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The paper claims that swapping a classical extension for a quantum-depth-infused layer in the Allegro interatomic potential improves force-prediction accuracy on a copper-lithium dataset by about 13%, and that the hybrid architecture stays

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Inserting a quantum depth-infused layer into Allegro gives the best force accuracy on a copper-lithium dataset, about 13% better than a classical MLP variant, but not on other datasets.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection Useful architecture work undercut by a single-run, validation-selected 13% claim; the QDI-vs-MLP comparison is not yet demonstrated. the 4 major comments →

arxiv 2602.16908 v2 pith:72NZYSSI submitted 2026-02-18 cond-mat.mtrl-sci cs.LGquant-ph

Multi-objective optimization and quantum hybridization of equivariant deep learning interatomic potentials

G. Laskaris , D. Morozov , D. Tarpanov , A. Seth , J. Procelewska , G. Sai Gautam , A. Sagingalieva , R. Brasher
show 1 more author
A. Melnikov
This is my paper
classification cond-mat.mtrl-sci cs.LGquant-ph
keywords machine learning interatomic potentialsequivariant neural networksquantum-classical hybridquantum depth-infused layermulti-objective hyperparameter optimizationforce prediction accuracycopper-lithium datasetvariational quantum circuit
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper tests whether quantum-classical hybrid layers can improve the accuracy of machine-learned interatomic potentials. The authors take Allegro, an E(3)-equivariant neural network for predicting atomic energies and forces, and build two modified versions: one adds a classical MLP layer, the other replaces that layer with a quantum depth-infused (QDI) variational circuit. Using multi-objective hyperparameter optimization over force error and inference time, they find both variants beat the baseline on force prediction across QM9, rMD17-aspirin, rMD17-benzene, and a newly generated copper-lithium dataset. The central result: on copper-lithium, the quantum-hybrid variant reaches the best force accuracy, about 13% better than the classical variant, and it does so even though its hyperparameters were transferred from another dataset rather than tuned on copper-lithium.

Core claim

On the paper's own terms, the claim is that inserting a quantum depth-infused layer into Allegro, in place of an equivalent classical MLP extension, yields the best overall force-prediction accuracy on the new Cu-Li dataset (validation force MAE 37.8 meV/Å vs 43.3 meV/Å for the classical variant and 45.9 for vanilla Allegro). The hybrid also achieves competitive force errors on aspirin, benzene, and QM9 using hyperparameters transferred from Cu-Li, suggesting the QDI layer's representational capacity transfers across chemistries. The authors attribute the gain to the QDI layer's data re-uploading and entangling variational structure, which lets a few qubits process many input features and ca

What carries the argument

The central mechanism is the quantum depth-infused (QDI) layer, a variational quantum circuit that encodes input features sequentially onto a small qubit lattice using data re-uploading, interleaved with entangling gates and trainable variational layers. It replaces the two hidden MLP layers in the classical Allegro+MLP variant, followed by a fully connected layer. The QDI layer is characterized in the supplementary material as ZX-irreducible, with full parameter retention, a largely diagonal Fisher information matrix with no barren-plateau signature, and a Fourier spectrum with ~79% nonzero coefficients, indicating high expressivity. The other machinery is the multi-objective hyperparameter

Load-bearing premise

The central claim rests on the assumption that the ~13% force-error advantage of Allegro+QDI over Allegro+MLP on Cu-Li reflects the architecture itself, not run-to-run variance, validation-set luck, or the fact that only the QDI variant received full dataset-specific hyperparameter optimization on Cu-Li.

What would settle it

Retrain Allegro+MLP and Allegro+QDI on Cu-Li with the same optimization budget, multiple random seeds, and identical validation splits; if the mean force MAE gap shrinks to within run-to-run noise (or reverses), the claimed quantum advantage does not survive. An even cleaner test: replace the QDI circuit with a random-but-fixed classical nonlinear feature map of similar Fourier expressivity; if it matches the QDI accuracy, the improvement is not tied to the quantum structure.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Both Allegro+MLP and Allegro+QDI outperform baseline Allegro in force accuracy across the tested datasets, so adding depth beyond the standard architecture is a reliable accuracy lever.
  • The QDI variant achieves the best force MAE on Cu-Li among all variants, indicating quantum-classical hybridization can push MLIP accuracy on inorganic systems.
  • The transfer of Cu-Li hyperparameters to organic datasets without retuning still yields competitive results, suggesting the QDI architecture is relatively robust to hyperparameter choice.
  • The optimizer exposes a clear accuracy-inference-time trade-off for each model, and the QDI variant's inference time is only slightly worse than classical variants, making it a practical candidate for production use.

Where Pith is reading between the lines

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

  • The 13% Cu-Li advantage over the classical variant may not replicate on other datasets: QDI was only fully optimized on Cu-Li, while the MLP variant was optimized per dataset, so the comparison on other datasets is not a controlled test of the architecture.
  • The gain could stem from the nonlinear data-re-uploading encoding rather than fundamentally quantum effects; a classical circuit with similar Fourier expressivity might match it, which would be a testable alternative.
  • If the transferred-hyperparameter result generalizes, QDI offers a practical advantage: dataset-specific tuning is expensive, and a quantum layer that works across chemistries with a single configuration would simplify deployment.
  • The supplementary Fisher/ZX/Fourier analysis provides a template for screening future quantum layer designs before expensive training runs, but its predictive value for force accuracy is not demonstrated in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper applies multi-objective hyperparameter optimization (SAMO-COBRA) to the Allegro equivariant interatomic potential, comparing the vanilla model with two architectural variants: Allegro+MLP (extra classical MLP layers) and Allegro+QDI (a quantum-depth-infused variational circuit). Evaluation is performed on QM9, rMD17-aspirin, rMD17-benzene, and a self-generated Cu-Li slab/interface dataset. The central claim is that the QDI variant achieves the best force prediction on Cu-Li, outperforming the classical MLP variant by approximately 13% (37.8 vs 43.3 meV/Å validation force MAE), while being competitive on the other datasets with hyperparameters transferred from Cu-Li. The paper also reports Pareto fronts for accuracy vs inference time and includes a supplementary analysis of the QDI circuit via ZX-calculus, Fisher information, and Fourier expressivity.

Significance. If the central result were robust, it would provide evidence that quantum-classical hybrid layers can improve the accuracy of equivariant MLIPs on inorganic systems, a direction with potential practical value. The paper also demonstrates a systematic multi-objective HPO workflow and releases code for Allegro+MLP, which is useful. However, the headline gain rests on a single best-run validation metric with no error bars, a comparison that is not matched in optimization effort or model capacity, and a proprietary dataset that is not released. The significance is therefore conditional: the claim is plausible but not currently established.

major comments (4)
  1. [Table IV, Section IV] The central claim of a ~13% Cu-Li force-MAE advantage of Allegro+QDI (37.8 vs 43.3 meV/Å) rests on single best-run validation numbers without error bars or repeated seeds. With batch size 1 and 100 epochs (Tables II, III), the 5.5 meV/Å margin is plausibly within run-to-run noise. Moreover, because SAMO-COBRA's objective is exactly this validation force MAE (Section III.E), selecting the best validation run from the HPO history introduces selection bias. Please report mean ± std over multiple seeds and evaluate on a held-out test split, or the 13% claim is not supported.
  2. [Section III.E, Section IV, Tables II-III] The QDI and MLP comparisons are not matched. QDI was optimized only on force MAE, not on inference time, due to a technical issue, while MLP was optimized on both objectives; the QDI Pareto front was manually calculated. The best configurations also differ substantially in capacity: Cu-Li QDI has final MLP width 128, batch 5, lr 4.02e-03, plus QDI depth 3 and 4 input features, whereas Cu-Li MLP has final MLP [4,16], batch 1, lr 9.35e-04. The reported advantage may therefore reflect differential optimization effort or model capacity rather than quantum hybridization. A matched comparison with the same optimization protocol and controlled classical parameter budget is required.
  3. [Data Availability] The Cu-Li dataset, which is the only dataset where QDI was fully optimized and the basis of the headline result, is not released, and no link is given for the Allegro+QDI implementation. This prevents external verification and reproducibility of the central claim. The generation protocol in Section III.B is detailed, but the dataset itself and code should be made available, or at least the raw validation/test predictions for multiple runs.
  4. [Abstract, Table V] The abstract's claim that the hybrid variant is 'competitive on the remaining datasets' is weakened by the QM9 energy MAE of 23.3 meV vs 10.3 for vanilla Allegro and 21.4 for Allegro+MLP (Table V). On rMD17-aspirin, QDI (9.9 meV/Å) is also slightly worse than Allegro+MLP (9.0). Given that QDI hyperparameters were transferred from Cu-Li without dataset-specific optimization, the text should either temper 'competitive' or provide error bars that establish statistical equivalence.
minor comments (5)
  1. [Section II.B / References] References [22] and [23] are the same paper. Also, 'our implementation of SAMO-COBRA' should clarify the relation to the original implementation and whether the specific implementation is available.
  2. [Figure 2] Axis labels are inconsistent ('validation F mae', 'validation f mae') and units appear only in the caption. Standardize the labels and include units on the axes.
  3. [Section V / Discussion] The statement that variants offer 'both better accuracy and time' is not supported by Fig. 2: on rMD17-aspirin and rMD17-benzene, vanilla Allegro has faster inference, and on QM9 the MLP variant has worse energy accuracy. Please revise to reflect the Pareto trade-off.
  4. [Section III.D] The phrase 'weights initialization and nonlinearity methods constitute two hyperparameters that we included to the configuration of the model but did not optimize' conflates hyperparameters with fixed configuration choices. Rephrase for clarity.
  5. [Tables IV-V] Clarify whether the 'Allegro [13]' column values are copied from the original paper or obtained from a re-trained common configuration. The text says a common configuration file was used, but the column label suggests literature values.

Circularity Check

0 steps flagged

No circular derivation: central claim is an empirical benchmark; self-citations are background/implementation citations, not load-bearing.

full rationale

The paper's central result is the Cu-Li force-MAE comparison in Table IV (Allegro+QDI 37.8 meV/A vs Allegro+MLP 43.3 meV/A). This is not a derivation that reduces to its own inputs: both variants were trained and evaluated on the same external data, and the QDI architecture is not defined in terms of the measured MAE. The hyperparameters were selected by SAMO-COBRA using the validation force MAE as one objective, but Allegro+MLP was optimized on the same validation metric (forces MAE and inference time; Sec. III.E); the 13% margin is therefore an empirical difference, not a tautology. No equation in the paper defines a predicted quantity as equal to a fitted parameter. The self-citations [14]-[21] supply the QDI layer and prior hybrid-QML applications, and [22,23,51] the SAMO-COBRA algorithm/code; these are implementation and background citations, not a uniqueness theorem or a load-bearing derivation. The paper itself discloses limitations (Sec. III.E: QDI optimization only on Cu-Li due to a 'technical issue'; Sec. IV: QDI Pareto front 'manually calculated'; Data Availability omits the self-generated Cu-Li dataset; no repeated seeds or error bars in Table IV). These limit robustness and external verifiability but are not circularity. The only circularity-adjacent feature is minor self-citation, which is not load-bearing; hence score 2 rather than 0.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 1 invented entities

The central benchmark rests on hyperparameters fitted to validation data, DFT-generated Cu-Li labels, the base Allegro model, and an assumption that single-split validation MAE is reliable. The QDI layer is the only new architectural ingredient and it is the authors' own prior design.

free parameters (4)
  • Allegro+QDI Cu-Li hyperparameters = rmax=5.0; lr=4.02e-3; batch=5; P=12; layers=3; edge_MLP=4; final_MLP=128; QDI_depth=3; QDI_input=4
    Selected by SAMO-COBRA on the Cu-Li validation force MAE and inference time; the central 13% claim depends on this configuration.
  • Allegro+MLP Cu-Li hyperparameters = rmax=5.0; lr=9.35e-4; batch=1; P=10.3; layers=3; edge_MLP=16; final_MLP=[4,16]
    The classical counterpart used for the 13% comparison; also fitted to the same Cu-Li validation objective.
  • Allegro baseline Cu-Li hyperparameters = rmax=12.0; lr=2e-3; batch=2; P=5; layers=2; edge_MLP=128
    Vanilla Allegro configuration selected by HPO and used as baseline in Table IV.
  • Transferred QDI hyperparameters for rMD17/QM9 = Varies in Table III; text says QDI-exclusive hyperparameters are from Cu-Li
    Because of the stated technical issue, QDI was not fully optimized on the other datasets; performance there depends on hyperparameters fitted elsewhere.
axioms (5)
  • domain assumption Allegro is a valid E(3)-equivariant interatomic potential baseline
    The paper starts from Musaelian et al. [13] and assumes the base architecture is appropriate for energy/force prediction.
  • domain assumption DFT labels from VASP/R2SCAN are ground truth for the Cu-Li dataset
    The self-generated Cu-Li dataset is built from 11,635 DFT structures; all accuracy claims inherit the accuracy of these labels.
  • domain assumption The QDI layer can be trained and evaluated as described, on a simulator or hardware
    The paper does not state whether QDI runs on a quantum simulator or real hardware, but claims an inference-time comparison for it.
  • domain assumption Validation MAE on one split is a reliable performance measure without error bars
    All reported 'best run' scores are validation errors; no repeated seeds or test-set results are given.
  • domain assumption A common configuration file across models makes the comparison fair
    The authors state they could not replicate the exact Allegro [13] config and instead used a common config, changing only optimized hyperparameters.
invented entities (1)
  • Quantum depth-infused (QDI) layer inside Allegro no independent evidence
    purpose: Replaces the final MLP block with a data-re-uploading variational quantum circuit; the paper claims it improves force accuracy on Cu-Li.
    The QDI concept comes from the same group's earlier papers and is here applied to interatomic potentials. No falsifiable prediction outside this benchmark is given; the only evidence is the reported Cu-Li validation MAE.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Multi-objective optimization and quantum hybridization of equivariant deep learning interatomic potentials." pith.science (2026). https://pith.science/paper/72NZYSSI

@misc{pith2026260216908,
  author       = {Pith},
  title        = {Pith review of: Multi-objective optimization and quantum hybridization of equivariant deep learning interatomic potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/72NZYSSI}},
  note         = {Machine review of arXiv:2602.16908}
}
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read the original abstract

Allegro is a machine learning interatomic potential model designed to predict atomic properties in molecules using E(3) equivariant neural networks. When training this model, there tends to be a trade-off between accuracy and inference time. For this reason, we apply multi-objective hyperparameter optimization to both objectives. Additionally, we experiment with modified architectures by constructing variants of Allegro: one extended with additional classical layers and one incorporating quantum-classical hybrid layers. We evaluate all models on QM9, rMD17-aspirin, rMD17-benzene, and a self-generated dataset of copper-lithium structures. As results, both variants surpass Allegro in force prediction accuracy across multiple datasets. The classical variant consistently improves over the baseline, while the quantum-classical hybrid variant achieves the best overall force prediction accuracy on the Cu-Li dataset, where it was fully optimized, outperforming the classical variant by approximately 13%. Notably, the hybrid variant also achieves competitive results on the remaining datasets despite using hyperparameters transferred from Cu-Li without dataset-specific optimization, suggesting that quantum-classical hybridization is a promising direction for enhancing MLIP architectures.

Figures

Figures reproduced from arXiv: 2602.16908 by A. Melnikov, A. Sagingalieva, A. Seth, D. Morozov, D. Tarpanov, G. Laskaris, G. Sai Gautam, J. Procelewska, R. Brasher.

Figure 1
Figure 1. Figure 1: FIG. 1: Quantum depth-infused layer [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Pareto fronts as they generated by our implementation of SAMO-COBRA during multi-objective [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Training curves of all Allegro variants on QM9 dataset using the best hyperparameters, as achieved from [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Training curves for all Allegro variants including Allegro with the parameters reported by Anders et. al. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

61 extracted references · 2 canonical work pages

  1. [1]

    V. L. Deringer, M. A. Caro, and G. Cs´ anyi, Machine learning interatomic potentials as emerging tools for ma- terials science, Advanced Materials31, 1902765 (2019)

    2019

  2. [2]

    D. M. Anstine and O. Isayev, Machine learning inter- atomic potentials and long-range physics, The Journal of Physical Chemistry A127, 2417 (2023)

    2023

  3. [3]

    Mueller, A

    T. Mueller, A. Hernandez, and C. Wang, Machine learn- ing for interatomic potential models, The Journal of chemical physics152(2020)

    2020

  4. [4]

    Argaman and G

    N. Argaman and G. Makov, Density functional theory: 7 Valemae Allegro Allegro [13] Allegro+MLP Allegro+QDI aspirin 3.62.32.6 3.2 benzene 0.21 0.20 0.210.19 CuLi 809.0 - 698.9495.7 QM910.313.4 21.4 23.3 TABLE V: Valemae values for best runs in (meV) for up to 300 epochs on QM9, 100 for CuLi and 3000 for aspirin and benzene. An introduction, American Journa...

    2000

  5. [5]

    M. Orio, D. A. Pantazis, and F. Neese, Density functional theory, Photosynthesis research102, 443 (2009)

    2009

  6. [6]

    K. T. Sch¨ utt, H. E. Sauceda, P.-J. Kindermans, A. Tkatchenko, and K.-R. M¨ uller, Schnet–a deep learn- ing architecture for molecules and materials, The Journal of Chemical Physics148(2018)

    2018

  7. [7]

    J. Fu, S. Yang, F. He, L. He, Y. Li, J. Zhang, and X. Xiong, Sch-net: a deep learning architecture for auto- matic detection of schizophrenia, BioMedical Engineering OnLine20, 75 (2021)

    2021

  8. [8]

    A. P. Bart´ ok, M. C. Payne, R. Kondor, and G. Cs´ anyi, Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons, Physical re- view letters104, 136403 (2010)

    2010

  9. [9]

    V. L. Deringer, A. P. Bart´ ok, N. Bernstein, D. M. Wilkins, M. Ceriotti, and G. Cs´ anyi, Gaussian process regression for materials and molecules, Chemical Reviews 121, 10073 (2021)

    2021

  10. [10]

    Drautz, Atomic cluster expansion for accurate and transferable interatomic potentials, Physical Review B 99, 014104 (2019)

    R. Drautz, Atomic cluster expansion for accurate and transferable interatomic potentials, Physical Review B 99, 014104 (2019)

    2019

  11. [11]

    Lysogorskiy, C

    Y. Lysogorskiy, C. v. d. Oord, A. Bochkarev, S. Menon, M. Rinaldi, T. Hammerschmidt, M. Mrovec, A. Thomp- son, G. Cs´ anyi, C. Ortner,et al., Performant implemen- tation of the atomic cluster expansion (pace) and appli- cation to copper and silicon, npj computational materials 7, 97 (2021)

    2021

  12. [12]

    Batzner, A

    S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Kornbluth, N. Molinari, T. E. Smidt, and B. Kozinsky, E (3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials, Nature communications13, 2453 (2022)

    2022

  13. [13]

    Musaelian, S

    A. Musaelian, S. Batzner, A. Johansson, L. Sun, C. J. Owen, M. Kornbluth, and B. Kozinsky, Learning local equivariant representations for large-scale atomistic dy- namics, Nature Communications14, 579 (2023)

    2023

  14. [14]

    Sagingalieva, M

    A. Sagingalieva, M. Kordzanganeh, N. Kenbayev, D. Kosichkina, T. Tomashuk, and A. Melnikov, Hybrid quantum neural network for drug response prediction, Cancers15, 2705 (2023)

    2023

  15. [15]

    Sagingalieva, S

    A. Sagingalieva, S. Komornyik, A. Senokosov, A. Joshi, C. Mansell, O. Tsurkan, K. Pinto, M. Pflitsch, and A. Melnikov, Photovoltaic power forecasting using quan- tum machine learning, Solar Energy302, 114016 (2025)

    2025

  16. [16]

    Lusnig, A

    L. Lusnig, A. Sagingalieva, M. Surmach, T. Protase- vich, O. Michiu, J. McLoughlin, C. Mansell, G. de’Petris, D. Bonazza, F. Zanconati,et al., Hybrid quantum image classification and federated learning for hepatic steatosis diagnosis, Diagnostics14, 558 (2024)

    2024

  17. [17]

    Sedykh, M

    A. Sedykh, M. Podapaka, A. Sagingalieva, K. Pinto, M. Pflitsch, and A. Melnikov, Hybrid quantum physics- informed neural networks for simulating computational fluid dynamics in complex shapes, Machine Learning: Science and Technology5, 025045 (2024)

    2024

  18. [18]

    Anoshin, A

    M. Anoshin, A. Sagingalieva, C. Mansell, D. Zhiganov, V. Shete, M. Pflitsch, and A. Melnikov, Hybrid quantum cycle generative adversarial network for small molecule generation, IEEE Transactions on Quantum Engineering 5, 2500514 (2024)

    2024

  19. [19]

    Sagingalieva, L

    A. Sagingalieva, L. Lusnig, F. Cavalli, and A. Melnikov, Hybrid quantum neural networks for computer-aided sex diagnosis in forensic and physical anthropology, Infor- matics in Medicine Unlocked58, 101682 (2025)

    2025

  20. [20]

    Tsurkan, A

    O. Tsurkan, A. Konstantinova, A. Sedykh, A. Senokosov, D. Tarpanov, M. Anoshin, A. Sagingalieva, and A. Melnikov, Hybrid quantum recurrent neural net- work for remaining useful life prediction, arXiv preprint arXiv:2504.20823 (2026)

    arXiv 2026

  21. [21]

    Kurkin, J

    A. Kurkin, J. Hegemann, M. Kordzanganeh, and A. Mel- nikov, Forecasting steam mass flow in power plants using the parallel hybrid network, Engineering Applications of Artificial Intelligence160, 111912 (2025)

    2025

  22. [22]

    R. d. Winter, B. v. Stein, T. B¨ ack, H. Ishibuchi, Q. Zhang, R. Cheng, K. Li, H. Li, H. Wang, and A. Zhou, Samo-cobra: a fast surrogate assisted con- strained multi-objective optimization algorithm, inEvo- lutionary Multi-Criterion Optimization(Springer Nature Switzerland AG 2021, 2021) pp. 270–282

    2021

  23. [23]

    de Winter, B

    R. de Winter, B. van Stein, and T. B¨ ack, Samo-cobra: a fast surrogate assisted constrained multi-objective opti- mization algorithm, inInternational conference on evolu- tionary multi-criterion optimization(Springer, 2021) pp. 270–282

    2021

  24. [24]

    Ruddigkeit, R

    L. Ruddigkeit, R. Van Deursen, L. C. Blum, and J.- L. Reymond, Enumeration of 166 billion organic small molecules in the chemical universe database gdb-17, Journal of chemical information and modeling52, 2864 (2012)

    2012

  25. [25]

    Ramakrishnan, P

    R. Ramakrishnan, P. O. Dral, M. Rupp, and O. A. Von Lilienfeld, Quantum chemistry structures and prop- erties of 134 kilo molecules, Scientific data1, 1 (2014)

    2014

  26. [26]

    A. S. Christensen and A. V. lilienfeld, Revised MD17 dataset (rMD17) 10.6084/m9.figshare.12672038.v3 (2020)

    work page doi:10.6084/m9.figshare.12672038.v3 2020

  27. [27]

    Grisafi, D

    A. Grisafi, D. M. Wilkins, M. J. Willatt, and M. Ceri- otti, Atomic-scale representation and statistical learning of tensorial properties, inMachine Learning in Chem- istry: Data-Driven Algorithms, Learning Systems, and Predictions(ACS Publications, 2019) pp. 1–21

    2019

  28. [28]

    Zhang, J

    L. Zhang, J. Han, H. Wang, R. Car, and W. E, Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics, Physical review letters 120, 143001 (2018)

    2018

  29. [29]

    J. S. Smith, O. Isayev, and A. E. Roitberg, Ani-1: an extensible neural network potential with dft accuracy at force field computational cost, Chemical science8, 3192 8 (2017)

    2017

  30. [30]

    Kozinsky, A

    B. Kozinsky, A. Musaelian, A. Johansson, and S. Batzner, Scaling the leading accuracy of deep equivari- ant models to biomolecular simulations of realistic size, inProceedings of the International Conference for High Performance Computing, Networking, Storage and Anal- ysis(2023) pp. 1–12

    2023

  31. [31]

    X. Fu, A. Musaelian, A. Johansson, T. Jaakkola, and B. Kozinsky, Learning interatomic potentials at multiple scales, arXiv preprint arXiv:2310.13756 (2023)

    Pith/arXiv arXiv 2023

  32. [32]

    Lim and B

    L.-H. Lim and B. J. Nelson, What is an equivariant neural network, arXiv preprint arXiv:2205.07362 , 5987 (2022)

    Pith/arXiv arXiv 2022

  33. [33]

    Cohen and M

    T. Cohen and M. Welling, Group equivariant convolu- tional networks, inInternational conference on machine learning(PMLR, 2016) pp. 2990–2999

    2016

  34. [34]

    V. G. Satorras, E. Hoogeboom, and M. Welling, E (n) equivariant graph neural networks, inInternational con- ference on machine learning(PMLR, 2021) pp. 9323– 9332

    2021

  35. [35]

    Navon, A

    A. Navon, A. Shamsian, I. Achituve, E. Fetaya, G. Chechik, and H. Maron, Equivariant architectures for learning in deep weight spaces, inInternational Confer- ence on Machine Learning(PMLR, 2023) pp. 25790– 25816

    2023

  36. [36]

    S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson, and G. Ceder, Python materials genomics (py- matgen): A robust, open-source python library for mate- rials analysis, Computational Materials Science68, 314 (2013)

    2013

  37. [37]

    Zur and T

    A. Zur and T. C. McGill, Lattice match: An ap- plication to heteroepitaxy, Journal of Applied Physics 55, 378 (1984), https://pubs.aip.org/aip/jap/article- pdf/55/2/378/18404059/378 1 online.pdf

    1984

  38. [38]

    W. G. Hoover, Canonical dynamics: Equilibrium phase- space distributions, Phys. Rev. A31, 1695 (1985)

    1985

  39. [39]

    N. Shuichi, Constant Temperature Molecular Dynamics Methods, Progress of Theoretical Physics Supplement 103, 1 (1991), https://academic.oup.com/ptps/article- pdf/doi/10.1143/PTPS.103.1/5351100/103-1.pdf

    work page doi:10.1143/ptps.103.1/5351100/103-1.pdf 1991

  40. [40]

    S. Nos´ e, A unified formulation of the con- stant temperature molecular dynamics meth- ods, The Journal of Chemical Physics81, 511 (1984), https://pubs.aip.org/aip/jcp/article- pdf/81/1/511/18949062/511 1 online.pdf

    1984

  41. [41]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational Materials Science 6, 15 (1996)

    1996

  42. [42]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B54, 11169 (1996)

    1996

  43. [43]

    Kresse and D

    G. Kresse and D. Joubert, From ultrasoft pseudopoten- tials to the projector augmented-wave method, Phys. Rev. B59, 1758 (1999)

    1999

  44. [44]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996)

    1996

  45. [45]

    J. W. Furness, A. D. Kaplan, J. Ning, J. P. Perdew, and J. Sun, Accurate and numerically efficient r2scan meta-generalized gradient approximation, The Journal of Physical Chemistry Letters11, 8208 (2020), pMID: 32876454, https://doi.org/10.1021/acs.jpclett.0c02405

    doi:10.1021/acs.jpclett.0c02405 2020

  46. [46]

    Chmiela, H

    S. Chmiela, H. E. Sauceda, K.-R. M¨ uller, and A. Tkatchenko, Revised md17 dataset (rmd17), https://figshare.com/articles/dataset/Revised_ MD17_dataset_rMD17_/12672038(2020), accessed April 2025

    2020

  47. [47]

    Ramakrishnan, P

    R. Ramakrishnan, P. O. Dral, M. Rupp, and O. A. von Lilienfeld, Qm9 molecular dataset,https:// paperswithcode.com/dataset/qm9or(2014), accessed April 2025

    2014

  48. [48]

    Batzner, A

    S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Kornbluth, N. Molinari, T. E. Smidt, and B. Kozinsky, E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials, Nature Communications13, 2453 (2022)

    2022

  49. [49]

    Musaelian, S

    A. Musaelian, S. Batzner, A. Johansson, L. Sun, J. Guo, and B. Kozinsky, Learning local equivariant representa- tions for large-scale atomistic dynamics, Nature Commu- nications14, 621 (2023)

    2023

  50. [50]

    Allegro+MLP: A modification of Allegro to include MLP components,https://github.com/glatq/allegro (2025), GitHub repository

    2025

  51. [51]

    de Zomer and M

    R. de Zomer and M. Dijkstra, SAMO-COBRA: Combin- ing on-the-fly active learning and efficient sampling for training machine-learned potentials,https://github. com/RoydeZomer/SAMO-COBRA(2023), GitHub repository

    2023

  52. [52]

    Coecke and R

    B. Coecke and R. Duncan, Interacting quantum observ- ables, Automata, Languages and Programming13, 298 (2008)

    2008

  53. [53]

    van de Wetering, ZX-calculus for the working quan- tum computer scientist, arXiv preprint arXiv:2012.13966 (2020)

    J. van de Wetering, ZX-calculus for the working quan- tum computer scientist, arXiv preprint arXiv:2012.13966 (2020)

    Pith/arXiv arXiv 2012

  54. [54]

    Abbas, D

    A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, The power of quantum neural networks, arXiv preprint arXiv:2011.00027 (2020)

    Pith/arXiv arXiv 2011

  55. [55]

    Amari, Natural gradient works efficiently in learning, Neural Computation10, 251 (1998)

    S.-i. Amari, Natural gradient works efficiently in learning, Neural Computation10, 251 (1998)

    1998

  56. [56]

    J. R. McClean, S. Boixo, and V. N. Smelyanskiy, Barren plateaus in quantum neural network training landscapes, Nature Communications9(2018)

    2018

  57. [57]

    Larocca, N

    M. Larocca, N. Ju, D. Garc ´ ıa-Mart ´ ın, P. J. Coles, and M. Cerezo, Theory of overparametrization in quantum neural networks, Nature Computational Science3, 542 (2023)

    2023

  58. [58]

    Luxenberg and S

    E. Luxenberg and S. Boyd, Exponentially weighted mov- ing models, arXiv preprint arXiv:2404.08136 (2024). 9 SUPPLEMENT AR Y MA TERIAL A. Quantum Circuits Analysis In this section, we analyze the quantum circuit used in this paper. There are three metrics for evaluation: •Redundancy analysis with ZX calculus •Trainability analysis with Fisher information •...

    Pith/arXiv arXiv 2024

  59. [59]

    After this ZX graph can be simplified [52] using language’s set of rules, thus taking out redundant elements [53]

    Redundancy analysis with ZX calculus ZX calculus is a graphical language used to simplify quantum circuits using specific notation, where the cir- cuit is transformed into a sequence of nodes, called ”spi- ders”, and edges to connect them. After this ZX graph can be simplified [52] using language’s set of rules, thus taking out redundant elements [53]. Af...

  60. [60]

    Given a subsetSof labeled samples from this distribution, we aim to optimize our model to accurately approximatef(ˆx)

    Trainability analysis with Fisher information A supervised machine learning task can be framed as constructing a hypothesis modelh θ(ˆx) based on a labeled dataset (x, y)∈X×Yto approximate the natural data distributionf(x). Given a subsetSof labeled samples from this distribution, we aim to optimize our model to accurately approximatef(ˆx). This involves ...

  61. [61]

    For a feature vector of lengthN, this Fourier series is expressed as a function of the fea- ture vectorxand the trainable parametersθ: fθ(x) = X ω1∈Ω1

    Expressivity analysis with Fourier series It has been demonstrated that the output of a param- eterized quantum circuit can be represented as a trun- cated Fourier series. For a feature vector of lengthN, this Fourier series is expressed as a function of the fea- ture vectorxand the trainable parametersθ: fθ(x) = X ω1∈Ω1 ... X ωN ∈ΩN cω1,...ωN (θ)e−iωx (7...

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.