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REVIEW 4 major objections 5 minor 44 references

GQWformer: A Quantum-based Transformer for Graph Representation Learning

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read GQWformer injects attribute-aware quantum-walk distances into transformer attention and reports state-of-the-art graph classification accuracy on five benchmarks.

desk verdict A quantum-walk variant of Random Walk Conformer whose reported gains over RWC are within noise on every dataset, and whose core encoding step is undefined. read the letter →

arxiv 2412.02285 v1 pith:AIUBR6XP submitted 2024-12-03 cs.LG cs.AI

Lei Yu , Hongyang Chen , Jingsong Lv , Linyao Yang This is my paper · ORCID
classification cs.LGcs.AI
keywords GraphTransformerQuantumWalkStructuralEncodingInductiveBiasClassificationSelf-AttentionRecurrentModuleAttribute-AwareCoins
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

GQWformer is a graph transformer for graph classification whose central claim is that discrete-time quantum walks, made sensitive to node attributes through learnable feature-dependent coin operators, can supply the structural inductive bias that plain self-attention lacks. The paper argues that this attribute-aware quantum-walk encoding captures both topology and node features, and that adding the resulting pairwise distances as an additive bias inside the attention softmax, together with a bidirectional recurrent module over the walk sequence, improves classification. On five public benchmarks the authors report accuracies of 95.2% on MUTAG, 76.7% on PTC, 80.7% on PROTEINS, 79.3% on IMDB-B, and 55.3% on IMDB-M, each above the random-walk transformer baseline they compare against. If correct, this establishes quantum-walk encodings as a practical structural prior for graph transformers rather than a purely theoretical device.

What carries the argument

The attribute-aware graph quantum walk is the load-bearing object. It is a discrete-time quantum walk on an arbitrary graph with a position Hilbert space over nodes and a coin Hilbert space of dimension equal to the maximum degree; the coin at each node is a Householder reflection built from a trainable attention vector $g(v_i)$ over that node's neighbors, making the walk sensitive to node features as well as topology. The paper's key identity is the evolution $U = S(I \otimes C)$, where $C$ is the collection of feature-dependent coin operators and $S$ is the shift operator; iterating $U$ for $T$ steps yields the superposition tensor $|\phi_t\rangle$, and the extracted pairwise bias matrix $M_T$ is what carries structural information into the self-attention bias of Eq. (10).

What would settle it

Compute the exact map from $|\phi_t\rangle$ to $M_T$ as the authors intend and re-run the five classification benchmarks under the stated ten-fold protocol; the central claim is false if the reported accuracies cannot be reproduced, or if zeroing out the quantum-walk bias $p_{ij}$ changes accuracy by less than one standard deviation.

Watch

Extended reading notes

Core claim

The central claim is that a graph transformer can be made structure-aware by running multiple non-interacting discrete-time quantum walks on the node-attributed graph, one starting at each node, and using the resulting superposition sequence as an inductive bias. A learnable, feature-dependent coin operator $C_i = I - 2g(v_i)g(v_i)^\top/(g(v_i)^\top g(v_i))$ is generated from neighbor attention scores, the walk is evolved for $T$ steps, and the squared spin amplitudes are aggregated into matrices $\{M_0, \dots, M_T\}$. The matrix $M_T$ is treated as a pairwise distance between nodes and added as a bias $p_{ij}$ inside the attention softmax, so attention weights reflect both semantic similarity $q_i^\top k_j$ and the quantum-walk distance. A bidirectional GRU then consumes the sequence $\{M_0, \dots, M_T\}$ to strengthen local, order-sensitive information, and its pooled output is combined with the attention output. The paper reports that this design outperforms state-of-the-art graph classifiers on MUTAG, PTC, PROTEINS, IMDB-B, and IMDB-M.

Load-bearing premise

The whole construction assumes that a well-defined pairwise distance matrix can be extracted from the quantum-walk superposition and used as a calibrated attention bias, but the paper does not specify how $M_t$ is computed from $|\phi_t\rangle$, how the shift operator handles nodes with degree below the maximum, or how the feature-dependent coin is built for variable-degree neighborhoods.

Editorial extensions

If this is right

  • Graph transformers gain a principled structural bias by feeding quantum-walk distances into the attention bias, so node pairs that are close in the quantum-walk metric receive different attention than pairs with similar features but different topology.
  • The walk length $T$ controls the model's receptive field; the sensitivity analysis shows performance peaks at $T=4$ on PTC, so choosing $T$ is a real hyperparameter affecting the global-local balance.
  • The ablation results imply both modules are needed: GQW-Attn captures global pairwise structure and GQW-Recu captures local, order-sensitive structure, and removing either one lowers PTC accuracy.
  • Attribute-aware coins matter: the proposed coin function outperforms both feature-blind and invariant quantum-walk encodings on PTC.
  • Consequently, quantum-walk encodings are a viable alternative to classical random-walk or shortest-path encodings in graph transformers, not just a theoretical curiosity.

Reading between the lines

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

  • If the encoding is as transferable as it appears, the same $M_T$ bias could be dropped into other attention-based graph models, extending the result beyond the exact GQWformer stack.
  • A classical diffusion encoding with a learnable pairwise bias might reproduce most of the gain; comparing the two would isolate what the quantum machinery adds.
  • Because the paper fixes $T$ and the block count for all datasets, per-dataset tuning of walk length and number of blocks could push the reported accuracies higher.
  • The reported gap over the random-walk transformer baseline is sharpest on PTC, so a replication study focused on PTC would be the cheapest way to test the mechanism's contribution.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes GQWformer, a graph transformer that injects structural information into self-attention via attribute-aware discrete-time quantum walks (QWs). A learnable coin operator, constructed with Householder reflections from an attention function over node features, drives T steps of a multi-walker QW; the resulting sequence of matrices is used both as an additive attention bias (GQW-Attn, Eq. 10) and as input to a bidirectional GRU module (GQW-Recu). The model is evaluated on five TUDataset graph classification benchmarks, with a claim of consistent and significant improvement over the RWC baseline. The manuscript also includes an ablation study on PTC and a sensitivity analysis over walk length.

Significance. If the method were fully specified and the empirical gains were significant, the idea of using learnable quantum-walk encodings as structural biases for graph transformers would be a potentially interesting contribution to graph representation learning. The paper identifies a real limitation of standard graph transformers — the lack of attribute-aware structural inductive biases — and proposes a mechanism that is conceptually novel. However, the contribution as presented is not yet substantiated: the central empirical claim is contradicted by the paper's own reported error bars, the core quantum-walk encoding pipeline is underspecified at several load-bearing steps, and no code is provided to disambiguate the construction. The ablation and sensitivity studies are useful in intent but are reported without standard deviations or repeated runs, limiting their evidentiary value.

major comments (4)
  1. [Experiments, Table 1] The claim that GQWformer 'consistently and significantly outperforms RWC on all five datasets by a large margin' is not supported by the numbers in Table 1. The accuracy differences over RWC are: MUTAG 95.2±3.0 vs 94.7±3.7 (diff 0.5), PTC 76.7±4.2 vs 74.4±4.4 (diff 2.3), PROTEINS 80.7±4.3 vs 79.4±4.7 (diff 1.3), IMDB-B 79.3±1.7 vs 78.8±3.1 (diff 0.5), IMDB-M 55.3±2.1 vs 54.5±2.9 (diff 0.8). Using the reported standard deviations in a two-sample comparison with n=10, the resulting t-values are approximately 0.33, 1.20, 0.64, 0.42, and 0.66 respectively; none approach conventional significance levels. No significance tests, confidence intervals, or seed-level variance information are reported. Furthermore, the baseline numbers appear to be copied from prior publications rather than obtained by rerunning the baselines under identical conditions; this is not stated in the paper. Since the abstract's headline claim is that GQWformer 'outperforms existing state-of-the-art graph classification algorithms,' the reported evidence does not establish this.
  2. [Method, 'Attribute-aware Graph Quantum Walk'] The construction of the encoding matrices {M0, ..., MT} is not specified. The paper states that 'by summing the squares of the spin states of each superposition state, we derive a sequence of matrices' and then treats MT as a structural encoding matrix whose (i,j)-th element p_ij 'encodes the distance between node vi and node vj.' No equation defines M_t from the superposition tensor |ϕt⟩ ∈ C^{n×n×d}, nor is it stated whether p_ij is real-valued, normalized, or bounded. Since p_ij is added directly to the attention logit q_i^T k_j in Eq. (10) without any scaling or calibration, the magnitude and sign of p_ij can dominate or be negligible relative to the semantic term depending on the unspecified construction. Without a precise definition of M_t and p_ij, the central mechanism of GQW-Attn is not well-defined and the method cannot be reproduced.
  3. [Method, Eqs. (8)-(9), and shift operator] The quantum-walk update is underspecified for graphs whose nodes have degrees smaller than the maximum degree d. The coin space H_c is defined as the span of {|i⟩, i ∈ [1,...,d]}, and in Eq. (9) g(vi) is computed against Xi_d, a matrix with d rows equal to the feature of vi. For a node of degree deg(vi) < d, it is not explained how the attention function a handles nonexistent neighbors, whether features are padded, or how the Householder coin C_i = I - 2g(vi)g(vi)^T/(g(vi)^T g(vi)) is formed when the effective dimension of g(vi) is deg(vi) rather than d. Similarly, the shift operator S is said to 'swap the states of vertices connected by edges,' but for a node with degree less than d, the coin basis includes directions that do not correspond to any edge; the action of S on those components is not defined. These are not cosmetic details: they determine whether the walk is a well-defined unitary process on the given graph.
  4. [Method, 'Attribute-aware Graph Quantum Walk'; Introduction; Ablation] The paper repeatedly describes the QW encoding as an 'inductive bias' that provides 'tailored inductive bias' and 'structural bias.' However, the coin operator C_i is generated from g(vi), which depends on the learnable weight matrix W (Eq. 9), and W is trained end-to-end on the classification objective. Thus the structural encoding is fitted to the labels rather than being a fixed, independent prior. This weakens the conceptual claim that QWs inject an a priori structural inductive bias; the encoding is better described as a learned structural feature. This issue should be explicitly acknowledged, and the ablation rows 'vanilla QW' and 'QW(inv)' only partially address it because they compare against other learned or fixed coin variants, not against the absence of the QW bias with the same number of parameters.
minor comments (5)
  1. [Throughout] The manuscript contains numerous typos and grammatical errors, including 'chanllenging', 'Auther', 'intrinsic', 'similarly', 'infromation', 'precess', 'oppsite', 'datsets', 'captical', 'drived', and 'concatebate'. A careful proofreading pass is needed.
  2. [Preliminary, Eq. (1)] The self-attention formulation in Eq. (1) is nonstandard: a_i^l is a weighted sum of value vectors v_j^(l-1), but then h_i^l = a_i^l + γ h_i^(l-1) omits the usual output projection and residual connection structure. The notation also overloads a_i^l as an attention vector rather than a scalar attention weight. Clarify the exact transformer variant being used.
  3. [Ablation Studies, Table 2] The ablation results are reported as single accuracy numbers without standard deviations or repeated trials. Given that the main results carry ±1.7 to ±4.3 percentage point standard deviations, single-run ablations are not informative for comparing modules or encoding variants.
  4. [Sensitivity Analysis, Table 3] The sensitivity analysis varies walk length from 3 to 8 on PTC only and reports one run per setting. The differences, e.g., 76.7 at length 4 vs 73.0 at length 3, are not accompanied by variance information, so it is unclear whether the pattern is stable or noise.
  5. [Readout] The virtual-node readout is mentioned briefly but not specified: how the virtual node is initialized, whether its features are learned, and how its hidden state is obtained after the GQWformer blocks are not described.

Circularity Check

1 steps flagged · score 6.0 of 10

The 'quantum-walk structural distance' used as the attention bias is explicitly a trainable parameter optimized on the classification objective, so the claimed QW structural inductive bias reduces to a fitted pairwise bias; the reported accuracy gains over RWC are additionally within one standard deviation on all five datasets.

  1. fitted input called prediction [Method, 'Attribute-aware Graph Quantum Walk' and 'Graph Quantum Walk Self-attention Module', Eqs. (8)-(10)]
    "By summing the squares of the spin states of each superposition state, we derive a sequence of matrices {M0, M1, ...,MT }... we interpret MT as a structural encoding matrix, where the (i, j)-th element of MT , denoted as pij, encodes the distance between node vi and node vj. As previous described, this distance is a trainable parameter... Ci = I − 2g(vi)g(vi)⊤/(g(vi)⊤g(vi)). Here, g(vi) is a function related to the features of node vi... g(vi) =a(WXN (vi), WXi d), where ... W ∈ CF ×F′ is a learnable weight matrix."

    The attention bias p_ij in Eq. (10) is presented as a structural 'distance' derived from quantum walks, but it is computed from coin operators generated by g(vi), which depends on a learnable weight matrix W and an attention function a (Eqs. 8-9). The paper itself calls p_ij 'a trainable parameter.' Since W and a are optimized by gradient descent on the same graph-classification loss whose accuracy is the paper's headline result, the 'structural encoding' is fitted to the labels rather than independently derived from the graph. Any measured benefit of this bias is therefore, by construction, a benefit of a learned pairwise attention bias; it cannot separately confirm the claim that QW states 'encapsulate rich structural attributes' or provide a fixed inductive bias.

full rationale

The only load-bearing circularity I can exhibit from the paper's own equations is the treatment of the QW encoding as a fixed structural inductive bias when, in fact, it is a learnable function of node features optimized for classification. Specifically, p_ij is called a 'distance' and a 'structural encoding matrix,' but Eq. (8) and Eq. (9) make it depend on trainable W, and the paper explicitly calls it 'a trainable parameter,' so the claimed structural prior is fitted to the task rather than supplied a priori. This is a fitted-input-called-prediction pattern: the model's attention bias is learned, and the paper then credits the quantum-walk 'inductive bias' for the resulting performance. I did not score the reported superiority over RWC as circularity: the mean differences are within one standard deviation on all five datasets (e.g., MUTAG 95.2±3.0 vs 94.7±3.7), but that is a statistical-evidence concern, not a circular-derivation concern. There is no self-citation chain; the citations to Dernbach et al., Rohde et al., and Yeh et al. are external prior work. The underspecified computation of M_t from |ϕt⟩ is a reproducibility defect, not circularity. Overall, the central mechanism partially reduces by construction to a learned pairwise bias, so the circularity score is 6 rather than 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central construction rests on a learnable weight matrix W that defines the coin operators (Eq. 9) and on at least two unproven modeling choices: the existence of a well-defined shift operator for arbitrary graphs with a single coin dimension, and the interpretation of summed spin-state probabilities as pairwise attention biases. No new physical entities are introduced; the quantum walk is a classical simulation.

free parameters (2)
  • W = trained, not reported
    Learnable weight matrix in Eq. (9) transforms node features to produce attention-based g(v_i) that defines each Householder coin operator; hence the entire QW encoding is trained rather than fixed.
  • walk length T = 4
    Set to 4 in Implementation Details; sensitivity on PTC shows 76.7 at T=4 and lower at other values, suggesting the choice is tuned and affects the attention bias range.
assumptions (3)
  • domain assumption The discrete-time quantum walk shift operator S is well-defined for arbitrary graphs with a fixed coin-space dimension d equal to the maximum degree, including for nodes of degree less than d.
    Introduced in 'Graph Quantum Walk' via operator S, but no construction is given for dangling coin directions or degree mismatch; relies on Kendon (2006) without specifying edge labeling or boundary behavior.
  • domain assumption The matrices M_t obtained by summing squared spin-state amplitudes are meaningful pairwise distances suitable as additive attention biases.
    Stated in Method: 'By summing the squares of the spin states ... derive a sequence of matrices ... take the sequence as input.' No derivation establishes that these entries encode a calibrated distance or affinity.
  • standard math Householder reflection matrices with e = g(v_i) are valid coin operators for the QW and remain differentiable through backpropagation.
    Eqs. (7) and (8) use Householder reflections, which are unitary; the standard math is correct, but the domain-specific choice of g(v_i) as a learnable attention vector is an additional unverified modeling assumption.

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Cite this review

Pith. "Pith review of GQWformer: A Quantum-based Transformer for Graph Representation Learning." pith.science (2026). https://pith.science/paper/AIUBR6XP

@misc{pith2026241202285,
  author       = {Pith},
  title        = {Pith review of: GQWformer: A Quantum-based Transformer for Graph Representation Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AIUBR6XP}},
  note         = {Machine review of arXiv:2412.02285}
}
read the original abstract

Graph Transformers (GTs) have demonstrated significant advantages in graph representation learning through their global attention mechanisms. However, the self-attention mechanism in GTs tends to neglect the inductive biases inherent in graph structures, making it chanllenging to effectively capture essential structural information. To address this issue, we propose a novel approach that integrate graph inductive bias into self-attention mechanisms by leveraging quantum technology for structural encoding. In this paper, we introduce the Graph Quantum Walk Transformer (GQWformer), a groundbreaking GNN framework that utilizes quantum walks on attributed graphs to generate node quantum states. These quantum states encapsulate rich structural attributes and serve as inductive biases for the transformer, thereby enabling the generation of more meaningful attention scores. By subsequently incorporating a recurrent neural network, our design amplifies the model's ability to focus on both local and global information. We conducted comprehensive experiments across five publicly available datasets to evaluate the effectiveness of our model. These results clearly indicate that GQWformer outperforms existing state-of-the-art graph classification algorithms. These findings highlight the significant potential of integrating quantum computing methodologies with traditional GNNs to advance the field of graph representation learning, providing a promising direction for future research and applications.

Figures

Figures reproduced from arXiv: 2412.02285 by the authors.

Figure 1
Figure 1. The comparison of encoding strategies. In graphs [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. GQWformer. The model is composed of several GQWformer blocks. Each block consists of a GQW Generator, a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The architecture of the Graph Quantum Walk Re [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reference graph

Works this paper leans on

44 extracted references · 27 canonical work pages

  1. [1]

    Aharonov, Y.; Davidovich, L.; and Zagury, N. 1993. Quantum random walks. Physical Review A, 48(2): 1687

    work page 1993

  2. [2]

    Ambainis, A. 2003. Quantum walks and their algorithmic applications. International Journal of Quantum Information, 1(04): 507--518

    work page 2003

  3. [3]

    Y.; and Hancock, E

    Bai, L.; Jiao, Y.; Cui, L.; Rossi, L.; Wang, Y.; Philip, S. Y.; and Hancock, E. R. 2021. Learning graph convolutional networks based on quantum vertex information propagation. IEEE Transactions on Knowledge and Data Engineering, 35(2): 1747--1760

    work page 2021

  4. [4]

    F.; and Bronstein, M

    Bodnar, C.; Frasca, F.; Otter, N.; Wang, Y.; Li \`o , P.; Montufar, G. F.; and Bronstein, M. 2021. Weisfeiler and Lehman Go Cellular: CW Networks. Advances in Neural Information Processing Systems, 34: 2625--2640

    work page 2021

  5. [5]

    Bouritsas, G.; Frasca, F.; Zafeiriou, S.; and Bronstein, M. M. 2022. Improving graph neural network expressivity via subgraph isomorphism counting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(1): 657--668

    2022

  6. [6]

    Chen, D.; O’Bray, L.; and Borgwardt, K. 2022. Structure-aware transformer for graph representation learning. In International Conference on Machine Learning, 3469--3489. PMLR

    2022

  7. [7]

    Cong, I.; Choi, S.; and Lukin, M. D. 2019. Quantum convolutional neural networks. Nature Physics, 15(12): 1273--1278

    work page 2019

  8. [8]

    S.; and Welling, M

    de Haan, P.; Cohen, T. S.; and Welling, M. 2020. Natural Graph Networks. Advances in Neural Information Processing Systems, 33: 3636--3646

    work page 2020

Show all 44 references
  1. [9]

    Dernbach, S.; Mohseni-Kabir, A.; Pal, S.; Gepner, M.; and Towsley, D. 2019. Quantum walk neural networks with feature dependent coins. Applied Network Science, 4: 1--16

    2019

  2. [10]

    Farhi, E.; and Gutmann, S. 1998. Quantum computation and decision trees. Physical Review A, 58(2): 915

    1998

  3. [11]

    S.; Riley, P

    Gilmer, J.; Schoenholz, S. S.; Riley, P. F.; Vinyals, O.; and Dahl, G. E. 2017. Neural message passing for Quantum chemistry. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, 1263--1272

    2017

  4. [12]

    Huang, H.-Y.; Broughton, M.; Mohseni, M.; Babbush, R.; Boixo, S.; Neven, H.; and McClean, J. R. 2021. Power of data in quantum machine learning. Nature communications, 12(1): 2631

    2021

  5. [13]

    Huang, X.; Song, Q.; Li, Y.; and Hu, X. 2019. Graph recurrent networks with attributed random walks. In 25th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2019, 732--740. Association for Computing Machinery

    2019

  6. [14]

    Kendon, V. 2006. Quantum walks on general graphs. International Journal of Quantum Information, 4(05): 791--805

    2006

  7. [15]

    B.; and Goldstein, T

    Kong, K.; Chen, J.; Kirchenbauer, J.; Ni, R.; Bruss, C. B.; and Goldstein, T. 2023. GOAT: A global transformer on large-scale graphs. In International Conference on Machine Learning, 17375--17390. PMLR

    2023

  8. [16]

    Kreuzer, D.; Beaini, D.; Hamilton, W.; L \'e tourneau, V.; and Tossou, P. 2021. Rethinking graph transformers with spectral attention. Advances in Neural Information Processing Systems, 34: 21618--21629

    2021

  9. [17]

    Liu, C.; Zhan, Y.; Ma, X.; Ding, L.; Tao, D.; Wu, J.; and Hu, W. 2023. Gapformer: graph transformer with graph pooling for node classification. In Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, 2196--2205

    2023

  10. [18]

    Liu, Y.; Arunachalam, S.; and Temme, K. 2021. A rigorous and robust quantum speed-up in supervised machine learning. Nature Physics, 17(9): 1013--1017

    2021

  11. [19]

    Maron, H.; Ben-Hamu, H.; Serviansky, H.; and Lipman, Y. 2019. Provably Powerful Graph Networks. Advances in Neural Information Processing Systems, 32

    2019

  12. [20]

    Maron, H.; Ben-Hamu, H.; Shamir, N.; and Lipman, Y. 2018. Invariant and Equivariant Graph Networks. In International Conference on Learning Representations

    2018

  13. [21]

    Mialon, G.; Chen, D.; Selosse, M.; and Mairal, J. 2021. Graphit: Encoding graph structure in transformers. arXiv preprint arXiv:2106.05667

    2021 arXiv

  14. [22]

    M.; Bause, F.; Kersting, K.; Mutzel, P.; and Neumann, M

    Morris, C.; Kriege, N. M.; Bause, F.; Kersting, K.; Mutzel, P.; and Neumann, M. 2020. Tudataset: A collection of benchmark datasets for learning with graphs. arXiv preprint arXiv:2007.08663

    2020 arXiv

  15. [23]

    Nikolentzos, G.; and Vazirgiannis, M. 2020. Random walk graph neural networks. Advances in Neural Information Processing Systems, 33: 16211--16222

    2020

  16. [24]

    P.; Schreiber, A.; S tefa n \'a k, M.; Jex, I.; and Silberhorn, C

    Rohde, P. P.; Schreiber, A.; S tefa n \'a k, M.; Jex, I.; and Silberhorn, C. 2011. Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation. New Journal of Physics, 13(1): 013001

    2011

  17. [25]

    Rong, Y.; Bian, Y.; Xu, T.; Xie, W.; WEI, Y.; Huang, W.; and Huang, J. 2020. Self-Supervised Graph Transformer on Large-Scale Molecular Data. Advances in Neural Information Processing Systems, 33: 12559--12571

    2020

  18. [26]

    Schuld, M.; and Killoran, N. 2019. Quantum machine learning in feature hilbert spaces. Physical review letters, 122(4): 040504

    2019

  19. [27]

    Schuster, M.; and Paliwal, K. K. 1997. Bidirectional recurrent neural networks. IEEE transactions on Signal Processing, 45(11): 2673--2681

    1997

  20. [28]

    Tang, Y.; Yan, J.; and Edwin, H. 2022. From quantum graph computing to quantum graph learning: A survey. arXiv preprint arXiv:2202.09506

    2022 arXiv

  21. [29]

    Toenshoff, J.; Ritzert, M.; Wolf, H.; and Grohe, M. 2021. Graph learning with 1d convolutions on random walks. arXiv preprint arXiv:2102.08786

    2021 arXiv

  22. [30]

    N.; Kaiser, .; and Polosukhin, I

    Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.; Gomez, A. N.; Kaiser, .; and Polosukhin, I. 2017. Attention is all you need. Advances in neural information processing systems, 30

    2017

  23. [31]

    P.; and Yan, J

    Wu, Q.; Zhao, W.; Li, Z.; Wipf, D. P.; and Yan, J. 2022. Nodeformer: A scalable graph structure learning transformer for node classification. Advances in Neural Information Processing Systems, 35: 27387--27401

    2022

  24. [32]

    Wu, Q.; Zhao, W.; Yang, C.; Zhang, H.; Nie, F.; Jiang, H.; Bian, Y.; and Yan, J. 2024. Simplifying and empowering transformers for large-graph representations. Advances in Neural Information Processing Systems, 36

    2024

  25. [33]

    Xing, Y.; Wang, X.; Li, Y.; Huang, H.; and Shi, C. 2024. Less is More: on the Over-Globalizing Problem in Graph Transformers. arXiv preprint arXiv:2405.01102

    2024 arXiv

  26. [34]

    Xu, K.; Hu, W.; Leskovec, J.; and Jegelka, S. 2018. How Powerful are Graph Neural Networks? In International Conference on Learning Representations

    2018

  27. [35]

    Yan, G.; Tang, Y.; and Yan, J. 2022. Towards a native quantum paradigm for graph representation learning: A sampling-based recurrent embedding approach. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 2160--2168

    2022

  28. [36]

    Yeh, P.-K.; Chen, H.-W.; and Chen, M.-S. 2023. Random walk conformer: Learning graph representation from long and short range. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, 10936--10944

    2023

  29. [37]

    Ying, C.; Cai, T.; Luo, S.; Zheng, S.; Ke, G.; He, D.; Shen, Y.; and Liu, T.-Y. 2021. Do Transformers Really Perform Badly for Graph Representation? Advances in Neural Information Processing Systems, 34: 28877--28888

    2021

  30. [38]

    Yu, S.; Peng, C.; Wang, Y.; Shehzad, A.; Xia, F.; and Hancock, E. R. 2023. Quantum Graph Learning: Frontiers and Outlook. arXiv preprint arXiv:2302.00892

    2023 arXiv

  31. [39]

    Zhang, M.; Cui, Z.; Neumann, M.; and Chen, Y. 2018. An end-to-end deep learning architecture for graph classification. In Proceedings of the AAAI conference on artificial intelligence, volume 32

    2018

  32. [40]

    Zhang, Z.; Chen, D.; Wang, J.; Bai, L.; and Hancock, E. R. 2019. Quantum-based subgraph convolutional neural networks. Pattern Recognition, 88: 38--49

    2019

  33. [41]

    Zhang, Z.; Liu, Q.; Hu, Q.; and Lee, C.-K. 2022. Hierarchical graph transformer with adaptive node sampling. Advances in Neural Information Processing Systems, 35: 21171--21183

    2022

  34. [42]

    Zhao, L.; Jin, W.; Akoglu, L.; and Shah, N. 2022. From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awareness. In International Conference on Learning Representations

    2022

  35. [43]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all...

  36. [44]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.