arxiv: 2605.09226 · v1 · submitted 2026-05-09 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Quantum Injection Pathways for Implicit Graph Neural Networks

Hausi A. M\"uller, Luis F. Rivera, Pengyuan Xu, Tristan Zaborniak

Pith reviewed 2026-05-12 02:27 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum deep equilibrium modelsgraph neural networksimplicit networksquantum machine learningfixed-point solversinjection pathways

0 comments

The pith

Independent injection of quantum signals into graph deep equilibrium models yields superior accuracy and efficiency on classification benchmarks.

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

This paper formulates three methods for injecting quantum signals into deep equilibrium models applied to graphs. The methods differ in whether the quantum signal is computed once and held fixed or recomputed at each step of the fixed-point solver. Contraction guarantees are provided for all three under assumptions on the Lipschitz constants of the classical and quantum components. Empirical results on the NCI1, PROTEINS, and MUTAG datasets show that the independent injection method achieves the best test accuracy while converging in fewer iterations than the classical baseline and the dependent injection methods. This approach matters for readers interested in combining the memory efficiency of implicit models with quantum computation without the trainability issues of deep parameterized circuits.

Core claim

The paper establishes three quantum injection pathways for graph DEQs—independent, state-dependent, and backbone-dependent—and demonstrates through theory and experiment that independent injection, which computes the quantum signal once per graph and holds it fixed during the solve, provides the highest test accuracy on standard benchmarks while requiring fewer forward-solver iterations.

What carries the argument

The fixed-point operator of the graph deep equilibrium model, modified by one of three quantum signal injection strategies that differ in the timing and target of the quantum computation within the iterative solve.

If this is right

Independent injection requires fewer iterations to reach the fixed point compared to both the classical DEQ and the state- and backbone-dependent quantum injections.
All variants admit unique fixed points when the Lipschitz constants satisfy the contraction conditions derived in the paper.
The quantum DEQ framework achieves the expressive power of deep networks at the cost of a single layer's memory.
Quantum signals can be coupled to implicit graph models without requiring repeated quantum evaluations at every solver step.

Where Pith is reading between the lines

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

The independent injection method could be particularly advantageous on quantum hardware where each quantum computation is costly, as it minimizes the number of quantum calls.
The contraction mapping results provide a theoretical foundation that may extend to other implicit models incorporating quantum components.
Testable extensions include applying these injection pathways to larger graph datasets or different quantum signal encodings.

Load-bearing premise

The contraction guarantees and thus the well-posedness of the fixed-point equations depend on the classical backbone and the quantum signal having sufficiently small Lipschitz constants.

What would settle it

Observing that independent injection fails to achieve the highest accuracy or uses more iterations than the baselines on the NCI1, PROTEINS, or MUTAG benchmarks would directly challenge the main empirical claim.

Figures

Figures reproduced from arXiv: 2605.09226 by Hausi A. M\"uller, Luis F. Rivera, Pengyuan Xu, Tristan Zaborniak.

**Figure 1.** Figure 1: Quantum injection pathways in a shared implicit graph framework. Panels (a)–(c) show the ID, SD, and BD variants as [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: Training-time forward-solver effort on NCI1, PRO [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Deep XYZ Ansatz used in the reported experiments. The appendix schematic shows one repetition of the circuit, [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

read the original abstract

Deep Equilibrium Models (DEQs) replace a stack of explicit layers with a single operator whose fixed point defines the output, giving the expressive power of an arbitrarily deep network at the memory cost of a single layer. Quantum Deep Equilibrium Models (QDEQs) bring this idea to quantum machine learning, offering an alternative to Parameterized Quantum Circuits (PQCs), whose depth is limited by hardware coherence and trainability. Here, we introduce, formulate, and compare three ways of coupling a quantum signal to graph DEQs, differing in where the signal enters the fixed-point operator. \textit{Independent} injection computes the quantum signal once per graph and forward fixed-point solve, and holds it fixed throughout the solve. \textit{State-dependent} injection instead recomputes the signal at every solver step and applies it to the current iterate. \textit{Backbone-dependent} injection likewise recomputes at every iteration but applies the signal to the classical backbone's output evaluated at the current iterate. We establish contraction guarantees for each variant under explicit assumptions on the Lipschitz constants of the classical backbone and the quantum signal. On the TU Dortmund graph-classification benchmarks NCI1, PROTEINS, and MUTAG, independent injection achieves the best test accuracy while using fewer forward-solver iterations than both the classical equilibrium baseline and the two dependent variants.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

Independent injection of a fixed quantum signal into graph DEQs gives the best accuracy and fewest iterations on three small benchmarks, but the contraction guarantees rest on unverified Lipschitz bounds for the quantum part.

read the letter

The main point is that the paper defines three concrete ways to couple a quantum signal into a graph deep equilibrium model and finds that computing the quantum output once per graph and holding it fixed during the fixed-point solve beats both the classical DEQ baseline and the two variants that recompute the signal at every iteration. On NCI1, PROTEINS, and MUTAG the independent version reaches higher test accuracy while needing fewer solver steps to converge.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces three quantum injection pathways—independent, state-dependent, and backbone-dependent—for coupling quantum signals to graph Deep Equilibrium Models (DEQs). It derives contraction guarantees for the fixed-point iterations of each variant under assumptions on the Lipschitz constants of the classical backbone and quantum signal. Empirically, on the NCI1, PROTEINS, and MUTAG graph classification tasks, the independent injection variant is reported to achieve the highest test accuracy while requiring fewer solver iterations than the classical DEQ baseline and the dependent injection methods.

Significance. If the contraction guarantees hold under the stated assumptions and the empirical superiority is robust, this work offers a memory-efficient hybrid quantum-classical approach to graph neural networks that sidesteps the depth limitations of parameterized quantum circuits. The explicit derivation of contraction conditions for each injection variant and the head-to-head comparison on standard TU Dortmund benchmarks constitute clear strengths that could guide future design of implicit QML models.

major comments (1)

[Theoretical analysis of contraction guarantees] The contraction guarantees (stated in the abstract as holding 'under explicit assumptions' on the Lipschitz constants of the classical backbone and quantum signal) are load-bearing for the central claim that independent injection uses fewer forward-solver iterations. The manuscript provides no evidence that these Lipschitz bounds were measured, enforced, or satisfied by the trained models evaluated on NCI1, PROTEINS, and MUTAG. If the quantum-signal Lipschitz constant exceeds the threshold required for a contraction factor <1, the reported iteration advantage disappears and the comparison to the classical DEQ baseline is no longer valid.

minor comments (2)

[Experimental evaluation] The experimental section should report standard deviations or error bars on test accuracies and specify the precise train/validation/test splits and random seeds used for the NCI1, PROTEINS, and MUTAG benchmarks.
Implementation details for the quantum signal (circuit architecture, embedding into the graph DEQ, and how the signal is computed in each injection variant) should be expanded to enable reproduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment regarding the contraction guarantees below.

read point-by-point responses

Referee: The contraction guarantees (stated in the abstract as holding 'under explicit assumptions' on the Lipschitz constants of the classical backbone and quantum signal) are load-bearing for the central claim that independent injection uses fewer forward-solver iterations. The manuscript provides no evidence that these Lipschitz bounds were measured, enforced, or satisfied by the trained models evaluated on NCI1, PROTEINS, and MUTAG. If the quantum-signal Lipschitz constant exceeds the threshold required for a contraction factor <1, the reported iteration advantage disappears and the comparison to the classical DEQ baseline is no longer valid.

Authors: We agree that explicit verification of the Lipschitz assumptions in the trained models would strengthen the connection between the theoretical guarantees and the empirical iteration counts. The contraction analysis provides sufficient conditions under which each injection variant is guaranteed to converge, and the independent variant is predicted to do so under milder conditions on the quantum-signal Lipschitz constant. The reported iteration numbers, however, are direct empirical measurements obtained from the converged solver runs on the trained models for NCI1, PROTEINS, and MUTAG; all methods reached a fixed point within the stated iteration budgets. In the revised manuscript we will add a supplementary analysis that estimates the Lipschitz constants of the quantum signal and classical backbone for the final trained models on each dataset. This will allow us to check whether the contraction thresholds are met in practice and to discuss any implications for the observed iteration advantages. We view this as a partial revision that directly addresses the referee's concern while preserving the manuscript's core empirical and theoretical contributions. revision: partial

Circularity Check

0 steps flagged

No circularity: definitions and guarantees are independent of reported outcomes

full rationale

The three injection variants are introduced via explicit procedural distinctions (independent vs. state-dependent vs. backbone-dependent recomputation of the quantum signal) that do not presuppose the accuracy or iteration-count results. Contraction guarantees are derived from standard fixed-point arguments under explicit Lipschitz-constant assumptions on the backbone and quantum signal; these assumptions are not fitted to the NCI1/PROTEINS/MUTAG benchmarks and do not reduce the empirical claims to tautologies. No self-citations appear as load-bearing premises, and the benchmark numbers are presented as direct observations rather than predictions forced by the model definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on standard contraction-mapping assumptions for fixed-point existence and on empirical performance on three graph datasets; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Lipschitz constants of the classical backbone and quantum signal are bounded such that each injection variant contracts
Invoked to guarantee unique fixed points for independent, state-dependent, and backbone-dependent injection.

pith-pipeline@v0.9.0 · 5540 in / 1214 out tokens · 30789 ms · 2026-05-12T02:27:24.254770+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We establish contraction guarantees for each variant under explicit assumptions on the Lipschitz constants of the classical backbone and the quantum signal (Theorems III.1–III.3)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear
Lip_F(Φ_ID_Θ) ≤ κ; Lip_F(Φ_SD_Θ) ≤ κ + α L_q; Lip_F(Φ_BD_Θ) ≤ κ(1 + α L_q)

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · 6 internal anchors

[1]

Graph neural networks: A review of methods and applications,

J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, and M. Sun, “Graph neural networks: A review of methods and applications,”AI Open, vol. 1, pp. 57–81, 2020. [Online]. Available: https://doi.org/10.1016/j.aiopen.2021.01.001

work page doi:10.1016/j.aiopen.2021.01.001 2020
[2]

Graph neural networks,

G. Corso, H. Stark, S. Jegelka, T. Jaakkola, and R. Barzilay, “Graph neural networks,”Nature Reviews Methods Primers, vol. 4, no. 1, p. 17, mar 2024. [Online]. Available: https://doi.org/10.1038/ s43586-024-00294-7

work page 2024
[3]

Semi-Supervised Classification with Graph Convolutional Networks

T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” inInternational Conference on Learning Representations (ICLR), 2017. [Online]. Available: https: //doi.org/10.48550/arXiv.1609.02907

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1609.02907 2017
[4]

Inductive Representation Learning on Large Graphs,

W. L. Hamilton, R. Ying, and J. Leskovec, “Inductive representation learning on large graphs,” 2018. [Online]. Available: https://doi.org/10. 48550/arXiv.1706.02216

work page arXiv 2018
[5]

How Powerful are Graph Neural Networks?

K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph neural networks?” inInternational Conference on Learning Representations (ICLR), 2019. [Online]. Available: https://doi.org/10. 48550/arXiv.1810.00826

work page internal anchor Pith review arXiv 2019
[6]

Graph neural networks exponentially lose expressive power for node classification,

K. Oono and T. Suzuki, “Graph neural networks exponentially lose expressive power for node classification,” inInternational Conference on Learning Representations (ICLR), 2020. [Online]. Available: https://openreview.net/forum?id=S1ldO2EFPr

work page 2020
[7]

Zico Kolter, and Vladlen Koltun

S. Bai, J. Z. Kolter, and V . Koltun, “Deep equilibrium models,” in Advances in Neural Information Processing Systems, 2019. [Online]. Available: https://doi.org/10.48550/arXiv.1909.01377

work page doi:10.48550/arxiv.1909.01377 2019
[8]

Implicit graph neural networks,

F. Gu, H. Chang, W. Zhu, S. Sojoudi, and L. El Ghaoui, “Implicit graph neural networks,” inAdvances in Neural Information Processing Systems, H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, Eds., vol. 33. Curran Associates, Inc., 2020, pp. 11 984– 11 995. [Online]. Available: https://proceedings.neurips.cc/paper files/ paper/2020/file/8b5c84...

work page 2020
[9]

Torchdeq: A library for deep equilibrium models,

Z. Geng and J. Z. Kolter, “Torchdeq: A library for deep equilibrium models,” 2023. [Online]. Available: https://doi.org/10.48550/arXiv.2310. 18605

work page doi:10.48550/arxiv.2310 2023
[10]

Monotone operator equilibrium networks,

E. Winston and J. Z. Kolter, “Monotone operator equilibrium networks,” inAdvances in Neural Information Process- ing Systems, vol. 33, 2020, pp. 10 718–10 728. [Online]. Available: https://proceedings.neurips.cc/paper files/paper/2020/hash/ 798d1c2813cbdf8bcdb388db0e32d496-Abstract.html

work page 2020
[11]

Nature Reviews Physics3(9), 625–644 (2021) https: //doi.org/10.1038/s42254-021-00348-9

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, “Variational quantum algorithms,”Nature Reviews Physics, vol. 3, no. 9, pp. 625–644, Aug. 2021. [Online]. Available: https://doi.org/10.1038/s42254-021-00348-9

work page doi:10.1038/s42254-021-00348-9 2021
[12]

Quantum Science and Technology4(4), 043001 (2019) https://doi.org/10.1088/2058-9565/ab4eb5

M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, “Parameterized quantum circuits as machine learning models,”Quantum Science and Technology, vol. 4, no. 4, p. 043001, Nov. 2019. [Online]. Available: https://doi.org/10.1088/2058-9565/ab4eb5

work page doi:10.1088/2058-9565/ab4eb5 2019
[13]

Embedding learning in hybrid quantum-classical neural networks,

M. Liu, J. Liu, R. Liu, H. Makhanov, D. Lykov, A. Apte, and Y . Alexeev, “Embedding learning in hybrid quantum-classical neural networks,” in2022 IEEE International Conference on Quantum Computing and Engineering (QCE). IEEE, Sep. 2022, pp. 79–86. [Online]. Available: https://doi.org/10.1109/QCE53715.2022.00026

work page doi:10.1109/qce53715.2022.00026 2022
[14]

GraphQNTK: Quantum neural tangent kernel for graph data,

Y . Tang and J. Yan, “GraphQNTK: Quantum neural tangent kernel for graph data,” inAdvances in Neural Information Processing Systems, S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, Eds., vol. 35. Curran Associates, Inc., 2022, pp. 6104–6118. [Online]. Available: https://doi.org/10.5555/3600270.3600712

work page doi:10.5555/3600270.3600712 2022
[15]

Seitz, M

A. Ray, D. Madan, S. Patil, P. Pati, M. Rapsomaniki, A. Kohlakala, T. R. Dlamini, S. J. Muller, K. Rhrissorrakrai, F. Utro, and L. Parida, “Hybrid quantum-classical graph neural networks for tumor classification in digital pathology,” in2024 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 01, 2024, pp. 1611–1616. [Online]. A...

work page doi:10.1109/qce60285.2024.00188 2024
[16]

Seitz, M

M. Cui, L. Chang, A. Chau, H. Mekuria, L. Adwankar, S. Pendyala, and L. McMahan, “Efficient and optimized small organic molecular graph generation pathway using a quantum generative adversarial network,” in2024 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 01, 2024, pp. 1565–1570. [Online]. Available: https://doi.org/10.11...

work page doi:10.1109/qce60285.2024.00183 2024
[18]

Nature Communications9(1) (2018) https:// doi.org/10.1038/s41467-018-07090-4

J. R. McClean, S. Boixo, V . N. Smelyanskiy, R. Babbush, and H. Neven, “Barren plateaus in quantum neural network training landscapes,”Nature Communications, vol. 9, no. 1, Nov. 2018. [Online]. Available: https://doi.org/10.1038/s41467-018-07090-4

work page doi:10.1038/s41467-018-07090-4 2018
[19]

A lie algebraic theory of barren plateaus for deep parameterized quantum circuits,

M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Ortiz Marrero, M. Larocca, and M. Cerezo, “A lie algebraic theory of barren plateaus for deep parameterized quantum circuits,”Nature Communications, vol. 15, no. 1, Aug. 2024. [Online]. Available: https://doi.org/10.1038/ s41467-024-49909-3

work page 2024
[20]

Characterizing barren plateaus in quantum ans ¨atze with the adjoint representation,

E. Fontana, D. Herman, S. Chakrabarti, N. Kumar, R. Yalovetzky, J. Heredge, S. H. Sureshbabu, and M. Pistoia, “Characterizing barren plateaus in quantum ans ¨atze with the adjoint representation,”Nature Communications, vol. 15, no. 1, Aug. 2024. [Online]. Available: https://doi.org/10.1038/s41467-024-49910-w

work page doi:10.1038/s41467-024-49910-w 2024
[21]

Evaluating analytic gradients on quantum hardware,

M. Schuld, V . Bergholm, C. Gogolin, J. Izaac, and N. Killoran, “Evaluating analytic gradients on quantum hardware,”Physical Review A, vol. 99, no. 3, p. 032331, 2019. [Online]. Available: https://doi.org/10.1103/PhysRevA.99.032331

work page doi:10.1103/physreva.99.032331 2019
[22]

The power of quantum neural networks,

A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, “The power of quantum neural networks,”Nature Computational Science, vol. 1, no. 6, pp. 403–409, Jun. 2021. [Online]. Available: https://doi.org/10.1038/s43588-021-00084-1

work page doi:10.1038/s43588-021-00084-1 2021
[23]

Quantum deep equilibrium models,

P. Schleich, M. Skreta, L. B. Kristensen, R. Vargas-Hernandez, and A. Aspuru-Guzik, “Quantum deep equilibrium models,” inThe Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. [Online]. Available: https://openreview.net/forum?id= CWhwKb0Q4k

work page 2024
[24]

Implicit deep learning,

L. E. Ghaoui, F. Gu, B. Travacca, A. Askari, and A. Y . Tsai, “Implicit deep learning,” 2020. [Online]. Available: https: //doi.org/10.48550/arXiv.1908.06315

work page doi:10.48550/arxiv.1908.06315 2020
[25]

CerDEQ: Certifiable deep equilibrium model,

M. Li, Y . Wang, and Z. Lin, “CerDEQ: Certifiable deep equilibrium model,” inProceedings of the 39th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu, and S. Sabato, Eds., vol. 162. PMLR, 17–23 Jul 2022, pp. 12 998–13 013. [Online]. Available: https://proc...

work page 2022
[26]

Lyapunov-stable deep equilibrium models,

H. Chu, S. Wei, T. Liu, Y . Zhao, and Y . Miyatake, “Lyapunov-stable deep equilibrium models,” inProceedings of the AAAI Conference on Artificial Intelligence, vol. 38, no. 10, 2024, pp. 11 615–11 623. [Online]. Available: https://doi.org/10.1609/aaai.v38i10.29044

work page doi:10.1609/aaai.v38i10.29044 2024
[27]

Implicit graph neural networks: A monotone operator viewpoint,

J. Baker, Q. Wang, C. D. Hauck, and B. Wang, “Implicit graph neural networks: A monotone operator viewpoint,” inProceedings of the 40th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 202. PMLR, 2023, pp. 1521–1548. [Online]. Available: https://proceedings.mlr.press/v202/baker23a.html

work page 2023
[28]

Eignn: Efficient infinite-depth graph neural networks,

J. Liu, K. Kawaguchi, B. Hooi, Y . Wang, and X. Xiao, “Eignn: Efficient infinite-depth graph neural networks,” inAdvances in Neural Information Processing Systems, vol. 34, 2021, pp. 18 762– 18 773. [Online]. Available: https://proceedings.neurips.cc/paper/2021/ hash/9bd5ee6fe55aaeb673025dbcb8f939c1-Abstract.html

work page 2021
[29]

Mgnni: Multiscale graph neural networks with implicit layers,

J. Liu, B. Hooi, K. Kawaguchi, and X. Xiao, “Mgnni: Multiscale graph neural networks with implicit layers,” 2022. [Online]. Available: https://doi.org/10.48550/arXiv.2210.08353

work page doi:10.48550/arxiv.2210.08353 2022
[30]

Optimization- induced graph implicit nonlinear diffusion,

Q. Chen, Y . Wang, Y . Wang, J. Yang, and Z. Lin, “Optimization- induced graph implicit nonlinear diffusion,” 2022. [Online]. Available: https://doi.org/10.48550/arXiv.2206.14418

work page doi:10.48550/arxiv.2206.14418 2022
[31]

URLhttps://doi.org/10.48550/arXiv

J. Zheng, Q. Gao, and Y . Lv, “Quantum graph convolutional neural networks,” 2021. [Online]. Available: https://doi.org/10.48550/arXiv. 2107.03257

work page internal anchor Pith review doi:10.48550/arxiv 2021
[32]

On the design of quantum graph convolutional neural network in the nisq-era and beyond,

Z. Hu, J. Li, Z. Pan, S. Zhou, L. Yang, C. Ding, O. Khan, T. Geng, and W. Jiang, “On the design of quantum graph convolutional neural network in the nisq-era and beyond,” in Proceedings - 2022 IEEE 40th International Conference on Computer Design, ICCD 2022, 2022, pp. 290–297. [Online]. Available: https://doi.org/10.1109/ICCD56317.2022.00050

work page doi:10.1109/iccd56317.2022.00050 2022
[33]

Quantum graph neural networks,

G. Verdon, T. McCourt, E. Luzhnica, V . Singh, S. Leichenauer, and J. Hidary, “Quantum graph neural networks,” 2019, arXiv preprint. [Online]. Available: https://doi.org/10.48550/arXiv.1909.12264 11

work page doi:10.48550/arxiv.1909.12264 2019
[34]

On designing general and expressive quantum graph neural networks with applications to MILP instance representation,

X. Ye, H. Xiong, J. Huang, Z. Chen, J. Wang, and J. Yan, “On designing general and expressive quantum graph neural networks with applications to MILP instance representation,” inThe Thirteenth International Conference on Learning Representations, 2025. [Online]. Available: https://openreview.net/forum?id=IQi8JOqLuv

work page 2025
[35]

Inductive graph representation learning with quantum graph neural networks,

A. M. Faria, I. F. Gra ˜na, and S. Varsamopoulos, “Inductive graph representation learning with quantum graph neural networks,” 2026. [Online]. Available: https://arxiv.org/abs/2503.24111

work page arXiv 2026
[36]

From graphs to qubits: A critical review of quantum graph neural networks,

A. Ceschini, F. Mauro, F. D. Falco, A. Sebastianelli, A. Verdone, A. Rosato, B. L. Saux, M. Panella, P. Gamba, and S. L. Ullo, “From graphs to qubits: A critical review of quantum graph neural networks,”

work page
[37]

Available: https://doi.org/10.48550/arXiv.2408.06524

[Online]. Available: https://doi.org/10.48550/arXiv.2408.06524

work page doi:10.48550/arxiv.2408.06524
[38]

Hardware-Aware Quantum Kernel Design Based on Graph Neural Networks

Y . Liu, F. Meng, L. Wang, Y . Hu, S. Li, X. Yu, and Z. Zhang, “Haqgnn: Hardware-aware quantum kernel design based on graph neural networks,” 2025. [Online]. Available: https://doi.org/10.48550/ arXiv.2506.21161

work page internal anchor Pith review Pith/arXiv arXiv 2025
[39]

FiLM : Visual reasoning with a general conditioning layer

E. Perez, F. Strub, H. de Vries, V . Dumoulin, and A. Courville, “Film: Visual reasoning with a general conditioning layer,”Proceedings of the AAAI Conference on Artificial Intelligence, vol. 32, no. 1, Apr. 2018. [Online]. Available: https://doi.org/10.1609/aaai.v32i1.11671

work page doi:10.1609/aaai.v32i1.11671 2018
[40]

A hybrid quantum-classical neural network with deep residual learning,

Y . Liang, W. Peng, Z.-J. Zheng, O. Silv ´en, and G. Zhao, “A hybrid quantum-classical neural network with deep residual learning,” 2021. [Online]. Available: https://doi.org/10.48550/arXiv.2012.07772

work page doi:10.48550/arxiv.2012.07772 2021
[41]

Resqnets: A residual approach for mitigating barren plateaus in quantum neural networks,

M. Kashif and S. Al-kuwari, “Resqnets: A residual approach for mitigating barren plateaus in quantum neural networks,”EPJ Quantum Technology, vol. 11, p. 4, 2024. [Online]. Available: https://doi.org/10.1140/epjqt/s40507-023-00216-8

work page doi:10.1140/epjqt/s40507-023-00216-8 2024
[42]

On optimizing hyperparameters for quantum neural networks,

S. Herbst, V . De Maio, and I. Brandic, “On optimizing hyperparameters for quantum neural networks,” in2024 IEEE International Conference on Quantum Computing and Engineering (QCE). IEEE, Sep. 2024, pp. 1478–1489. [Online]. Available: https://doi.org/10.1109/QCE60285. 2024.00174

work page doi:10.1109/qce60285 2024
[43]

Data re-uploading for a universal quantum classifier,

A. P ´erez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, “Data re-uploading for a universal quantum classifier,” Quantum, vol. 4, p. 226, Feb. 2020. [Online]. Available: https: //doi.org/10.22331/q-2020-02-06-226

work page doi:10.22331/q-2020-02-06-226 2020
[44]

Rudin,Principles of mathematical analysis

W. Rudin,Principles of mathematical analysis. McGraw-Hill, 1976, vol. 3

work page 1976
[45]

R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed. Cambridge University Press, 2012. [Online]. Available: https://doi.org/10.1017/ 9781139020411

work page 2012
[46]

Attention Is All You Need

A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” inAdvances in Neural Information Processing Systems (NeurIPS), vol. 30, 2017, pp. 5998–6008. [Online]. Available: https://doi.org/10.48550/arXiv.1706.03762

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1706.03762 2017
[47]

ArXiv abs/2007.08663(2020)

C. Morris, N. M. Kriege, F. Bause, K. Kersting, P. Mutzel, and M. Neumann, “Tudataset: A collection of benchmark datasets for learning with graphs,” inICML 2020 Workshop on Graph Representation Learning and Beyond (GRL+ 2020), 2020. [Online]. Available: https://doi.org/10.48550/arXiv.2007.08663

work page doi:10.48550/arxiv.2007.08663 2020
[48]

A fair comparison of graph neural networks for graph classification,

F. Errica, M. Podda, D. Bacciu, and A. Micheli, “A fair comparison of graph neural networks for graph classification,” inInternational Conference on Learning Representations (ICLR), 2020. [Online]. Available: https://openreview.net/forum?id=HygDF6NFPB

work page 2020
[49]

Benchmarking graph neural networks,

V . P. Dwivedi, C. K. Joshi, A. T. Luu, T. Laurent, Y . Bengio, and X. Bresson, “Benchmarking graph neural networks,” 2020. [Online]. Available: https://doi.org/10.48550/arXiv.2003.00982

work page doi:10.48550/arxiv.2003.00982 2020
[50]

Decoupled weight decay regularization,

I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,”

work page
[51]

Decoupled Weight Decay Regularization

[Online]. Available: https://doi.org/10.48550/arXiv.1711.05101 12

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1711.05101