identifiable circular step is the experimental single-qubit comparison: the PINNverse parameters are fitted by minimizing the data-matching loss, and the same experimental data are then used to compute MAE and claim that PINNverse 'predicts' the data better than the analytical model. Because the analytical model is a restricted version (J1 = J3 = 0) of the PINNverse model, the lower in-sample MAE is a direct consequence of the additional fitted degrees of freedom, not evidence of predictive advantage. However, this step is supporting rather than central: the main claim of simultaneous Hamiltonian and decay-rate identification is validated by parameter-recovery experiments that are self-contained. Overall circularity score is therefore 2.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The model does not introduce new physical entities. The main free parameters are the Hamiltonian coefficients and decay rates that are the targets of inference; the only additional degrees of freedom beyond the baseline are the two extra J parameters in the experimental fit. The Markovian Lindblad form and the identifiability of the parameters from the chosen observables are the key assumptions.

free parameters (1)

Extra Hamiltonian coefficients J1 and J3 in the single-qubit experimental fit = J1 = 2.4e-2 MHz, J3 = -1.08e-2 MHz
These coefficients are set to zero in the reference model of Ref [22] but are fit by PINNverse, adding two degrees of freedom that make a lower training-data MAE expected.

assumptions (4)

domain assumption The two-qubit system follows a Markovian Lindblad master equation with local dephasing (sigma3) and amplitude-damping (sigma-) channels.
Invoked in Section II when defining the Lindblad operators L1..L4; not justified from the underlying device physics.
standard math The Heisenberg-picture adjoint master equation (Eq. 3) with dissipator (Eq. 4) governs the observable dynamics.
Standard result for Lindblad dynamics; used as the physics constraint in the loss.
domain assumption The 15 non-identity Pauli observables provide sufficient information to uniquely determine all Hamiltonian coefficients and decay rates.
Assumed implicitly; no identifiability or uniqueness analysis is provided, though the observed recovery at about 1% MAPE suggests empirical identifiability for the tested random instances.
domain assumption For the experimental single-qubit data, the noise is described by one dephasing channel sigma3 and two amplitude-damping channels sigma- and sigma+ at finite temperature.
Taken from Ref [22]; the PINNverse model uses the same three Lindblad operators.

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Cite this review

Pith. "Pith review of Inverse Physics-informed neural networks procedure for detecting noise in open quantum systems." pith.science (2026). https://pith.science/paper/V7GKBPQP

@misc{pith2026250712552,
  author       = {Pith},
  title        = {Pith review of: Inverse Physics-informed neural networks procedure for detecting noise in open quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7GKBPQP}},
  note         = {Machine review of arXiv:2507.12552}
}

read the original abstract

Accurate characterization of quantum systems is essential for the development of quantum technologies, particularly in the noisy intermediate-scale quantum (NISQ) era. While traditional methods for Hamiltonian learning and noise characterization often require extensive measurements and scale poorly with system size, machine learning approaches offer promising alternatives. In this work, we extend the inverse physics-informed neural network (referred to as PINNverse) framework to open quantum systems governed by Lindblad master equations. By incorporating both coherent and dissipative dynamics into the neural network training, our method enables simultaneous identification of Hamiltonian parameters and decay rates from noisy experimental data. We demonstrate the effectiveness and robustness of the approach through numerical simulations of two-qubit open systems. Our results show that PINNverse provides a scalable and noise-resilient framework for quantum system identification, with potential applications in quantum control and error mitigation.

Figures

Figures reproduced from arXiv: 2507.12552 by the authors.

**Figure 2.** illustrates the sensitivity of the parameter estimation performance of the PINNverse to the level of noise added to the simulated experimental data. Specifically, we report the MAPE for the mean dissipation coefficient γmean = P k γ4/4 and the Hamiltonian coupling Jmean, as a function of the standard deviation of Gaussian noise. The results are averaged over multiple independent PINN runs, and the error bars repres… view at source ↗

**Figure 3.** FIG. 3. Expectation values as a function of time for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. The MAPE plotted on a logarithmic scale for each pa [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 7.** Figure 7: (a) show that both theoretical models show good agreement with the experimental data, although the PINNverse framework offers a slightly better fit up to the time 6µs. This can be clearly seen in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 6.** Figure 6: FIG. 6. (a) The observable [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [13]

L. K. Castelano, I. Cunha, F. S. Luiz, R. de Jesus Napoli- tano, M. V. d. S. Prado, and F. F. Fanchini, Combin- ing physics-informed neural networks with the freezing mechanism for general hamiltonian learning, Phys. Rev. A 110, 032607 (2024)

work page 2024

[2] [22]

Ficheux, S

Q. Ficheux, S. Jezouin, Z. Leghtas, and B. Huard, Dy- namics of a qubit while simultaneously monitoring its re- laxation and dephasing, Nature Communications 9, 1926 (2018)

work page 2018

[3] [1]

Wittler, F

N. Wittler, F. Roy, K. Pack, M. Werninghaus, A. S. Roy, D. J. Egger, S. Filipp, F. K. Wilhelm, and S. Machnes, Integrated tool set for control, calibration, and charac- terization of quantum devices applied to superconducting qubits, Physical review applied 15, 034080 (2021)

work page 2021

[4] [2]

B. Rost, L. Del Re, N. Earnest, et al., Long-time error- mitigating simulation of open quantum systems on near term quantum computers, npj Quantum Information 11, 10 (2025)

work page 2025

[5] [3]

Erhard, J

A. Erhard, J. J. Wallman, L. Postler, P. Schindler, T. Monz, J. Emerson, and R. Blatt, Characterizing large- scale quantum computers via cycle benchmarking, Na- ture Communications 10, 5347 (2019), published: 25 November 2019; Received: 22 March 2019; Accepted: 08 October 2019

work page 2019

[6] [4]

I. L. Chuang and M. A. Nielsen, Prescription for ex- perimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44, 2455 (1997)

work page 1997

[7] [5]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Ancilla-assisted quantum process tomogra- phy, Phys. Rev. Lett. 90, 193601 (2003)

work page 2003

[8] [6]

S. T. Merkel, J. M. Gambetta, J. A. Smolin, S. Poletto, A. D. C´ orcoles, B. R. Johnson, C. A. Ryan, and M. Stef- fen, Self-consistent quantum process tomography, Phys. Rev. A 87, 062119 (2013)

2013