Pith. sign in

REVIEW 4 major objections 5 minor 41 references

Inserting a small variational quantum circuit into the readout of a classical interatomic potential improves molecular-energy predictions over an equivalent classical layer, but only when the pretrained network has room to improve.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 00:26 UTC pith:CT5TDOUW

load-bearing objection A careful, modest numerical study of quantum transfer learning for ANI that shows a small, possibly significant RMSE gain in a low-data regime; the effect is real-looking but not yet statistically pinned down. the 4 major comments →

arxiv 2607.27841 v1 pith:CT5TDOUW submitted 2026-07-30 quant-ph

Quantum machine learning interatomic potential: Application of variational quantum algorithm

Kohei Numata , Wataru Mizukami , Kosuke Mitarai , Keisuke Fujii , Yutaka Imamura This is my paper
classification quant-ph
keywords quantum circuit learningmachine learning interatomic potentialsneural network potentialsquantum transfer learningvariational quantum circuitsmolecular energy predictionANI modelhybrid quantum-classical machine learning
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 paper tries to establish that a hybrid quantum-classical machine-learning interatomic potential can beat its fully classical counterpart, and that the benefit is tied to the state of the pretrained classical network. Using the ANI neural-network potential and a quantum transfer-learning scheme, the authors train a small variational quantum circuit Q on top of a frozen classical embedding and compare it with an identity-initialized classical layer of the same size. On the single-heavy-atom dataset with a shallow embedding, the RzRy quantum circuit reaches a test RMSE of 1.48 kcal/mol versus 1.80 kcal/mol for the classical layer, and the improvement over pretraining is largest when the pretrained model is weakest (l=1). The paper also shows that widening the classical layer or pretraining it more accurately shrinks the gap, and that the quantum-transfer model reproduces the DFT energy ordering on cholesterol and its isomers. The authors conclude that quantum features can fill representational gaps left by undertrained classical networks, though they stop short of claiming practical quantum advantage.

Core claim

On its own terms, the discovery is that a variational quantum circuit placed in the readout path of the ANI interatomic potential outperforms a classically equivalent layer under precisely those conditions where the classical pretraining is not already at its ceiling. The headline number is the D1, l=1, n_q=4 case: the RzRy circuit gets 1.48 kcal/mol test RMSE while the classical single-layer control gets 1.80 kcal/mol, and improvements over the pretrained models are 1.55, 0.39, 0.14, and 0.13 kcal/mol for embedding depths l=1 through 4. On cholesterol, epicholesterol, cholestanol, and 7-dehydrocholesterol, the quantum-transfer model beats the original ANI potential on three of the four and

What carries the argument

The load-bearing object is the quantum transfer-learning architecture in which a pretrained classical ANI network is split into L_in and L_out, and a variational circuit Q—the quantum circuit learning module—is inserted between them. Q maps the n_q-dimensional embedding to itself using single-qubit rotations; the two encodings tested are Ry(-arcsin(x))H and Rz(arccos(x))Ry(arccos(x)), followed by parameterized single-qubit rotations with all parameters initialized to zero so Q starts as the identity. The controls are a fully classical network with an identity-initialized single layer L_mid in place of Q, and the original pretrained network itself. The identity-at-init trick is what makes the

Load-bearing premise

The central comparison assumes that an identity-initialized classical fully connected layer is the right null model for what a classical layer can do; if a classical layer with the quantum circuit's Fourier-style features matches its accuracy, the claimed quantum benefit disappears.

What would settle it

Train an identity-initialized classical layer with the same Fourier feature spectrum as the RzRy circuit (e.g., random Fourier features with matching frequencies) on the D1, l=1, n_q=4 task, and run paired trials: if its test RMSE meets or beats 1.48 kcal/mol, or if the 1.48-vs-1.80 gap is not significant across paired seeds, the quantum-layer advantage is not established.

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

If this is right

  • A frozen classical embedding plus a small trained quantum layer is a viable architecture for molecular-energy prediction, with the qubit count set by the compressed embedding rather than by molecular size.
  • Quantum-enhanced MLIPs give their largest gains in low-data, underparameterized regimes; once pretraining is accurate, the insertion of Q adds little.
  • The RzRy encoding is systematically better than Ry, indicating that the choice of quantum feature map matters more than the number of circuit parameters.
  • Because classical networks improve faster with width than the quantum circuits do with qubits, the practical advantage is condition-dependent and currently narrow.
  • Energy ordering among closely related sterol isomers is correctly captured by the quantum-transfer model, supporting use of such hybrid potentials for relative-stability screening.

Where Pith is reading between the lines

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

  • The paper does not test a classical layer with the same Fourier/trigonometric inductive bias as the RzRy circuit; if random Fourier features match the quantum circuit's 1.48 kcal/mol, the observed gain is a function-class effect, not a quantum effect.
  • A natural extension left implicit is to apply the same identity-initialized quantum correction at earlier layers of any MLIP, not only the final readout, to see whether the correction is additive across representational bottlenecks.
  • The zero-initialization protocol makes Q a residual correction term; this suggests the practical niche for near-term quantum hardware is small, shallow circuits patching specific underperforming classical models rather than full quantum replacements.
  • The data-dependence pattern (D1 benefit, D3/D4 no benefit) predicts that the advantage should reappear whenever a pretrained model is deliberately undertrained or data is scarce; this is a testable, quantifiable prediction.

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 proposes a hybrid quantum-classical machine learning interatomic potential (MLIP) by inserting a quantum circuit learning (QCL) module into a pretrained ANI network, following the transfer-learning scheme of Mari et al. The classical network is first pretrained, then a parameterized quantum circuit is inserted between the final two classical layers and trained while the classical parameters are frozen. Using a noiseless quantum circuit simulator, the authors compare this quantum-transfer model against a classical-transfer model with an identity-initialized dense layer, varying the number of qubits n_q, the number of pretrained layers l, the circuit depth d, and the dataset D1–D4. The headline result is that on D1 with l=1, n_q=4, d=1, the RzRy circuit reaches a test RMSE of 1.48 kcal/mol versus 1.80 kcal/mol for the classical layer (Sec. 3.4.1). The authors also report that the improvement is largest when the pretrained model has less capacity, and they demonstrate transferability to cholesterol and isomers by comparing against B3LYP/6-31G(d) DFT energies.

Significance. If the statistical comparison is robust, the paper makes a useful empirical contribution: it shows a concrete, small-scale architecture in which a frozen classical embedding plus a trainable quantum circuit improves MLIP accuracy in a low-data or shallow-pretraining regime. The claims in the abstract and conclusion are appropriately hedged, and the paper does not overclaim a practical quantum advantage. The use of an external dataset (ANI-1x via TorchANI) and DFT reference calculations is appropriate. The main value is as a feasibility study for quantum transfer learning in MLIPs; its significance depends on whether the reported improvement is distinguishable from noise and whether the classical baseline is the right comparator.

major comments (4)
  1. [Sec. 3.3 and Sec. 3.4.1, Fig. 3a]
  2. [Sec. 3.3 and Sec. 3.4.1]
  3. [Sec. 3.4.2 and Sec. 3.4.4]
  4. [Sec. 3.5, Fig. 7]
minor comments (5)
  1. [Sec. 2.2]
  2. [Sec. 3.1 and Table 1]
  3. [Eq. (3)]
  4. [Sec. 3.4.3 and Fig. 4]
  5. [Sec. 3.3]

Circularity Check

0 steps flagged

No significant circularity: the central quantum-vs-classical accuracy comparison is an empirical benchmark, not a constructed identity.

full rationale

The paper's central claim—that inserting a QCL circuit into the ANI readout yields slightly higher RMSE than a classical layer under certain conditions—is an empirical result obtained by training both models on the same ANI-derived data and comparing held-out RMSE. There is no step in which a fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in as an external fact. The identity initialization is applied symmetrically to both the quantum circuit and the classical L_mid layer, so the comparison is not forced by construction; the quantum model could have performed worse. The architecture follows Mari et al. and QCL follows Mitarai et al., but those citations provide background methods, not the evidence for the claimed improvement. The concern raised by the reviewer—that the 1.48 vs 1.80 kcal/mol gap lacks a significance test across five trials—is a statistical robustness issue, not circularity. The claim is externally benchmarked against the ANI-1x dataset and B3LYP/6-31G(d) DFT calculations, giving it independent content. No self-referential or load-bearing reduction to inputs was found.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The paper is an empirical architecture comparison and does not derive new physics. The ledger records the domain assumptions behind the ground-truth labels and the simulator rather than hidden fitted parameters. The scanned hyperparameters (n_q, d, l, learning rate) are an openly disclosed protocol, not degrees of freedom concealed to force the result.

free parameters (2)
  • Circuit/transfer hyperparameter scan (n_q, d, ansatz) = n_q in {4, 8, 12}; d in {1, 5}; V(x) in {Ry, RzRy}
    The reported quantum advantage appears only for a corner of this openly disclosed scan (l=1, n_q=4, d=1, RzRy). No significance or sensitivity analysis is given for selecting that corner; the conclusion is conditional on these choices.
  • Optimization schedule = learning rate 0.001; 100 pretraining epochs; transfer-set epochs not stated
    Chosen by hand. The paper notes d=5 training is unstable at the start, indicating the conclusions depend on the optimization schedule for deeper circuits.
axioms (3)
  • domain assumption ANI-1x energies from the TorchANI library are accurate ground-truth labels for molecular energies.
    Both pretraining and transfer training (Sec 3.2) use energies from the ANI model library (Ref. 37), not new electronic-structure calculations; any systematic error in those labels is inherited by the comparison.
  • domain assumption B3LYP/6-31G(d) DFT values for cholesterol and isomers are the correct reference for the transferability test.
    Sec 3.5 uses B3LYP/6-31G(d) on Gaussian 16 as ground truth; this is a mid-level DFT functional/basis, not a higher-level coupled-cluster reference, and relative isomer energies may carry method error comparable to the model errors being discussed.
  • domain assumption PennyLane StateVectorSimulator exactly represents the noiseless variational quantum circuit.
    The numerical assessment (Sec 3.1) assumes noiseless evolution; shot noise and hardware noise are explicitly excluded, so the results say nothing about device-level performance.

pith-pipeline@v1.3.0-daily-deepseek · 10970 in / 15284 out tokens · 139848 ms · 2026-08-01T00:26:47.239442+00:00 · methodology

0 comments
read the original abstract

This study applied quantum circuit learning, a commonly used hybrid quantum-classical machine learning algorithm, to a machine learning interatomic potential (MLIP) for predicting the energies of molecules in molecular datasets. We retrained the ANI model using the quantum transfer learning architecture [Mari et al., Quantum, 4:340, 2020] and evaluated numerical accuracy with a quantum circuit simulator. The evaluation confirmed that inserting a quantum circuit into the classical neural network of the MLIP yielded slightly higher accuracy than the fully classical neural network under certain conditions. In particular, the model incorporating a quantum circuit was more effective when the pretraining model had room for improvement in accuracy. These findings may contribute to advancing the application of quantum machine learning for MLIPs.

Figures

Figures reproduced from arXiv: 2607.27841 by Keisuke Fujii, Kohei Numata, Kosuke Mitarai, Wataru Mizukami, Yutaka Imamura.

Figure 1
Figure 1. Figure 1: MLIP with a quantum circuit. This hybrid architecture is based on the study by Mari et [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit for the model Q at nq = 4. V (x) is defined in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: RMSEs (kcal/mol) of transferred models for dataset [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Prediction-accuracy transitions for validation data. The upper-row panels ((a)–(d)) show [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Improvements (kcal/mol) of transferred models on the test data for datasets [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Skeletal structures of cholesterol and three closely related sterol isomers used as a practical [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Prediction errors (kcal/mol) from DFT-calculated values for cholesterol and its isomers. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 1 canonical work pages

  1. [1]

    Generalized neural-network representation of high- dimensional potential-energy surfaces.Physical review letters, 98(14):146401, 2007.https: //doi.org/10.1103/PhysRevLett.98.146401

    J¨ org Behler and Michele Parrinello. Generalized neural-network representation of high- dimensional potential-energy surfaces.Physical review letters, 98(14):146401, 2007.https: //doi.org/10.1103/PhysRevLett.98.146401

    doi:10.1103/physrevlett.98.146401 2007

  2. [2]

    Bart´ ok, Mike C

    Albert P. Bart´ ok, Mike C. Payne, Risi Kondor, and G´ abor Cs´ anyi. Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons.Physical Review Letters, 104(13):136403, 2010.https://doi.org/10.1103/PhysRevLett.104.136403

    doi:10.1103/physrevlett.104.136403 2010

  3. [3]

    Unke, Stefan Chmiela, Huziel E

    Oliver T. Unke, Stefan Chmiela, Huziel E. Sauceda, Michael Gastegger, Igor Poltavsky, Kristof T. Sch¨ utt, Alexandre Tkatchenko, and Klaus-Robert M¨ uller. Machine learning force fields.Chemical Reviews, 121(16):10142–10186, 2021.https://doi.org/10.1021/acs.chemrev.0c01111

    doi:10.1021/acs.chemrev.0c01111 2021

  4. [4]

    Ryan Jacobs, Dane Morgan, Siamak Attarian, Jun Meng, Chen Shen, Zhenghao Wu, Clare Yijia Xie, Julia H. Yang, Nongnuch Artrith, Ben Blaiszik, Gerbrand Ceder, Kamal Choudhary, Gabor Csanyi, Ekin Dogus Cubuk, Bowen Deng, Ralf Drautz, Xiang Fu, Jonathan Godwin, Vasant Honavar, Olexandr Isayev, et al. A practical guide to machine learning interatomic potential...

    arXiv 2025

  5. [5]

    Simon Batzner, Albert Musaelian, Lixin Sun, Mario Geiger, Jonathan P Mailoa, Mordechai Ko- rnbluth, Nicola Molinari, Tess E Smidt, and Boris Kozinsky. E (3)-equivariant graph neural net- 11 works for data-efficient and accurate interatomic potentials.Nature communications, 13(1):2453, 2022.https://doi.org/10.1038/s41467-022-29939-5

    doi:10.1038/s41467-022-29939-5 2022

  6. [6]

    Ilyes Batatia, David Peter Kovacs, Gregor N. C. Simm, Christoph Ortner, and Gabor Csanyi. MACE: Higher order equivariant message passing neural networks for fast and accurate force fields. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors,Ad- vances in Neural Information Processing Systems, 2022.https://openreview.net/forum?id= ...

    2022

  7. [7]

    Ilyes Batatia, Simon Batzner, D´ avid P´ eter Kov´ acs, Albert Musaelian, Gregor N. C. Simm, Ralf Drautz, Christoph Ortner, Boris Kozinsky, and G´ abor Cs´ anyi. The design space of e(3)- equivariant atom-centered interatomic potentials, 2022.https://arxiv.org/abs/2205.06643

    Pith/arXiv arXiv 2022

  8. [8]

    A universal graph deep learning interatomic potential for the periodic table.Nature Computational Science, 2(11):718–728, November 2022.https://doi

    Chi Chen and Shyue Ping Ong. A universal graph deep learning interatomic potential for the periodic table.Nature Computational Science, 2(11):718–728, November 2022.https://doi. org/10.1038/s43588-022-00349-3

    doi:10.1038/s43588-022-00349-3 2022

  9. [9]

    Bartel, and Gerbrand Ceder

    Bowen Deng, Peichen Zhong, KyuJung Jun, Janosh Riebesell, Kevin Han, Christopher J. Bartel, and Gerbrand Ceder. CHGNet as a pretrained universal neural network potential for charge- informed atomistic modelling.Nature Machine Intelligence, 5:1031–1041, 2023.https://doi. org/10.1038/s42256-023-00716-3

    doi:10.1038/s42256-023-00716-3 2023

  10. [10]

    Elena, D´ avid P

    Ilyes Batatia, Philipp Benner, Yuan Chiang, Alin M. Elena, D´ avid P. Kov´ acs, Janosh Riebesell, et al. A foundation model for atomistic materials chemistry.The Journal of Chemical Physics, 163(18):184110, 2025.https://doi.org/10.1063/5.0297006

    doi:10.1063/5.0297006 2025

  11. [11]

    Schoenholz, Muratahan Aykol, Gowoon Cheon, and Ekin Dogus Cubuk

    Amil Merchant, Simon Batzner, Samuel S. Schoenholz, Muratahan Aykol, Gowoon Cheon, and Ekin Dogus Cubuk. Scaling deep learning for materials discovery.Nature, 2023.https://doi. org/10.1038/s41586-023-06735-9

    doi:10.1038/s41586-023-06735-9 2023

  12. [12]

    Orb: A fast, scalable neural network potential, 2024

    Mark Neumann, James Gin, Benjamin Rhodes, Steven Bennett, Zhiyi Li, Hitarth Choubisa, Arthur Hussey, and Jonathan Godwin. Orb: A fast, scalable neural network potential, 2024. https://arxiv.org/abs/2410.22570

    Pith/arXiv arXiv 2024

  13. [13]

    Orb-v3: Atomistic simulation at scale, 2025.https://arxiv

    Benjamin Rhodes, Sander Vandenhaute, Vaidotas ˇSimkus, James Gin, Jonathan Godwin, Tim Duignan, and Mark Neumann. Orb-v3: Atomistic simulation at scale, 2025.https://arxiv. org/abs/2504.06231

    Pith/arXiv arXiv 2025

  14. [14]

    Mattersim: A deep learning atomistic model across elements, temperatures and pressures, 2024.https://arxiv.org/abs/2405.04967

    Han Yang, Chenxi Hu, Yichi Zhou, Xixian Liu, Yu Shi, Jielan Li, Guanzhi Li, Zekun Chen, Shuizhou Chen, Claudio Zeni, Matthew Horton, Robert Pinsler, Andrew Fowler, Daniel Z¨ ugner, Tian Xie, Jake Smith, Lixin Sun, Qian Wang, Lingyu Kong, Chang Liu, Hongxia Hao, and Ziheng Lu. Mattersim: A deep learning atomistic model across elements, temperatures and pre...

    Pith/arXiv arXiv 2024

  15. [15]

    Elena, Sam Walton Norwood, Thomas Wolf, and G´ abor Cs´ anyi

    Ilyes Batatia, Chen Lin, Joseph Hart, Elliott Kasoar, Alin M. Elena, Sam Walton Norwood, Thomas Wolf, and G´ abor Cs´ anyi. Cross learning between electronic structure theories for unifying molecular, surface, and inorganic crystal foundation force fields, 2025.https://arxiv.org/abs/ 2510.25380

    arXiv 2025

  16. [16]

    Taming multi-domain, -fidelity data: Towards foundation models for atomistic scale simulations, 2024

    Tomoya Shiota, Kenji Ishihara, Tuan Minh Do, Toshio Mori, and Wataru Mizukami. Taming multi-domain, -fidelity data: Towards foundation models for atomistic scale simulations, 2024. https://arxiv.org/abs/2412.13088. 12

    arXiv 2024

  17. [17]

    Expanding universal machine learning interatomic potentials to 97 elements towards nuclear applications, 2026.https: //arxiv.org/abs/2603.03223

    Naoya Kuroda, Kenji Ishihara, Tomoya Shiota, and Wataru Mizukami. Expanding universal machine learning interatomic potentials to 97 elements towards nuclear applications, 2026.https: //arxiv.org/abs/2603.03223

    arXiv 2026

  18. [18]

    Wood, Misko Dzamba, Xiang Fu, Meng Gao, Muhammed Shuaibi, Luis Barroso- Luque, Kareem Abdelmaqsoud, Vahe Gharakhanyan, John R

    Brandon M. Wood, Misko Dzamba, Xiang Fu, Meng Gao, Muhammed Shuaibi, Luis Barroso- Luque, Kareem Abdelmaqsoud, Vahe Gharakhanyan, John R. Kitchin, Daniel S. Levine, et al. UMA: A family of universal models for atoms, 2025.https://arxiv.org/abs/2506.23971

    arXiv 2025

  19. [19]

    Janosh Riebesell, Rhys E. A. Goodall, Philipp Benner, Yuan Chiang, Bowen Deng, Gerbrand Ceder, Mark Asta, Alpha A. Lee, Anubhav Jain, and Kristin A. Persson. A framework to evaluate machine learning crystal stability predictions.Nature Machine Intelligence, 7:836–847, 2025.https://doi.org/10.1038/s42256-025-01055-1

    doi:10.1038/s42256-025-01055-1 2025

  20. [20]

    Supervised learning with quantum-enhanced feature spaces.Nature, 567(7747):209–212, 2019.https://doi.org/10.1038/s41586-019-0980-2

    Vojtˇ ech Havl ´ ıˇ cek, Antonio D C´ orcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta. Supervised learning with quantum-enhanced feature spaces.Nature, 567(7747):209–212, 2019.https://doi.org/10.1038/s41586-019-0980-2

    doi:10.1038/s41586-019-0980-2 2019

  21. [21]

    Quantum neural networks force fields generation.Machine Learning: Science and Technology, 3(3):035004, 2022.https: //doi.org/10.1088/2632-2153/ac7d3c

    Oriel Kiss, Francesco Tacchino, Sofia Vallecorsa, and Ivano Tavernelli. Quantum neural networks force fields generation.Machine Learning: Science and Technology, 3(3):035004, 2022.https: //doi.org/10.1088/2632-2153/ac7d3c

    doi:10.1088/2632-2153/ac7d3c 2022

  22. [22]

    Symmetry-invariant quantum machine learning force fields, 2023.https://arxiv.org/abs/ 2311.11362

    Isabel Nha Minh Le, Oriel Kiss, Julian Schuhmacher, Ivano Tavernelli, and Francesco Tacchino. Symmetry-invariant quantum machine learning force fields, 2023.https://arxiv.org/abs/ 2311.11362

    Pith/arXiv arXiv 2023

  23. [23]

    Gabriele Lo Monaco, Marco Bertini, Salvatore Lorenzo, and G Massimo Palma. Quantum extreme learning of molecular potential energy surfaces and force fields.Machine Learning: Science and Technology, 5(3):035014, 2024.https://doi.org/10.1088/2632-2153/ad6120

    work page doi:10.1088/2632-2153/ad6120 2024

  24. [24]

    Harnessing disordered-ensemble quantum dynamics for machine learning.Phys

    Keisuke Fujii and Kohei Nakajima. Harnessing disordered-ensemble quantum dynamics for machine learning.Phys. Rev. Appl., 8:024030, Aug 2017.https://doi.org/10.1103/ PhysRevApplied.8.024030

    2017

  25. [25]

    Ma- chine learning with controllable quantum dynamics of a nuclear spin ensemble in a solid, 2018

    Makoto Negoro, Kosuke Mitarai, Keisuke Fujii, Kohei Nakajima, and Masahiro Kitagawa. Ma- chine learning with controllable quantum dynamics of a nuclear spin ensemble in a solid, 2018. https://arxiv.org/abs/1806.10910

    Pith/arXiv arXiv 2018

  26. [26]

    Boost- ing computational power through spatial multiplexing in quantum reservoir computing.Phys

    Kohei Nakajima, Keisuke Fujii, Makoto Negoro, Kosuke Mitarai, and Masahiro Kitagawa. Boost- ing computational power through spatial multiplexing in quantum reservoir computing.Phys. Rev. Appl., 11:034021, Mar 2019.https://doi.org/10.1103/PhysRevApplied.11.034021

    doi:10.1103/physrevapplied.11.034021 2019

  27. [27]

    ChangMo Yang, and Chang Woo Myung

    Soohaeng Yoo Willow, D. ChangMo Yang, and Chang Woo Myung. Hybrid quantum–classical machine learning potential with variational quantum circuits, 2025.https://arxiv.org/abs/ 2508.04098

    Pith/arXiv arXiv 2025

  28. [28]

    Ani-1: an extensible neural network potential with dft accuracy at force field computational cost.Chemical science, 8(4):3192–3203, 2017.https://doi.org/10.1039/C6SC05720A

    Justin S Smith, Olexandr Isayev, and Adrian E Roitberg. Ani-1: an extensible neural network potential with dft accuracy at force field computational cost.Chemical science, 8(4):3192–3203, 2017.https://doi.org/10.1039/C6SC05720A

    doi:10.1039/c6sc05720a 2017

  29. [29]

    Quantum circuit learn- ing.Physical Review A, 98(3):032309, 2018.https://doi.org/10.1103/PhysRevA.98.032309

    Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learn- ing.Physical Review A, 98(3):032309, 2018.https://doi.org/10.1103/PhysRevA.98.032309. 13

    doi:10.1103/physreva.98.032309 2018

  30. [30]

    Variational quan- tum algorithms.Nature Reviews Physics, 3(9):625–644, 2021.https://doi.org/10.1038/ s42254-021-00348-9

    Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quan- tum algorithms.Nature Reviews Physics, 3(9):625–644, 2021.https://doi.org/10.1038/ s42254-021-00348-9

    2021

  31. [31]

    Transfer learning in hybrid classical-quantum neural networks.Quantum, 4:340, 2020.https://doi.org/ 10.22331/q-2020-10-09-340

    Andrea Mari, Thomas R Bromley, Josh Izaac, Maria Schuld, and Nathan Killoran. Transfer learning in hybrid classical-quantum neural networks.Quantum, 4:340, 2020.https://doi.org/ 10.22331/q-2020-10-09-340

    doi:10.22331/q-2020-10-09-340 2020

  32. [32]

    Benjamin, and Xiao Yuan

    Sam McArdle, Suguru Endo, Al´ an Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. Quantum computational chemistry.Reviews of Modern Physics, 92(1):015003, 2020.https://doi.org/ 10.1103/RevModPhys.92.015003

    doi:10.1103/revmodphys.92.015003 2020

  33. [33]

    Olson, Matthias Degroote, Peter D

    Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, M´ aria Kieferov´ a, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Al´ an Aspuru-Guzik. Quantum chemistry in the age of quantum computing.Chemical Reviews, 119(19):10856–10915, 2019.https://doi.org/10.1021/acs.chemrev.8b00803

    doi:10.1021/acs.chemrev.8b00803 2019

  34. [34]

    Quantum algorithms for quantum chemistry and quantum materials science.Chemical Reviews, 120(22):12685–12717, 2020.https://doi.org/10.1021/acs.chemrev.9b00829

    Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan. Quantum algorithms for quantum chemistry and quantum materials science.Chemical Reviews, 120(22):12685–12717, 2020.https://doi.org/10.1021/acs.chemrev.9b00829

    doi:10.1021/acs.chemrev.9b00829 2020

  35. [35]

    A rigorous and robust quantum speed-up in supervised machine learning.Nature Physics, 17:1013–1017, jul 2021.https://doi

    Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme. A rigorous and robust quantum speed-up in supervised machine learning.Nature Physics, 17:1013–1017, jul 2021.https://doi. org/10.1038/s41567-021-01287-z

    doi:10.1038/s41567-021-01287-z 2021

  36. [36]

    Sohaib Alam, Guillermo Alonso-Linaje, B

    Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M. Sohaib Alam, Guillermo Alonso-Linaje, B. AkashNarayanan, Ali Asadi, Juan Miguel Arrazola, Utkarsh Azad, Sam Banning, Carsten Blank, Thomas R Bromley, Benjamin A. Cordier, Jack Ceroni, Alain Delgado, Olivia Di Matteo, Amintor Dusko, Tanya Garg, Diego Guala, Antho...

    Pith/arXiv arXiv 2022

  37. [37]

    Xiang Gao, Farhad Ramezanghorbani, Olexandr Isayev, Justin S Smith, and Adrian E Roitberg. Torchani: a free and open source pytorch-based deep learning implementation of the ani neural network potentials.Journal of chemical information and modeling, 60(7):3408–3415, 2020.https: //doi.org/10.1021/acs.jcim.0c00451

    doi:10.1021/acs.jcim.0c00451 2020

  38. [38]

    Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101, 2017.https://arxiv.org/abs/1711.05101

    Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101, 2017.https://arxiv.org/abs/1711.05101

    Pith/arXiv arXiv 2017

  39. [39]

    A stochastic approximation method.The annals of mathe- matical statistics, pages 400–407, 1951.https://doi.org/10.1214/aoms/1177729586

    Herbert Robbins and Sutton Monro. A stochastic approximation method.The annals of mathe- matical statistics, pages 400–407, 1951.https://doi.org/10.1214/aoms/1177729586. 14

    arXiv 1951

  40. [40]

    McClean, Masoud Mohseni, Patrick van der Smagt, and Martin Leib

    Andrea Skolik, Jarrod R. McClean, Masoud Mohseni, Patrick van der Smagt, and Martin Leib. Layerwise learning for quantum neural networks.Quantum Machine Intelligence, 3(1), January 2021.https://doi.org/10.1007/s42484-020-00036-4

    doi:10.1007/s42484-020-00036-4 2021

  41. [41]

    M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scal- mani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goin...

    2016