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Parity supervision, not the circuit architecture, drives generalization in IQP quantum generative models, a controlled benchmark shows.

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2026-08-02 14:22 UTC pith:HCANONXA

load-bearing objection A clean controlled study showing parity supervision beats MSE for generalization in IQP Born machines, but the key classical MaxEnt control may be undertrained and the benchmark builds in spectral alignment. the 3 major comments →

arxiv 2605.10258 v2 pith:HCANONXA submitted 2026-05-11 quant-ph

Parity Supervision as a Driver of Generalization in Quantum Generative Modeling

Markus Baumann , Daniel Hein , Steffen Udluft , Tobias Rohe , Claudia Linnhoff-Popien , Jonas Stein This is my paper
classification quant-ph MSC 81P6868Q12
keywords quantum circuit Born machinesIQP circuitsparity supervisionWalsh-Hadamard spectrumgenerative model generalizationunseen high-value state recoveryforward KL divergencespectral completion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks whether the parity moments used to train IQP circuit Born machines are only a computational convenience or also an inductive bias for generalization. In a small, exactly enumerable benchmark, it compares the same IQP circuit trained on Walsh–Hadamard parity moments versus coordinate-wise squared error, alongside classical controls. It finds that parity supervision yields a better exact forward-KL fit to the target and recovers more unseen high-value states, and that a parameter-free reconstruction from the parity moments already captures much of the effect. The conclusion is that when target, loss, and circuit share spectral structure, parity supervision transfers evidence from observed to unseen states.

Core claim

On the paper's own terms, the central discovery is an attribution result: replacing the training loss inside a fixed IQP circuit changes generalization more than the circuit choice alone. The same 24-parameter IQP circuit, trained on parity moments instead of coordinate-wise MSE, lowers mean forward KL from roughly 0.49 to 0.40 at the reference setting and achieves the lowest KL among all compared models in 190 of 200 paired instances. The parity-trained model also discovers about 53 distinct unseen high-value states per 1000 samples, versus 35–38 for trained classical baselines and 18 for a maximum-entropy model using the same parity moments. A parameter-free partial Walsh reconstruction bu

What carries the argument

The central object is the Walsh–Hadamard parity moment: for a mask α, the expectation E_x[r(x)](-1)^{α·x}, which measures how strongly a distribution aligns with a subset-parity pattern. Matching a sampled band of K such moments couples every state in the Boolean cube at once, because each state enters every moment through its parity fingerprint. The paper's key diagnostic is a parameter-free partial inverse reconstruction, clipped and renormalized into a valid distribution, showing that the matched band alone already biases mass toward unseen high-value states. The IQP circuit, with its commuting diagonal ZZ couplings, acts as a richer parametrized refinement of the same spectral scaffold,

Load-bearing premise

The benchmark is constructed so that the information determining unseen high-value states is concentrated on the sampled parity band, meaning the empirical parity moments actually carry the evidence the model needs; if a target's useful spectral information lay outside that band, the claimed transfer mechanism would not operate.

What would settle it

Run the same paired-instance protocol with a target on the same even-parity support but with the score function bit-permuted, so the longest-zero-block score depends on coordinates whose masks are far from the sampled band; if parity-trained IQP then fails to beat IQP-MSE on unseen-state recovery, the effect is an artifact of band–target alignment rather than a general property of parity supervision.

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

If this is right

  • Within a fixed IQP architecture, the choice of training objective can matter as much as the circuit itself, so future IQP training studies should compare parity supervision against pointwise losses under matched data and budget.
  • Parity supervision can serve as a finite-sample generalization lever when the target distribution's useful structure is concentrated in the sampled Walsh band.
  • Finite-budget discovery of unseen high-value states improves with parity training, which is directly relevant to combinatorial and structured generation tasks.
  • The effect survives device noise in a small hardware feasibility check, suggesting parity-supervised Born machines are not limited to noiseless simulation.
  • The parameter-free spectral proxy provides a no-training diagnostic for whether a chosen parity band will transfer evidence to unseen states.

Where Pith is reading between the lines

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

  • Editorial inference: if the mechanism is general, adaptive band selection—choosing masks whose empirical moments carry the most target signal—should amplify the effect beyond the paper's random sampled band.
  • Editorial inference: on targets whose useful structure is not Walsh-concentrated, such as long-range patterns not captured by low-weight masks, parity supervision would likely become neutral or harmful; this is a direct testable extension.
  • Editorial inference: the paper's alignment principle echoes a broader spectral-bias pattern in learning systems: match the training observables to the native frequency basis of the model class and the target's invariants.
  • Editorial inference: replacing exact enumeration with statistical estimators of KL and occupancy could extend the same attribution design to larger system sizes, where the paper's exact evaluation is no longer possible.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper reports a controlled benchmark study of parity (Walsh) supervision as an inductive bias for generalization in IQP circuit Born machines. The target is an exactly enumerable 12-qubit Boltzmann distribution on even-parity bit strings, with score given by the longest zero block; training uses m=200 samples. Within a fixed IQP circuit, the authors compare a parity-moment loss against coordinate-wise MSE, and also compare against classical controls (two Ising models, an autoregressive Transformer, and a maximum-entropy model constrained on the same parity moments), plus a parameter-free band-limited spectral reconstruction qspec. They find that parity supervision improves exact forward KL and unseen high-value-state recovery over IQP-MSE, that the tested classical controls do not match this, and that qspec already captures much of the recovery, with the IQP circuit providing additional refinement. A small IBM hardware experiment preserves the loss-level ordering. The paper is explicitly scoped as a mechanism study rather than a practical advantage claim.

Significance. If the attribution chain is fully supported, the paper makes a useful conceptual contribution: it isolates the training-signal effect from architecture and data-access effects in a quantum generative model, and it provides a concrete spectral-completion mechanism for generalization. The controlled design is a genuine strength: exact forward-KL evaluation by enumeration, paired instances with shared data and optimization budgets, a fixed circuit for the parity-vs-MSE comparison, a parameter-free diagnostic qspec, and a hardware feasibility check. The central paired IQP comparison appears internally sound. The main weaknesses are in the supporting attribution, particularly the adequacy of the MaxEnt control and the degree to which the benchmark's alignment is tested rather than assumed.

major comments (3)
  1. [§VI-B and Appendix C] MaxEnt-parity convergence is not verified. The 512-parameter convex log-partition objective is trained with Adam lr=0.05 for 600 full-objective steps, with no gradient-norm, duality-gap, or moment-matching diagnostic reported. For K=512 and m=200, 600 steps is not a priori sufficient to reach the optimum. Since MaxEnt-parity is the control supporting the statements that 'the same parity moments ... are insufficient' and that the IQP parameterization 'refines' the spectral scaffold, the reported KL=1.80±0.11 could reflect an unfinished optimization rather than an intrinsic property of the moments. Please report convergence diagnostics and, preferably, an exactly converged MaxEnt solution; the objective is convex and n=12 is enumerable, so this is straightforward.
  2. [§V-A and §IV-A] The alignment between the target and the supervised band is built into the construction rather than tested. The valid support (Eq. 17) is even parity, the sampled band Ω uses low-weight masks, and the high-value region (Eqs. 9–10) is defined on that same support. The paper asserts that ℓ(x)'s Walsh spectrum overlaps the sampled band, but no overlap measure is reported. Without a mismatched control—for example, a target with the same support whose high-value states are concentrated on Walsh coefficients outside Ω—the 'when spectrally aligned' caveat risks being a restatement of the construction. Please quantify the spectral overlap between the target and Ω, or add a misalignment ablation showing that the transfer mechanism fails when the overlap is removed.
  3. [§IV-D, Eqs. (13)–(16)] The visibility decomposition is derived for the unclipped qlin, but the experiments use qspec, the clipped and renormalized projection of Eq. (14). Eq. (16) does not by itself imply the same region-mass ordering for qspec once negative values are clipped and the normalization denominator changes. If Eq. (16) is used to explain why the band 'already generalizes' in §VI-D, the analysis should be repeated for qspec, or the qlin version of the recovery curves should be reported alongside; otherwise the analytic mechanism and the reported diagnostic are not the same object.
minor comments (4)
  1. [Eq. (8)] Exact forward KL is computed over S, but the text does not state how q(x)=0 is handled for states with p*(x)>0. All models used here may be strictly positive on S, but this should be made explicit given that exact enumeration is claimed.
  2. [Table IV / Fig. 5] Table IV reports 'KL wins' in 190/200 instances, while Fig. 5(a) states IQP-parity has lowest KL at every β value. Please clarify whether the former is per-instance and the latter is per-β averaging, and report paired uncertainty for the win counts.
  3. [Appendix B] The Transformer is selected by validation NLL, but the tiny variant achieves the lowest exact KL in the capacity ablation. This is a reasonable protocol, but the sensitivity should be discussed because exact KL is the headline metric.
  4. [Fig. 6] The qspec and IQP-parity recovery curves in panel (a) appear to lack error bars; please state whether they are a single representative seed or averaged, and, if averaged, include the seed spread.

Circularity Check

0 steps flagged

No significant circularity: controlled benchmark; qspec diagnostic is a definitional decomposition, not a fitted prediction.

full rationale

No circular step is load-bearing. The paper's central comparison is an empirical benchmark: the same IQP circuit is trained with two losses, and the parity-trained model is evaluated by exact forward KL and finite-budget recovery against the known target. These metrics are not the training objective's target values, and the improvement is not a fitted constant renamed as a result. The qspec diagnostic (Eqs. 13-14) is constructed from the same empirical parity moments it is used to explain, but it is explicitly a parameter-free decomposition rather than a 'prediction': its unseen-state recovery is an empirical property of the sampled data and would not hold for arbitrary moment vectors, so the mechanistic claim is not equivalent to the construction by definition. The benchmark's even-parity support and low-weight mask band are acknowledged scope choices, and the paper explicitly disclaims general applicability and states that the target, objective, and circuit are 'structurally aligned'; this is deliberate controlled-setup design, not hidden circularity. The primary weakness in the attribution chain is the MaxEnt-parity control's unverified convergence (Appendix C: 600 Adam steps on a 512-parameter convex objective, no convergence diagnostic), which is a missing-support issue in the empirical comparison, not a circularity. No self-citations are used as load-bearing evidence.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

No new physical entities are postulated. The central claim rests on benchmark hyperparameters (band, sample size, threshold), standard Walsh analysis, and the deliberate choice of a parity-defined support, which is the main circularity-relevant construction.

free parameters (6)
  • band width K = 512 reference; 128/256/512 in ablation
    Number of sampled Walsh masks matched; larger K should improve fit, and the paper shows the ordering depends on K.
  • mask sampling scale σ = 1 reference; 0.5, 2, 3 in ablation
    Controls the mask-weight distribution; σ=1 gives expected mask weight ~2.4, biasing the band toward low-order Walsh coefficients.
  • training sample size m = 200
    Sets the data-scarce regime (<10% of the valid support); generalization difficulty depends on this choice.
  • optimizer step budgets and learning rates = 600 steps; LR 0.05 for parity/Ising/MaxEnt, 1e-3 for Transformer
    Used for fairness across models, but MaxEnt convergence is not verified, and its poor KL may stem from underfitting.
  • β sharpness sweep = 0.1 to 2.0 in steps of 0.1
    Difficulty axis of the target family; conclusions are drawn across this sweep, and large β concentrates mass on few states.
  • high-value threshold τ = 0.1
    Defines the high-value region and unseen elite; recovery results depend on this threshold.
axioms (5)
  • standard math Walsh–Hadamard characters form an orthonormal basis and the full moment spectrum uniquely determines the distribution (Eqs. (4)–(5)).
    Basis for interpreting parity moments as spectral supervision and for the partial inverse reconstruction qlin.
  • domain assumption Born rule defines the QCBM distribution and the IQP ZZ-diagonal circuit can be simulated exactly for n=12 (Eqs. (1)–(2)).
    Standard quantum generative modeling assumptions; exact enumeration is used only for evaluation and attribution.
  • domain assumption Empirical parity moments from m=200 i.i.d. samples are informative proxies for the target moments.
    The transfer mechanism depends on sample moments carrying enough target information; finite-sample noise is not modeled.
  • ad hoc to paper Even-parity support S (Eq. (17)) is a fair proxy for a structured validity constraint.
    The validity constraint is itself parity-defined, aligning the target with parity supervision and favoring the tested effect.
  • ad hoc to paper The score ℓ (longest zero block) is not supplied to the loss, but its Walsh spectrum overlaps the sampled low-weight band.
    Needed for qspec transfer to unseen high-value states; not established beyond this benchmark.

pith-pipeline@v1.3.0-alltime-deepseek · 15593 in / 12656 out tokens · 147296 ms · 2026-08-02T14:22:38.505298+00:00 · methodology

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read the original abstract

Generative models learn probability distributions in order to produce new samples beyond a finite training set. Their usefulness therefore depends on assigning probability to valid but previously unseen states. In a controlled benchmark, we test whether parity-based training provides an inductive bias for this kind of generalization in instantaneous quantum polynomial-time (IQP) circuit Born machines. We compare the same IQP circuit trained with parity supervision and coordinate-wise mean-squared error (MSE), together with classical controls. Parity supervision improves exact distributional fit and recovery of unseen high-value states over IQP-MSE. A circuit-free spectral reconstruction shows that the matched parity moments already transfer evidence from observed samples to structurally compatible unseen states, while the IQP circuit further refines this structure. These results identify parity supervision as both a tractable training signal and a generalization mechanism when the target distribution, training objective, and circuit architecture are spectrally aligned.

Figures

Figures reproduced from arXiv: 2605.10258 by Claudia Linnhoff-Popien, Daniel Hein, Jonas Stein, Markus Baumann, Steffen Udluft, Tobias Rohe.

Figure 1
Figure 1. Figure 1: How parity supervision can transfer evidence to unseen states. An unobserved valid state ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: IQP ansatz used throughout the paper. Hadamard layers surround [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Target mass per score level across the β sweep for n = 12. Larger β concentrates mass on fewer high-score states. The analytic tractability of qlin yields a clean decomposition of the mass it assigns to any region A ⊆ S. Defining ϕ¯A(α) := 2−n X x∈A (−1)α·x (15) and summing Eq. (13) over A gives the uniform-plus-visibility decomposition qlin(A) = |A| 2 n |{z} uniform baseline + X K k=1 pbtrain(αk) ϕ¯A(αk) … view at source ↗
Figure 4
Figure 4. Figure 4: Exact KL diagnostics at the representative fixed- [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Cross-class diagnostics under the reference parity band [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mechanism at β=0.9. Cumulative recovery Rq(Q) on the unseen high-value set Eτ . (a) The parameter-free spectral proxy qspec already captures much of the recovery gain of the best IQP-parity model over uniform sampling. (b) Within the IQP family, parity-trained variants recover Eτ faster than IQP-MSE. (c) The parameter-free proxy exhibits the same (σ, K) spread, showing that the band moments—not circuit opt… view at source ↗
Figure 7
Figure 7. Figure 7: Exact forward KL at fixed β = 0.9 across n ∈ {10, . . . , 20}, shown as paired-instance point clouds for all baselines. IQP-parity achieves the lowest median KL at every tested size without per-n retuning. Table V. Representative median forward KL DKL(p ⋆∥q) at β = 0.9 across 10 matched seeds; lower is better. Model n = 10 n = 15 n = 20 IQP-parity 0.338 0.638 1.256 AR Transformer 0.720 0.957 1.318 Ising+fi… view at source ↗

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Reference graph

Works this paper leans on

33 extracted references · 5 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Inverse molecular design using machine learning: Generative models for matter engineering,

    B. Sanchez-Lengeling and A. Aspuru-Guzik, “Inverse molecular design using machine learning: Generative models for matter engineering,” Science, vol. 361, no. 6400, pp. 360–365, 2018

    2018

  2. [2]

    GuacaMol: Benchmarking models for de novo molecular design,

    N. Brown, M. Fiscato, M. H. S. Segler, and A. C. Vaucher, “GuacaMol: Benchmarking models for de novo molecular design,”Journal of Chemical Information and Modeling, vol. 59, no. 3, pp. 1096–1108, 2019

    2019

  3. [3]

    Molecular sets (MOSES): A benchmarking platform for molecular generation models,

    D. Polykovskiy, A. Zhebrak, B. Sanchez-Lengeling, S. Golovanov, O. Tatanovet al., “Molecular sets (MOSES): A benchmarking platform for molecular generation models,”Frontiers in Pharmacology, vol. 11, p. 565644, 2020

    2020

  4. [4]

    Machine learning for combinatorial optimization: A methodological tour d’horizon,

    Y . Bengio, A. Lodi, and A. Prouvost, “Machine learning for combinatorial optimization: A methodological tour d’horizon,”European Journal of Operational Research, vol. 290, no. 2, pp. 405–421, 2021

    2021

  5. [5]

    A note on the evalua- tion of generative models,

    L. Theis, A. van den Oord, and M. Bethge, “A note on the evalua- tion of generative models,” inInternational Conference on Learning Representations, 2016

    2016

  6. [6]

    Assessing generative models via precision and recall,

    M. S. M. Sajjadi, O. Bachem, M. Lucic, O. Bousquet, and S. Gelly, “Assessing generative models via precision and recall,” inAdvances in Neural Information Processing Systems (NeurIPS), 2018

    2018

  7. [7]

    Improved precision and recall metric for assessing generative models,

    T. Kynkäänniemi, T. Karras, S. Laine, J. Lehtinen, and T. Aila, “Improved precision and recall metric for assessing generative models,” inAdvances in Neural Information Processing Systems (NeurIPS), 2019

    2019

  8. [8]

    Reliable fidelity and diversity metrics for generative models,

    M. F. Naeem, S. J. Oh, Y . Uh, Y . Choi, and J. Yoo, “Reliable fidelity and diversity metrics for generative models,” inProceedings of the 37th International Conference on Machine Learning (ICML), ser. Proceedings of Machine Learning Research, vol. 119, 2020, pp. 7176–7185

    2020

  9. [9]

    Generalization met- rics for practical quantum advantage in generative models,

    K. Gili, M. Mauri, and A. Perdomo-Ortiz, “Generalization met- rics for practical quantum advantage in generative models,” 2022, arXiv:2201.08770

    Pith/arXiv arXiv 2022

  10. [10]

    A framework for demonstrating practical quantum advantage: Comparing quantum against classical generative models,

    M. Hibat-Allah, M. Mauri, J. Carrasquilla, and A. Perdomo-Ortiz, “A framework for demonstrating practical quantum advantage: Comparing quantum against classical generative models,”Communications Physics, vol. 7, p. 68, 2024

    2024

  11. [11]

    Differentiable learning of quantum circuit Born machine,

    J.-G. Liu and L. Wang, “Differentiable learning of quantum circuit Born machine,”Physical Review A, vol. 98, no. 6, p. 062324, 2018

    2018

  12. [12]

    A generative modeling approach for benchmark- ing and training shallow quantum circuits,

    M. Benedetti, D. Garcia-Pintos, O. Perdomo, V . Leyton-Ortega, Y . Nam, and A. Perdomo-Ortiz, “A generative modeling approach for benchmark- ing and training shallow quantum circuits,”npj Quantum Information, vol. 5, p. 45, 2019

    2019

  13. [13]

    Training of quantum circuits on a hybrid quantum computer,

    D. Zhu, N. M. Linke, M. Benedetti, K. A. Landsman, N. H. Nguyen et al., “Training of quantum circuits on a hybrid quantum computer,” Science Advances, vol. 5, no. 10, p. eaaw9918, 2019

    2019

  14. [14]

    Do quantum circuit Born machines generalize?

    K. Gili, M. Hibat-Allah, M. Mauri, C. Ballance, and A. Perdomo-Ortiz, “Do quantum circuit Born machines generalize?”Quantum Science and Technology, vol. 8, no. 3, p. 035021, 2023

    2023

  15. [15]

    Trainability barriers and opportunities in quantum generative modeling,

    M. S. Rudolph, S. Lerch, S. Thanasilp, O. Kiss, O. Shayaet al., “Trainability barriers and opportunities in quantum generative modeling,” npj Quantum Information, vol. 10, p. 116, 2024

    2024

  16. [16]

    O’Donnell,Analysis of Boolean Functions

    R. O’Donnell,Analysis of Boolean Functions. Cambridge University Press, 2014

    2014

  17. [17]

    Terras,Fourier Analysis on Finite Groups and Applications

    A. Terras,Fourier Analysis on Finite Groups and Applications. Cam- bridge University Press, 1999

    1999

  18. [18]

    A kernel two-sample test,

    A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schölkopf, and A. J. Smola, “A kernel two-sample test,”Journal of Machine Learning Research, vol. 13, pp. 723–773, 2012

    2012

  19. [19]

    IQPopt: Fast optimization of instantaneous quantum polynomial circuits in JAX,

    E. Recio-Armengol and J. Bowles, “IQPopt: Fast optimization of instantaneous quantum polynomial circuits in JAX,”arXiv preprint arXiv:2501.04776, 2025

    Pith/arXiv arXiv 2025

  20. [20]

    Simulating quantum computers with probabilistic methods,

    M. van den Nest, “Simulating quantum computers with probabilistic methods,”Quantum Information & Computation, vol. 11, no. 9–10, pp. 784–812, 2011

    2011

  21. [21]

    Train on classical, deploy on quantum: Scaling generative quantum machine learning to a thousand qubits,

    E. Recio-Armengol, S. Ahmed, and J. Bowles, “Train on classical, deploy on quantum: Scaling generative quantum machine learning to a thousand qubits,” 2025, arXiv:2503.02934

    arXiv 2025

  22. [22]

    On the spectral bias of neural networks,

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Linet al., “On the spectral bias of neural networks,” inProceedings of the 36th International Conference on Machine Learning (ICML), ser. Proceedings of Machine Learning Research, vol. 97. PMLR, 2019, pp. 5301–5310

    2019

  23. [23]

    Frequency principle: Fourier analysis sheds light on deep neural networks,

    Z.-Q. J. Xu, Y . Zhang, T. Luo, Y . Xiao, and Z. Ma, “Frequency principle: Fourier analysis sheds light on deep neural networks,”Communications in Computational Physics, vol. 28, no. 5, pp. 1746–1767, 2020, first circulated as arXiv:1901.06523, 2019

    Pith/arXiv arXiv 2020

  24. [24]

    Frequency bias in neural networks for input of non-uniform density,

    R. Basri, M. Galun, A. Geifman, D. Jacobs, Y . Kasten, and S. Kritchman, “Frequency bias in neural networks for input of non-uniform density,” arXiv preprint arXiv:2003.04560, 2020

    Pith/arXiv arXiv 2003

  25. [25]

    Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks,

    A. Canatar, B. Bordelon, and C. Pehlevan, “Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks,”Nature Communications, vol. 12, p. 2914, 2021

    2021

  26. [26]

    Spectrum dependent learning curves in kernel regression and wide neural networks,

    B. Bordelon, A. Canatar, and C. Pehlevan, “Spectrum dependent learning curves in kernel regression and wide neural networks,” inProceedings of the 37th International Conference on Machine Learning (ICML), ser. Proceedings of Machine Learning Research, vol. 119. PMLR, 2020, pp. 1024–1034

    2020

  27. [27]

    Effect of data encoding on the expressive power of variational quantum-machine-learning models,

    M. Schuld, R. Sweke, and J. J. Meyer, “Effect of data encoding on the expressive power of variational quantum-machine-learning models,” Physical Review A, vol. 103, no. 3, p. 032430, 2021

    2021

  28. [28]

    Fourier anal- ysis of variational quantum circuits for supervised learning,

    M. Wiedmann, M. Periyasamy, and D. D. Scherer, “Fourier anal- ysis of variational quantum circuits for supervised learning,” 2024, arXiv:2411.03450

    Pith/arXiv arXiv 2024

  29. [29]

    Spectral bias in variational quantum machine learning,

    C. Duffy and M. Jastrzebski, “Spectral bias in variational quantum machine learning,” 2025, arXiv:2506.22555

    arXiv 2025

  30. [30]

    Evaluating generalization in GFlowNets for molecule design,

    A. C. Nica, M. Jain, E. Bengio, C.-H. Liu, M. Korablyovet al., “Evaluating generalization in GFlowNets for molecule design,” inICLR 2022 Workshop on Machine Learning for Drug Discovery, 2022

    2022

  31. [31]

    Towards understanding and improving GFlowNet training,

    M. W. Shen, E. Bengio, E. Hajiramezanali, A. Loukas, K. Cho, and T. Biancalani, “Towards understanding and improving GFlowNet training,” inProceedings of the 40th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 202. PMLR, 2023, pp. 30 956–30 975

    2023

  32. [32]

    Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws,

    A. V . Gnedin, B. Hansen, and J. Pitman, “Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws,”Probability Surveys, vol. 4, pp. 146–171, 2007

    2007

  33. [33]

    A brief introduction to Fourier analysis on the Boolean cube,

    R. de Wolf, “A brief introduction to Fourier analysis on the Boolean cube,”Theory of Computing, Graduate Surveys, vol. 1, pp. 1–20, 2008

    2008