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arxiv: 2606.26873 · v1 · pith:DT7IUV4Nnew · submitted 2026-06-25 · 🪐 quant-ph · cs.LG

Scalable Message-Passing Quantum Graph Neural Networks in the Weisfeiler-Leman Hierarchy

Snehal Raj , Brian Coyle , L\'eo Monbroussou , Andr\'e J. Ferreira-Martins , Renato M. S. Farias , Elham Kashefi This is my paper

Pith reviewed 2026-06-26 04:55 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG
keywords quantum graph neural networksmessage passingWeisfeiler-Leman hierarchypermutation equivariancevariational quantum circuitsgraph learningpre-trainingscalability
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The pith

A quantum graph neural network performs message passing, stays permutation equivariant, and reaches any chosen Weisfeiler-Leman level with readout cost independent of graph size.

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

The paper constructs a quantum graph neural network that implements the message-passing primitive while guaranteeing permutation equivariance and expressivity at a user-specified level of the Weisfeiler-Leman hierarchy. Training proceeds first on small graphs to ease optimization, after which the same parameters apply to larger instances. Simulations on up to 56 qubits confirm the approach on graphs that ordinary message-passing networks cannot separate, on molecular property tasks, and on the traveling salesperson problem. The readout step is designed so its cost does not grow with the number of nodes.

Core claim

A quantum graph neural network can be built to perform message passing, to be permutation equivariant, and to sit at a chosen level of the Weisfeiler-Leman hierarchy. Training can be done first on small graph instances, allowing for a pre-training that can mitigate usual training issues, and its output can be read out at a cost that stays low as the graph grows.

What carries the argument

Quantum circuit construction for message passing that enforces exact permutation equivariance and places the model at a target Weisfeiler-Leman level.

If this is right

  • Pre-training on small graphs transfers to larger graphs without retraining the full circuit.
  • The model distinguishes non-isomorphic graphs that standard message-passing networks cannot separate when placed at a higher Weisfeiler-Leman level.
  • The same architecture applies to molecular property prediction and combinatorial problems such as the traveling salesperson problem.
  • Simulation results hold for system sizes up to 56 qubits across synthetic, chemical, and optimization datasets.

Where Pith is reading between the lines

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

  • The fixed-cost readout may allow the same trained circuit to be reused on graphs whose size exceeds current classical simulation limits.
  • Hybrid quantum-classical pipelines could combine this quantum message-passing block with classical post-processing for larger relational datasets.
  • The pre-training route might transfer to other variational quantum models that currently suffer from trainability barriers on relational data.

Load-bearing premise

The circuit construction achieves exact equivariance and exact target Weisfeiler-Leman placement without introducing costs that grow with graph size.

What would settle it

A measurement showing that readout cost increases with node count or that separation power falls short of the chosen Weisfeiler-Leman level on graphs larger than those tested.

Figures

Figures reproduced from arXiv: 2606.26873 by Andr\'e J. Ferreira-Martins, Brian Coyle, Elham Kashefi, L\'eo Monbroussou, Renato M. S. Farias, Snehal Raj.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
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Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
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Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. 1 [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
read the original abstract

Graphs provide a natural language for relational data in chemistry, biology and optimisation. Graph neural networks (GNNs) have driven much of the recent progress in learning from such data through message passing, a single primitive that generalises convolution and attention. Quantum counterparts have been proposed, but with limited connection to message passing and few guarantees on performance or scalability. More broadly, the trainability of variational quantum circuits is a recognised bottleneck for their wide applicability, and pre-training has emerged as one way to address it. Yet for a quantum model to be useful, it must offer expressivity guarantees along with demonstrable scalability. Here we show how a quantum graph neural network can be built to perform message passing, to be permutation equivariant, and to sit at a chosen level of the Weisfeiler-Leman hierarchy, the standard measure of how finely a model can tell graphs apart. We show that, as for classical GNNs, the training can be done first on small graph instances, allowing for a pre-training that can mitigate usual training issues, and its output can be read out at a cost that stays low as the graph grows. We validate the framework in large-scale simulations of up to 56 qubits across three datasets, on synthetic graphs that ordinary message passing cannot separate, on molecular property prediction, and on the travelling salesperson problem. Our framework opens a path for near-term quantum algorithms with theoretical guarantees and practical scalability, bringing the principles of graph learning into quantum circuit design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a quantum graph neural network framework that performs message passing, achieves permutation equivariance, and can be positioned at a chosen level of the Weisfeiler-Leman hierarchy. It incorporates pre-training on small graphs to address trainability and a readout whose computational cost remains independent of graph size. The approach is validated via simulations on up to 56 qubits across synthetic graphs (where classical MPNNs fail), molecular property prediction, and the traveling salesperson problem.

Significance. If the central construction holds with the claimed exact properties and scalable readout, the work would offer a notable contribution by importing classical GNN principles (message passing and WL expressivity) into variational quantum circuits while providing concrete scalability and pre-training pathways. This could support near-term quantum algorithms for graph-structured data with theoretical guarantees.

minor comments (2)
  1. [Abstract and Experiments] The abstract states validation via large-scale simulations but does not specify error bars, baseline comparisons, or the procedure used to confirm WL-level placement; the full manuscript should include these details in the experimental section to support the central claims.
  2. [Methods] Notation for the quantum circuit construction and the mapping to WL levels should be clarified with explicit definitions to ensure readers can verify the permutation equivariance and hierarchy placement without ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its potential significance, and recommendation for minor revision. We are pleased that the core contributions—permutation-equivariant message-passing QGNNs positioned in the WL hierarchy, pre-training on small graphs, and scalable readout—are accurately reflected.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract and provided context describe a construction for a quantum GNN achieving message passing, permutation equivariance, and tunable Weisfeiler-Leman expressivity, with claims of scalable readout and small-graph pre-training resting on the explicit circuit design rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or steps are shown that reduce the target outputs to the inputs by construction. The empirical validations on synthetic, molecular, and TSP tasks supply independent support, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the construction is described at a high level without detailing any fitted constants or new postulated objects.

pith-pipeline@v0.9.1-grok · 5829 in / 1221 out tokens · 23545 ms · 2026-06-26T04:55:01.420013+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

57 extracted references · 28 canonical work pages · 1 internal anchor

  1. [1]

    Gilmer , author S

    author author J. Gilmer , author S. S. \ Schoenholz , author P. F. \ Riley , author O. Vinyals ,\ and\ author G. E. \ Dahl ,\ title title Neural message passing for quantum chemistry ,\ in\ @noop booktitle Proceedings of the 34th International Conference on Machine Learning \ ( organization PMLR ,\ year 2017 )\ pp.\ pages 1263--1272 NoStop

    2017

  2. [2]

    Highly accurate protein structure prediction with AlphaFold

    author author J. Jumper , author R. Evans , author A. Pritzel , author T. Green , author M. Figurnov , author O. Ronneberger , author K. Tunyasuvunakool , author R. Bates , author A. Z \' dek , author A. Potapenko , et al. ,\ title title Highly accurate protein structure prediction with AlphaFold ,\ https://doi.org/10.1038/s41586-021-03819-2 journal journ...

    work page doi:10.1038/s41586-021-03819-2 2021

  3. [3]

    author author C. K. \ Joshi , author T. Laurent ,\ and\ author X. Bresson ,\ title title An efficient graph convolutional network technique for the travelling salesman problem ,\ https://arxiv.org/abs/1906.01227 journal journal arXiv preprint arXiv:1906.01227 \ ( year 2019 ) ,\ https://arxiv.org/abs/1906.01227 arXiv:1906.01227 [cs.LG] NoStop

    arXiv 1906

  4. [4]

    Kool , author H

    author author W. Kool , author H. van Hoof ,\ and\ author M. Welling ,\ https://arxiv.org/abs/1803.08475 title Attention, learn to solve routing problems! ( year 2019 ),\ https://arxiv.org/abs/1803.08475 arXiv:1803.08475 [stat.ML] NoStop

    Pith/arXiv arXiv 2019

  5. [5]

    Xu , author W

    author author K. Xu , author W. Hu , author J. Leskovec ,\ and\ author S. Jegelka ,\ https://arxiv.org/abs/1810.00826 title How powerful are graph neural networks? ( year 2019 ),\ https://arxiv.org/abs/1810.00826 arXiv:1810.00826 [cs.LG] NoStop

    Pith/arXiv arXiv 2019

  6. [6]

    Ceschini , author F

    author author A. Ceschini , author F. Mauro , author F. D. \ Falco , author A. Sebastianelli , author A. Verdone , author A. Rosato , author B. L. \ Saux , author M. Panella , author P. Gamba ,\ and\ author S. L. \ Ullo ,\ https://arxiv.org/abs/2408.06524 title From graphs to qubits: A critical review of quantum graph neural networks ( year 2024 ),\ https...

    arXiv 2024

  7. [7]

    Verdon , author T

    author author G. Verdon , author T. McCourt , author E. Luzhnica , author V. Singh , author S. Leichenauer ,\ and\ author J. Hidary ,\ title title Quantum graph neural networks ,\ https://arxiv.org/abs/1909.12264 journal journal arXiv preprint arXiv:1909.12264 \ ( year 2019 ) ,\ https://arxiv.org/abs/1909.12264 arXiv:1909.12264 [quant-ph] NoStop

    arXiv 1909

  8. [8]

    Ai , author Z

    author author X. Ai , author Z. Zhang , author L. Sun , author J. Yan ,\ and\ author E. Hancock ,\ https://arxiv.org/abs/2201.05158 title Towards quantum graph neural networks: A n ego-graph learning approach ( year 2024 ),\ https://arxiv.org/abs/2201.05158 arXiv:2201.05158 [quant-ph] NoStop

    arXiv 2024

  9. [9]

    \ Ryu , author E

    author author J.-Y. \ Ryu , author E. Elala ,\ and\ author J.-K. K. \ Rhee ,\ https://doi.org/10.3390/ma16124300 title Quantum graph neural network models for materials search ( year 2023 ) NoStop

    work page doi:10.3390/ma16124300 2023

  10. [10]

    Hu , author J

    author author Z. Hu , author J. Li , author Z. Pan , author S. Zhou , author L. Yang , author C. Ding , author O. Khan , author T. Geng ,\ and\ author W. Jiang ,\ title title On the design of quantum graph convolutional neural network in the NISQ -era and beyond ,\ in\ https://doi.org/10.1109/ICCD56317.2022.00051 booktitle IEEE 40th International Conferen...

    work page doi:10.1109/iccd56317.2022.00051 2022

  11. [11]

    Zheng , author Q

    author author J. Zheng , author Q. Gao ,\ and\ author Y. L \"u ,\ title title Quantum graph convolutional neural networks ,\ https://arxiv.org/abs/2107.03257 journal journal arXiv preprint arXiv:2107.03257 \ ( year 2021 ) ,\ https://arxiv.org/abs/2107.03257 arXiv:2107.03257 [quant-ph] NoStop

    arXiv 2021

  12. [12]

    Mernyei , author K

    author author P. Mernyei , author K. Meichanetzidis ,\ and\ author I. I. \ Ceylan ,\ title title Equivariant quantum graph circuits ,\ in\ https://proceedings.mlr.press/v162/mernyei22a.html booktitle Proceedings of the 39th International Conference on Machine Learning ,\ series Proceedings of Machine Learning Research , Vol.\ volume 162 ,\ editor edited b...

    2022

  13. [13]

    Skolik , author M

    author author A. Skolik , author M. Cattelan , author S. Yarkoni , author T. B \"a ck ,\ and\ author V. Dunjko ,\ title title Equivariant quantum circuits for learning on weighted graphs ,\ https://doi.org/10.1038/s41534-023-00710-y journal journal npj Quantum Information \ volume 9 ,\ pages 47 ( year 2023 ) NoStop

    work page doi:10.1038/s41534-023-00710-y 2023

  14. [14]

    Schatzki , author M

    author author L. Schatzki , author M. Larocca , author Q. T. \ Nguyen , author F. Sauvage ,\ and\ author M. Cerezo ,\ title title Theoretical guarantees for permutation-equivariant quantum neural networks ,\ https://doi.org/10.1038/s41534-024-00804-1 journal journal npj Quantum Information \ volume 10 ,\ pages 12 ( year 2024 ) NoStop

    work page doi:10.1038/s41534-024-00804-1 2024

  15. [15]

    author author Q. T. \ Nguyen , author L. Schatzki , author P. Braccia , author M. Ragone , author P. J. \ Coles , author F. Sauvage , author M. Larocca ,\ and\ author M. Cerezo ,\ title title Theory for equivariant quantum neural networks ,\ @noop journal journal PRX Quantum \ volume 5 ,\ pages 020328 ( year 2024 ) NoStop

    2024

  16. [16]

    Larocca , author S

    author author M. Larocca , author S. Thanasilp , author S. Wang , author K. Sharma , author J. Biamonte , author P. J. \ Coles , author L. Cincio , author J. R. \ McClean , author Z. Holmes ,\ and\ author M. Cerezo ,\ title en title Barren plateaus in variational quantum computing ,\ https://doi.org/10.1038/s42254-025-00813-9 journal journal Nature Review...

    work page doi:10.1038/s42254-025-00813-9 2025

  17. [17]

    author author J. R. \ McClean , author S. Boixo , author V. N. \ Smelyanskiy , author R. Babbush ,\ and\ author H. Neven ,\ title title Barren plateaus in quantum neural network training landscapes ,\ https://doi.org/10.1038/s41467-018-07090-4 journal journal Nature Communications \ volume 9 ,\ pages 4812 ( year 2018 ) NoStop

    work page doi:10.1038/s41467-018-07090-4 2018

  18. [18]

    Ragone , author B

    author author M. Ragone , author B. N. \ Bakalov , author F. Sauvage , author A. F. \ Kemper , author C. Ortiz Marrero , author M. Larocca ,\ and\ author M. Cerezo ,\ title title A L ie algebraic theory of barren plateaus for deep parameterized quantum circuits ,\ https://doi.org/10.1038/s41467-024-49909-3 journal journal Nature Communications \ volume 15...

    work page doi:10.1038/s41467-024-49909-3 2024

  19. [19]

    author author M. M. \ Bronstein , author J. Bruna , author T. Cohen ,\ and\ author P. Veli c kovi \'c ,\ title title Geometric deep learning: Grids, groups, graphs, geodesics, and gauges ,\ https://arxiv.org/abs/2104.13478 journal journal arXiv preprint arXiv:2104.13478 \ ( year 2021 ) ,\ https://arxiv.org/abs/2104.13478 arXiv:2104.13478 [cs.LG] NoStop

    Pith/arXiv arXiv 2021

  20. [20]

    Maron , author H

    author author H. Maron , author H. Ben-Hamu , author N. Shamir ,\ and\ author Y. Lipman ,\ title title Invariant and equivariant graph networks ,\ in\ @noop booktitle International Conference on Learning Representations \ ( year 2019 ) NoStop

    2019

  21. [21]

    Morris , author M

    author author C. Morris , author M. Ritzert , author M. Fey , author W. L. \ Hamilton , author J. E. \ Lenssen , author G. Rattan ,\ and\ author M. Grohe ,\ title title Weisfeiler and L eman go neural: Higher-order graph neural networks ,\ in\ @noop booktitle Proceedings of the AAAI Conference on Artificial Intelligence ,\ Vol. volume 33 \ ( year 2019 )\ ...

    2019

  22. [22]

    Chen , author L

    author author Z. Chen , author L. Chen , author S. Villar ,\ and\ author J. Bruna ,\ title title Can graph neural networks count substructures? ,\ @noop journal journal Advances in neural information processing systems \ volume 33 ,\ pages 10383 ( year 2020 ) NoStop

    2020

  23. [23]

    Fontana , author D

    author author E. Fontana , author D. Herman , author S. Chakrabarti , author N. Kumar , author R. Yalovetzky , author J. Heredge , author S. H. \ Sureshbabu ,\ and\ author M. Pistoia ,\ title title Characterizing barren plateaus in quantum ans \"a tze with the adjoint representation ,\ https://doi.org/10.1038/s41467-024-49910-w journal journal Nature Comm...

    work page doi:10.1038/s41467-024-49910-w 2024

  24. [24]

    Cerezo , author M

    author author M. Cerezo , author M. Larocca , author D. Garc \'i a-Mart \'i n , author N. L. \ Diaz , author P. Braccia , author E. Fontana , author M. S. \ Rudolph , author P. Bermejo , author A. Ijaz , author S. Thanasilp , author E. R. \ Anschuetz ,\ and\ author Z. Holmes ,\ title title Does provable absence of barren plateaus imply classical simulabil...

    work page doi:10.1038/s41467-025-63099-6 2025

  25. [25]

    Monbroussou , author E

    author author L. Monbroussou , author E. Z. \ Mamon , author J. Landman , author A. B. \ Grilo , author R. Kukla ,\ and\ author E. Kashefi ,\ title title Trainability and expressivity of H amming-weight preserving quantum circuits for machine learning ,\ https://doi.org/10.22331/q-2025-05-15-1745 journal journal Quantum \ volume 9 ,\ pages 1745 ( year 202...

    work page doi:10.22331/q-2025-05-15-1745 2025

  26. [26]

    Recio-Armengol , author S

    author author E. Recio-Armengol , author S. Ahmed ,\ and\ author J. Bowles ,\ title title Train on classical, deploy on quantum: scaling generative quantum machine learning to a thousand qubits ,\ @noop journal journal arXiv preprint arXiv:2503.02934 \ ( year 2025 ) ,\ https://arxiv.org/abs/2503.02934 arXiv:2503.02934 [quant-ph] NoStop

    arXiv 2025

  27. [27]

    Bako , author Z

    author author B. Bako , author Z. Kolarovszki ,\ and\ author Z. Zimboras ,\ title title Fermionic born machines: Classical training of quantum generative models based on fermion sampling ,\ @noop journal journal arXiv preprint arXiv:2511.13844 \ ( year 2025 ) ,\ https://arxiv.org/abs/2511.13844 arXiv:2511.13844 [quant-ph] NoStop

    arXiv 2025

  28. [28]

    author author M. S. \ Rudolph , author J. Miller , author D. Motlagh , author J. Chen , author A. Acharya ,\ and\ author A. Perdomo-Ortiz ,\ title title Synergistic pretraining of parametrized quantum circuits via tensor networks ,\ https://doi.org/10.1038/s41467-023-43908-6 journal journal Nature Communications \ volume 14 ,\ pages 8367 ( year 2023 ) NoStop

    work page doi:10.1038/s41467-023-43908-6 2023

  29. [29]

    \ Cai , author M

    author author J.-Y. \ Cai , author M. F \"u rer ,\ and\ author N. Immerman ,\ title title An optimal lower bound on the number of variables for graph identification ,\ @noop journal journal Combinatorica \ volume 12 ,\ pages 389 ( year 1992 ) NoStop

    1992

  30. [30]

    Ramakrishnan , author P

    author author R. Ramakrishnan , author P. O. \ Dral , author M. Rupp ,\ and\ author O. A. \ von Lilienfeld ,\ title title Quantum chemistry structures and properties of 134 kilo molecules ,\ https://doi.org/10.1038/sdata.2014.22 journal journal Scientific Data \ volume 1 ,\ pages 140022 ( year 2014 ) NoStop

    work page doi:10.1038/sdata.2014.22 2014

  31. [31]

    Kerenidis , author J

    author author I. Kerenidis , author J. Landman ,\ and\ author A. Prakash ,\ title title Quantum algorithms for deep convolutional neural networks ,\ in\ https://arxiv.org/abs/1911.01117 booktitle International Conference on Learning Representations \ ( year 2020 )\ https://arxiv.org/abs/1911.01117 arXiv:1911.01117 [quant-ph] NoStop

    arXiv 1911

  32. [32]

    Landman , author N

    author author J. Landman , author N. Mathur , author Y. Y. \ Li , author M. Strahm , author S. Kazdaghli , author A. Prakash ,\ and\ author I. Kerenidis ,\ title title Quantum methods for neural networks and application to medical image classification ,\ https://doi.org/10.22331/q-2022-12-22-881 journal journal Quantum \ volume 6 ,\ pages 881 ( year 2022 ) NoStop

    work page doi:10.22331/q-2022-12-22-881 2022

  33. [33]

    author author E. A. \ Cherrat , author S. Raj , author I. Kerenidis , author A. Shekhar , author B. Wood , author J. Dee , author S. Chakrabarti , author R. Chen , author D. Herman , author S. Hu , author P. Minssen , author R. Shaydulin , author Y. Sun , author R. Yalovetzky ,\ and\ author M. Pistoia ,\ title title Quantum deep hedging ,\ https://doi.org...

    work page doi:10.22331/q-2023-11-29-1191 2023

  34. [34]

    Monbroussou , author J

    author author L. Monbroussou , author J. Landman , author L. Wang , author A. B. \ Grilo ,\ and\ author E. Kashefi ,\ title en title Subspace preserving quantum convolutional neural network architectures ,\ https://doi.org/10.1088/2058-9565/adbf43 journal journal Quantum Science and Technology \ volume 10 ,\ pages 025050 ( year 2025 b ) NoStop

    work page doi:10.1088/2058-9565/adbf43 2058

  35. [35]

    Raj \ and\ author B

    author author S. Raj \ and\ author B. Coyle ,\ https://arxiv.org/abs/2502.06916 title QuIC : Quantum-inspired compound adapters for parameter efficient fine-tuning ( year 2025 ),\ https://arxiv.org/abs/2502.06916 arXiv:2502.06916 [cs.LG] NoStop

    arXiv 2025

  36. [36]

    Veličković , author G

    author author P. Veličković , author G. Cucurull , author A. Casanova , author A. Romero , author P. Liò ,\ and\ author Y. Bengio ,\ https://arxiv.org/abs/1710.10903 title Graph attention networks ( year 2018 ),\ https://arxiv.org/abs/1710.10903 arXiv:1710.10903 [stat.ML] NoStop

    Pith/arXiv arXiv 2018

  37. [37]

    Morris , author G

    author author C. Morris , author G. Rattan ,\ and\ author P. Mutzel ,\ title title Weisfeiler and L eman go sparse: Towards scalable higher-order graph embeddings ,\ in\ @noop booktitle Advances in Neural Information Processing Systems ,\ Vol. volume 33 \ ( year 2020 )\ pp.\ pages 21824--21840 NoStop

    2020

  38. [38]

    Mathur , author B

    author author N. Mathur , author B. Coyle , author N. Jain , author S. Raj , author A. Tandon , author J. S. \ Krauser ,\ and\ author R. Stoessel ,\ title title Bayesian quantum orthogonal neural networks for anomaly detection ,\ in\ @noop booktitle 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) ,\ Vol. volume 1 \ ( organiza...

    2025

  39. [39]

    Coyle , author S

    author author B. Coyle , author S. Raj , author N. Mathur , author E. A. \ Cherrat , author N. Jain , author S. Kazdaghli ,\ and\ author I. Kerenidis ,\ title title Training-efficient density quantum machine learning ,\ @noop journal journal npj Quantum Information \ volume 11 ,\ pages 172 ( year 2025 ) NoStop

    2025

  40. [40]

    Scalable On-Hardware Training of Quantum Neural Networks and Application to Clinical Data Imputation

    author author N. Mathur , author P. K. \ Barkoutsos , author M. Yamada , author M. Roetteler ,\ and\ author I. Kerenidis ,\ https://doi.org/10.48550/arXiv.2606.03517 title Scalable On - Hardware Training of Quantum Neural Networks and Application to Clinical Data Imputation ( year 2026 ),\ note arXiv:2606.03517 [quant-ph] NoStop

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.03517 2026

  41. [41]

    Jain , author J

    author author N. Jain , author J. Landman , author N. Mathur ,\ and\ author I. Kerenidis ,\ title en title Quantum Fourier networks for solving parametric PDEs ,\ https://doi.org/10.1088/2058-9565/ad42ce journal journal Quantum Science and Technology \ volume 9 ,\ pages 035026 ( year 2024 ) NoStop

    work page doi:10.1088/2058-9565/ad42ce 2058

  42. [42]

    author author R. M. \ Farias , author T. O. \ Maciel , author G. Camilo , author R. Lin , author S. Ramos-Calderer ,\ and\ author L. Aolita ,\ title title Quantum encoder for fixed- H amming-weight subspaces ,\ https://link.aps.org/doi/10.1103/PhysRevApplied.23.044014 journal journal Phys. Rev. Appl. \ volume 23 ,\ pages 044014 ( year 2025 ) NoStop

    work page doi:10.1103/physrevapplied.23.044014 2025

  43. [43]

    Kerenidis \ and\ author A

    author author I. Kerenidis \ and\ author A. Prakash ,\ https://arxiv.org/abs/2202.00054 title Quantum machine learning with subspace states ( year 2022 ),\ https://arxiv.org/abs/2202.00054 arXiv:2202.00054 [quant-ph] NoStop

    arXiv 2022

  44. [44]

    P \'e rez-Salinas , author A

    author author A. P \'e rez-Salinas , author A. Cervera-Lierta , author E. Gil-Fuster ,\ and\ author J. I. \ Latorre ,\ title title Data re-uploading for a universal quantum classifier ,\ https://doi.org/10.22331/q-2020-02-06-226 journal journal Quantum \ volume 4 ,\ pages 226 ( year 2020 ) NoStop

    work page doi:10.22331/q-2020-02-06-226 2020

  45. [45]

    Arute , author K

    author author F. Arute , author K. Arya , author R. Babbush , author D. Bacon , author J. C. \ Bardin , author R. Barends , author S. Boixo , author M. Broughton , author B. B. \ Buckley , author D. A. \ Buell , et al. ,\ title title Hartree-- Fock on a superconducting qubit quantum computer ,\ https://doi.org/10.1126/science.abb9811 journal journal Scien...

    work page doi:10.1126/science.abb9811 2020

  46. [46]

    Wan , author W

    author author K. Wan , author W. J. \ Huggins , author J. Lee ,\ and\ author R. Babbush ,\ title title Matchgate shadows for fermionic quantum simulation ,\ https://doi.org/10.1007/s00220-023-04844-0 journal journal Communications in Mathematical Physics \ volume 404 ,\ pages 629 ( year 2023 ) NoStop

    work page doi:10.1007/s00220-023-04844-0 2023

  47. [47]

    Maron , author H

    author author H. Maron , author H. Ben-Hamu , author H. Serviansky ,\ and\ author Y. Lipman ,\ title title Provably powerful graph networks ,\ in\ @noop booktitle Advances in Neural Information Processing Systems \ ( year 2019 ) NoStop

    2019

  48. [48]

    Wang \ and\ author M

    author author Y. Wang \ and\ author M. Zhang ,\ title title An empirical study of realized GNN expressiveness ,\ in\ https://arxiv.org/abs/2304.07702 booktitle Proceedings of the 41st International Conference on Machine Learning (ICML) \ ( year 2024 )\ https://arxiv.org/abs/2304.07702 arXiv:2304.07702 [cs.LG] NoStop

    arXiv 2024

  49. [49]

    Liao , author X.-M

    author author Y. Liao , author X.-M. \ Zhang ,\ and\ author C. Ferrie ,\ title title Graph neural networks on quantum computers ,\ https://arxiv.org/abs/2405.17060 journal journal arXiv preprint arXiv:2405.17060 \ ( year 2024 ) ,\ https://arxiv.org/abs/2405.17060 arXiv:2405.17060 [quant-ph] NoStop

    arXiv 2024

  50. [50]

    Vinyals , author M

    author author O. Vinyals , author M. Fortunato ,\ and\ author N. Jaitly ,\ title title Pointer networks ,\ in\ https://proceedings.neurips.cc/paper_files/paper/2015/file/29921001f2f04bd3baee84a12e98098f-Paper.pdf booktitle Advances in Neural Information Processing Systems ,\ Vol. volume 28 ,\ editor edited by\ editor C. Cortes , editor N. Lawrence , edito...

    arXiv 2015

  51. [51]

    author author Google Quantum AI and Collaborators ,\ title title Quantum error correction below the surface code threshold ,\ https://doi.org/10.1038/s41586-024-08449-y journal journal Nature \ volume 638 ,\ pages 920 ( year 2025 ) NoStop

    work page doi:10.1038/s41586-024-08449-y 2025

  52. [52]

    author author G.-L. R. \ Anselmetti , author D. Wierichs , author C. Gogolin ,\ and\ author R. M. \ Parrish ,\ title title Local, expressive, quantum-number-preserving VQE ans\" a tze for fermionic systems ,\ https://doi.org/10.1088/1367-2630/ac2cb3 journal journal New Journal of Physics \ volume 23 ,\ pages 113010 ( year 2021 ) NoStop

    work page doi:10.1088/1367-2630/ac2cb3 2021

  53. [53]

    Monbroussou , author B

    author author L. Monbroussou , author B. Polacchi , author V. Yacoub , author E. Caruccio , author G. Rodari , author F. Hoch , author G. Carvacho , author N. Spagnolo , author T. Giordani , author M. Bossi , author A. Rajan , author N. D. \ Giano , author R. Albiero , author F. Ceccarelli , author R. Osellame , author E. Kashefi ,\ and\ author F. Sciarri...

    work page doi:10.1117/1.ap.7.6.066012 2025

  54. [54]

    Grohe ,\ https://doi.org/10.1017/9781139028868 title Descriptive Complexity, Canonisation, and Definable Graph Structure Theory ,\ series Lecture Notes in Logic , Vol

    author author M. Grohe ,\ https://doi.org/10.1017/9781139028868 title Descriptive Complexity, Canonisation, and Definable Graph Structure Theory ,\ series Lecture Notes in Logic , Vol. volume 47 \ ( publisher Cambridge University Press ,\ year 2017 ) NoStop

    work page doi:10.1017/9781139028868 2017

  55. [55]

    Cerezo , author A

    author author M. Cerezo , author A. Sone , author T. Volkoff , author L. Cincio ,\ and\ author P. J. \ Coles ,\ title title Cost function dependent barren plateaus in shallow parametrized quantum circuits ,\ https://doi.org/10.1038/s41467-021-21728-w journal journal Nature Communications \ volume 12 ,\ pages 1791 ( year 2021 ) NoStop

    work page doi:10.1038/s41467-021-21728-w 2021

  56. [56]

    Larocca , author N

    author author M. Larocca , author N. Ju , author D. Garc \'i a-Mart \'i n , author P. J. \ Coles ,\ and\ author M. Cerezo ,\ title title Theory of overparametrization in quantum neural networks ,\ https://doi.org/10.1038/s43588-023-00467-6 journal journal Nature Computational Science \ volume 3 ,\ pages 542 ( year 2023 ) NoStop

    work page doi:10.1038/s43588-023-00467-6 2023

  57. [57]

    D'Alessandro ,\ @noop title Introduction to Quantum Control and Dynamics ,\ Applied Mathematics and Nonlinear Science\ ( publisher Chapman & Hall/CRC ,\ year 2007 ) NoStop

    author author D. D'Alessandro ,\ @noop title Introduction to Quantum Control and Dynamics ,\ Applied Mathematics and Nonlinear Science\ ( publisher Chapman & Hall/CRC ,\ year 2007 ) NoStop

    2007