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REVIEW 2 major objections 4 minor 73 references

Learning complexity gradually in quantum machine learning models

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Hardest-first training lifts quantum ML accuracy from 77% to 93%

desk verdict Solid empirical demonstration of hard-example mining for QML phase recognition, with a theoretical appendix that doesn't survive scrutiny; worth reviewing. read the letter →

arxiv 2411.11954 v2 pith:4ERL5CVP submitted 2024-11-18 quant-ph cond-mat.quant-gascond-mat.stat-mech

Erik Recio-Armengol , Franz J. Schreiber , Jens Eisert , Carlos Bravo-Prieto This is my paper · ORCID
classification quant-phcond-mat.quant-gascond-mat.stat-mech MSC 81P6868T05 PACS 03.67.-a
keywords quantummachinelearningcurriculumhardexampleminingphaserecognitionbarrenplateausdata-centricinductivebiasconvolutionalneuralnetworkself-paced
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

This paper claims that how quantum training data is presented to a learning model is itself an inductive bias that can be tuned. The authors introduce a scoring-and-pacing framework, inspired by classical curriculum learning and hard-example mining, in which each quantum state is assigned a difficulty score and a pacing function controls how much of the dataset is accessible at each epoch. On quantum phase-recognition tasks for two spin-chain Hamiltonians, the framework's self-paced 'Hardest' strategy—training every epoch on the ten highest-loss states—reaches test accuracies of 92.6% and 93.5%, against 77.4% for standard random-order training on both models. The paper also gives two analytical results suggesting that score-ordered training can mitigate barren plateaus and, under convexity, improve convergence. If these results hold, data ordering becomes a third, largely unexploited design axis for quantum machine learning, alongside circuit architecture and parameter initialization.

What carries the argument

The load-bearing mechanism is the scoring-and-pacing training loop. A scoring function $s : \mathcal{H} \to [0,1]$ assigns each training state a difficulty value; a pacing function $p(t)$ sets the fraction of lowest-score data accessible in epoch $t$; mini-batches are then drawn from that restricted subset, so that simpler or harder examples are presented first depending on the ordering. The paper instantiates three families of scores: self-taught (loss of a pre-trained QCNN), self-paced (current loss of the training model, re-ranked every epoch), and physics-inspired ($s(\rho) = 1 - P_g(\rho)$, where $P_g$ is the g-purity of the state with respect to the dynamical Lie algebra of the circuit, linking the score to the barren-plateau variance formula). The QCNN on 8 qubits is the numerical testbed for the framework.

What would settle it

Recompute the Table II comparison with a different optimizer (for example, vanilla SGD instead of ADAM) or with fresh random seeds beyond ten, and check whether the Hardest strategy's roughly 15-point test-accuracy gap over Standard persists; a collapse of the gap would falsify the practical claim. A more direct test of the theory: estimate the gradient-estimator bias between mini-batches drawn from $D_t$ and the full training-set gradient on the actual 50-state training set; if it is significantly nonzero, Proposition 2's assumed unbiasedness, and hence its convergence conclusion, is falsified.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that a QML training loop can be made to improve by controlling the order in which training states are revealed, rather than by changing the circuit or the optimizer. Formally, the authors define a scoring function $s(|\psi\rangle)$ and a monotone pacing function $p(t)$; at epoch $t$ the learner samples mini-batches only from the score-restricted subset $\{s \leq z_t\}$. The strongest empirical claim is that the self-paced Hardest variant, which keeps the accessible set fixed to the ten states with the highest current loss, achieves 92.6% (generalized cluster) and 93.5% (bond-alternating XXZ) test accuracy versus 77.4% for standard training in both models, and also captures phase boundaries that the standard model misses. Theoretically, the paper claims that if the expected squared gradient norm is non-increasing in score, then prioritized presentation increases the expected gradient magnitude in early epochs (mitigating barren plateaus), and that under convexity and Lipschitz smoothness, score-prioritized sampling with increasing pacing yields expected empirical risk no larger than random-order training after $T$ epochs. The paper reports that the physics-inspired g-purity ordering is computationally hard to scale and does not consistently beat standard training on the XXZ task, while the Hardest strategy's predictions are less confident.

Load-bearing premise

The convergence guarantee in Proposition 2 assumes that mini-batches sampled from the score-restricted subset $D_t = \{s(\psi) \le z_t\}$ are unbiased gradient estimators of the full empirical risk, even though that subset was selected by score; if conditioning on scores biases the gradient, the claimed bound $E[R(\vartheta_T^{s,p})] \le E[R(\vartheta_T^{rand})]$ can fail, and the empirical results must carry the argument alone.

Editorial extensions

If this is right

  • Data ordering becomes a third, tunable inductive-bias axis for QML experiments, on top of circuit design and parameter initialization, and it can be combined with warm starts.
  • For quantum phase recognition, self-paced hard-example mining is a strong baseline: it beats standard random-order training by roughly 15 percentage points on both spin-chain benchmarks studied.
  • Score-based presentation can be justified as a barren-plateau mitigation when a score exists for which the expected squared gradient norm decreases with score, as with the g-purity ordering.
  • The framework transfers directly to classical-data QML pipelines such as data re-uploading, since the scoring function can operate on the original data or its embedding.

Reading between the lines

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

  • Editorial inference: the Hardest strategy's success despite its low exposure to the full dataset suggests it doubles as a coreset-selection method; one testable extension is to keep the hard-core set fixed and check whether the 50-sample training set can be pruned to fewer than ten states without losing the 92% test accuracy.
  • Editorial inference: because the paper reports that Hardest predictions are less confident (smaller probability gaps), the accuracy gain may come at the cost of calibration; a natural follow-up is to apply temperature scaling or label smoothing to the output probabilities and test whether confidence can be restored without sacrificing accuracy.
  • Editorial inference: the inconsistent behavior of the g-purity ordering across the two Hamiltonians indicates that the sufficient condition in Proposition 1 is not a constructive recipe; a hybrid score that combines g-purity with distance to phase boundaries could reproduce the Hardest strategy's gains more cheaply than per-epoch re-ranking.
  • Editorial inference: the proof of Proposition 2 rests on the assumption that mini-batches from the score-restricted subset $D_t$ are unbiased gradient estimates of the full empirical risk; since conditioning on scores generally biases the estimator, the convergence guarantee is the paper's most fragile component and the empirical comparison is the load-bearing evidence.
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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

2 major / 4 minor

Summary. The manuscript proposes a data-ordering and pacing framework for quantum machine learning: each training state is assigned a score, a pacing function controls the fraction of the dataset accessible at each epoch, and mini-batches are drawn from the accessible subset. The authors state two analytical claims: Proposition 1, that restricting early training to low-score points increases the expected squared gradient magnitude and mitigates barren-plateau effects, and Proposition 2, that score-restricted sampling with a monotonically increasing pacing function yields lower expected empirical risk after T epochs than random ordering. The numerical core is a QCNN phase-recognition study on the generalized cluster and bond-alternating XXZ spin chains, using an 8-qubit circuit, 50 training states, and 10 runs. The central empirical finding is that self-paced hard-example strategies, particularly the 'Hardest' strategy of always training on the ten highest-loss states, improve test accuracy from about 77.4% to 92.6-93.5% across both models (Table II). A physics-inspired ordering based on g-purity gives model-dependent results. Code and data are cited as available.

Significance. If the empirical result is robust, the paper makes a useful contribution by showing that data-centric ordering is a new inductive-bias axis for QML, complementary to architectural design and warm-start initialization. The numerical setup is transparent and reproducible: fixed 8-qubit QCNN, 50 training points, 10 runs, standard-error bands in the learning curves, and a code repository reference. The phase-boundary analysis in Fig. 6 is a valuable diagnostic that goes beyond aggregate accuracy. The main weakness is the theoretical component: Proposition 2's proof has a genuine bias gap, and Proposition 1 is a conditional statement whose premise is essentially assumed. The durable value of the paper is therefore the empirical phenomenon and the framework's practicality, not the stated convergence guarantee.

major comments (2)
  1. [Appendix B, Eqs. (B2)-(B7)] The proof of Proposition 2 does not establish Eq. (2). The SGD bound in Eq. (B5) is the standard guarantee for an unbiased gradient estimator, E[g_t | theta_t] = ∇R(theta_t), where the variance term is E||g_t - ∇R(theta_t)||^2. Under score-restricted sampling from D_t, however, E[g_t | theta_t] = ∇R_{D_t}(theta_t), not ∇R(theta_t). Therefore the quantity sigma^2_{s,p}(t) in Eq. (B3) is not the variance appearing in the standard bound; it equals the variance of the restricted estimator plus the squared norm of the bias ||∇R_{D_t}(theta_t) - ∇R(theta_t)||^2. Substituting Eq. (B3) into Eq. (B5) and comparing with sigma^2_rand(t) silently omits this bias. A valid comparison of E[R(theta_T)] would require a bound on the accumulated bias terms, which is not supplied. This is a load-bearing gap because the abstract advertises theoretical insights into faster convergence.
  2. [Section II.B.2 and Proposition 2] The numerical 'Hard' and 'Hardest' strategies rank data by the current loss ℓ(theta; (|psi>, y)), but Proposition 2 requires a scoring function that orders data by gradient variance. The paper does not show that loss ordering and gradient-variance ordering coincide for the MSE loss in Eq. (6). For a quadratic loss, high-loss points often have larger gradient norms and can therefore have larger, not smaller, stochastic-gradient variance. Consequently, the experiments do not demonstrably instantiate the assumptions of the convergence proposition, and the theory as stated does not explain the Hardest results. The empirical finding remains interesting, but the claimed link between the theory and the experiments needs either an additional argument or an explicit weakening of the claim.
minor comments (4)
  1. [Definition 1 and Section II.B (after Eq. (7))] The scoring function is defined in Definition 1 as s : X -> [0,1] and in the propositions as s : H -> [0,1], but the experimental section defines s : S -> R, where S is the training set. Please align the domain and codomain across the definitions and the experiments.
  2. [Section I.B] The sentence 'slower-growing functions texted training on high-scoring examples' appears to contain a typo; the intended meaning is likely 'delay training' or 'postpone exposure to high-scoring examples'.
  3. [Table III] Table III reports only mean best accuracies without any measure of spread, whereas Figs. 3-5 include standard-error bands. Given the cross-model reversal in the Pg-based strategies, reporting standard errors would help the reader judge whether the reversal is meaningful.
  4. [Section II.B.2, Hardest strategy] The 'Hardest' strategy uses a constant pacing function and always trains on the ten highest-loss examples; it is therefore not a gradual curriculum. The paper should explicitly state that Hardest is a hard-example-mining variant rather than an instance of gradual pacing, to avoid conflating two distinct mechanisms.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the empirical results are self-contained, and the theoretical propositions are conditional formalizations rather than fitted predictions.

full rationale

The paper's central numerical claims (Tables I-III) compare hard-example/curriculum training against standard training on the same test distributions; no fitted parameter is renamed as a prediction, and the reported test accuracies are not the scoring losses used to order the data. Proposition 1 and Proposition 2 are explicitly conditional: they assume a scoring function that orders points by gradient magnitude or gradient variance and then derive the corresponding effect on early-epoch gradient size or SGD risk. This is a mathematical formalization, not a circular derivation, because the scoring function is not defined in terms of the conclusion and the assumption is not the empirical result. The Appendix B proof does contain a genuine bias problem (mini-batches from D_t give a biased estimator of ∇R), but that is a correctness gap, not circularity. Self-citations are limited to the authors' code/data repository [57] and background QML literature [48]; neither is load-bearing, so the paper is not relying on an unverified self-citation chain.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The theoretical results are conditional on scoring functions with specified gradient properties and on the barren plateau variance formula; the numerical results depend on hand-chosen hyperparameters. No new particles, forces, or entities are introduced.

free parameters (3)
  • Hardest subset size (mini-batch size L=10) = 10
    The self-paced Hardest strategy trains on the ten most difficult examples at each epoch; this size equals the fixed mini-batch size and is hand-chosen. The observed benefit may depend on this size.
  • Number of training epochs T = 600
    All strategies are trained for 600 epochs; comparisons may change with a different training budget.
  • Pacing function details = not specified in paper (deferred to Ref. [57])
    The exact pacing functions used in the gradual strategies are not described in the text, so their shape is an unstated modeling choice that could affect the comparison.
assumptions (4)
  • domain assumption The QCNN is assumed to approximate a unitary 2-design over the unitary group generated by the dynamical Lie algebra (DLA) g, so the variance of the loss is given by Eq. (9).
    Invoked in Sec II B 3 to justify the g-purity scoring function; this is a standard but non-trivial assumption about the ensemble formed by the parameterized circuit.
  • domain assumption Barren plateau variance formula Var(loss) = Pg(rho)Pg(O)/dim(g) from Ref. [5] is taken as given, and the g-purity of the training state is assumed to correlate with gradient magnitude (not just loss variance).
    The scoring function s(rho)=1-Pg(rho) in Eq. (11) is justified by this correlation, which the paper itself notes is an asymptotic intuition not guaranteed for finite sizes.
  • domain assumption Convexity of the loss and Lipschitz continuous gradients are assumed for Proposition 2.
    Acknowledged by the authors as 'not always realistic'; the proof relies on the standard convex SGD bound from Ref. [69].
  • ad hoc to paper The gradient estimator over D_t is unbiased for the full empirical risk in the SGD analysis.
    This unstated assumption underlies the convergence bound in Appendix B; it is false because D_t is a score-restricted subset, so this axiom is introduced by the proof itself.

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Cite this review

Pith. "Pith review of Learning complexity gradually in quantum machine learning models." pith.science (2026). https://pith.science/paper/4ERL5CVP

@misc{pith2026241111954,
  author       = {Pith},
  title        = {Pith review of: Learning complexity gradually in quantum machine learning models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ERL5CVP}},
  note         = {Machine review of arXiv:2411.11954}
}
read the original abstract

Quantum machine learning is an emergent field that continues to draw significant interest for its potential to offer improvements over classical algorithms in certain areas. However, training quantum models remains a challenging task, largely because of the difficulty in establishing an effective inductive bias when solving high-dimensional problems. In this work, we propose a training framework that prioritizes informative data points over the entire training set. This approach draws inspiration from classical techniques such as curriculum learning and hard example mining to introduce an additional inductive bias through the training data itself. By selectively focusing on informative samples, we aim to steer the optimization process toward more favorable regions of the parameter space. This data-centric approach complements existing strategies such as warm-start initialization methods, providing an additional pathway to address performance challenges in quantum machine learning. We provide theoretical insights into the benefits of prioritizing informative data for quantum models, and we validate our methodology with numerical experiments on selected recognition tasks of quantum phases of matter. Our findings indicate that this strategy could be a valuable approach for improving the performance of quantum machine learning models.

Figures

Figures reproduced from arXiv: 2411.11954 by the authors.

Figure 1
Figure 1. The training framework begins by scoring quantum data [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Ground-state phase diagrams of (a) the generalized clus￾ter Hamiltonian in Eq. (3), exhibiting (I) symmetry-protected topo￾logical, (II) ferromagnetic, (III) anti-ferromagnetic, and (IV) trivial phases; and (b) the bond-alternating XXZ Hamiltonian in Eq. (4), displaying (I) trivial, (II) symmetry-broken antiferromagnetic, and (III) topological phases. applying single-qubit unitaries based on the measurement out￾come… view at source ↗
Figure 3
Figure 3. Average accuracy on the test set as a function of the training [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Average accuracy on the test set as a function of the training [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Average accuracy on the test set as a function of the train [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: (a) The generalized cluster Hamiltonian and (b) bond-alternating XXZ Hamiltonian for Standard (top) and Hardest (bottom) strategies. The background color of each panel indicates the true quantum phase of the system. A correct classification occurs when the highest prob…

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Works this paper leans on

73 extracted references · 55 canonical work pages

  1. [57]

    Verresen, R

    R. Verresen, R. Moessner, and F. Pollmann, One-dimensional symmetry protected topological phases and their transitions, Phys. Rev. B 96, 165124 (2017)

    work page 2017

  2. [1]

    This is usually referred to as self-taught learning

    Self-taught learning In this first experiment, we order the quantum data based on the loss of a pre-trained QCNN model with identical archi- tecture and trained without ordering. This is usually referred to as self-taught learning. At the beginning of the training, we order the training set according to the loss of each data point. We then gradually incre...

    work page

  3. [2]

    In this approach, we rank the training set by the current loss of each data point at each epoch

    Self-paced learning Self-paced learning dynamically prioritizes training data based on the performance of the current model, unlike the self-taught strategy which scores each point based on the loss with respect to the target hypothesis. In this approach, we rank the training set by the current loss of each data point at each epoch. We explore four strate...

    work page

  4. [3]

    At its core, this is a concentration of measure effect in high dimen- sions and can be captured in several different ways

    Physics-inspired learning A key difference between classical and quantum machine learning is, once again, the presence of barren plateaus, where the variation of the loss function over the parameter landscape is exponentially suppressed with the system size [6]. At its core, this is a concentration of measure effect in high dimen- sions and can be capture...

    work page

  5. [4]

    Vicente L´opez

    Accuracy at the cost of confidence In this section, we try to understand further the differ- ences between the Standard approach and the best-performing method within the proposed framework: the self-paced Hard- est learning strategy. We analyze the probabilities for the dif- ferent phases as determined by the QCNN along a specific cut in the phase diagra...

    work page

  6. [5]

    Sweke, J.-P

    R. Sweke, J.-P. Seifert, D. Hangleiter, and J. Eisert, On the quantum versus classical learnability of discrete distributions, Quantum 5, 417 (2021)

    work page 2021

  7. [6]

    Y . Liu, S. Arunachalam, and K. Temme, A rigorous and ro- bust quantum speed-up in supervised machine learning, Nature Phys. 17, 1013 (2021)

    work page 2021

  8. [7]

    Huang, R

    H.-Y . Huang, R. Kueng, and J. Preskill, Information-theoretic bounds on quantum advantage in machine learning, Phys. Rev. Lett. 126, 190505 (2021)

    work page 2021

Show all 73 references
  1. [8]

    Pirnay, R

    N. Pirnay, R. Sweke, J. Eisert, and J.-P. Seifert, Superpolyno- mial quantum-classical separation for density modeling, Phys. Rev. A 107, 042416 (2023)

    2023

  2. [9]

    Ragone, B

    M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Or- tiz Marrero, M. Larocca, and M. Cerezo, A Lie algebraic the- ory of barren plateaus for deep parameterized quantum circuits, Nature Comm. 15, 7172 (2024)

    2024

  3. [10]

    J. R. McClean, S. Boixo, V . N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Comm. 9, 4812 (2018)

    2018

  4. [11]

    Larocca, S

    M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Bia- monte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, A review of barren plateaus in variational quantum computing, arXiv preprint arXiv:2405.00781 (2024)

    2024 arXiv

  5. [12]

    E. R. Anschuetz and B. T. Kiani, Quantum variational algo- rithms are swamped with traps, Nature Comm.13, 7760 (2022)

    2022

  6. [13]

    A. A. Mele, A. Angrisani, S. Ghosh, S. Khatri, J. Eisert, D. S. Franc ¸a, and Y . Quek, Noise-induced shallow circuits and absence of barren plateaus, arXiv preprint arXiv:2403.13927 (2024)

    2024 arXiv

  7. [14]

    N. A. Nemkov, E. O. Kiktenko, and A. K. Fedorov, Barren plateaus are swamped with traps, arXiv preprint arXiv:2405.05332 (2024)

    2024 arXiv

  8. [15]

    Thanasilp, S

    S. Thanasilp, S. Wang, N. A. Nghiem, P. Coles, and M. Cerezo, Subtleties in the trainability of quantum machine learning mod- els, Quant. Mach. Int. 5, 21 (2023)

    2023

  9. [16]

    K ¨ubler, S

    J. K ¨ubler, S. Buchholz, and B. Sch¨olkopf, The inductive bias of quantum kernels, Advances in Neural Information Processing Systems 34, 12661 (2021)

    2021

  10. [17]

    Cerezo, G

    M. Cerezo, G. Verdon, H.-Y . Huang, L. Cincio, and P. J. Coles, Challenges and opportunities in quantum machine learning, Na- ture Comp. Sc. 2, 567 (2022)

    2022

  11. [18]

    Bowles, V

    J. Bowles, V . J. Wright, M. Farkas, N. Killoran, and M. Schuld, Contextuality and inductive bias in quantum machine learning, arXiv preprint arXiv:2302.01365 (2023)

    2023 arXiv

  12. [19]

    Larocca, F

    M. Larocca, F. Sauvage, F. M. Sbahi, G. Verdon, P. J. Coles, and M. Cerezo, Group-invariant quantum machine learning, PRX Quantum 3, 030341 (2022)

    2022

  13. [20]

    J. J. Meyer, M. Mularski, E. Gil-Fuster, A. A. Mele, F. Arzani, A. Wilms, and J. Eisert, Exploiting symmetry in variational quantum machine learning, PRX Quantum 4, 010328 (2023)

    2023

  14. [21]

    Bowles, D

    J. Bowles, D. Wierichs, and C.-Y . Park, Backpropagation scaling in parameterised quantum circuits, arXiv preprint arXiv:2306.14962 (2023)

    2023

  15. [22]

    Park and N

    C.-Y . Park and N. Killoran, Hamiltonian variational ansatz without barren plateaus, Quantum 8, 1239 (2024). 10

    2024

  16. [23]

    Jerbi, C

    S. Jerbi, C. Gyurik, S. C. Marshall, R. Molteni, and V . Dun- jko, Shadows of quantum machine learning, Nature Comm. 15, 5676 (2024)

    2024

  17. [24]

    Schatzki, M

    L. Schatzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, Theoretical guarantees for permutation-equivariant quantum neural networks, npj Quant. Inf. 10, 12 (2024)

    2024

  18. [25]

    Rodriguez-Grasa, Y

    P. Rodriguez-Grasa, Y . Ban, and M. Sanz, Training embedding quantum kernels with data re-uploading quantum neural net- works, arXiv preprint arXiv:2401.04642 (2024)

    2024 arXiv

  19. [26]

    Zhang, L

    K. Zhang, L. Liu, M.-H. Hsieh, and D. Tao, Escaping from the barren plateau via Gaussian initializations in deep variational quantum circuits, Advances in Neural Information Processing Systems 35, 18612 (2022)

    2022

  20. [27]

    H. R. Grimsley, G. S. Barron, E. Barnes, S. E. Economou, and N. J. Mayhall, Adaptive, problem-tailored variational quantum eigensolver mitigates rough parameter landscapes and barren plateaus, npj Quant. Inf. 9, 19 (2023)

    2023

  21. [28]

    Dborin, F

    J. Dborin, F. Barratt, V . Wimalaweera, L. Wright, and A. G. Green, Matrix product state pre-training for quantum machine learning, Quant. Sc. Tech. 7, 035014 (2022)

    2022

  22. [29]

    M. S. Rudolph, J. Miller, D. Motlagh, J. Chen, A. Acharya, and A. Perdomo-Ortiz, Synergistic pretraining of parametrized quantum circuits via tensor networks, Nature Comm. 14, 8367 (2023)

    2023

  23. [30]

    A. A. Mele, G. B. Mbeng, G. E. Santoro, M. Collura, and P. Torta, Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz, Phys. Rev. A106, L060401 (2022)

    2022

  24. [31]

    R. Puig, M. Drudis, S. Thanasilp, and Z. Holmes, Variational quantum simulation: a case study for understanding warm starts, arXiv preprint arXiv:2404.10044 (2024)

    2024 arXiv

  25. [32]

    Y . Wang, B. Qi, C. Ferrie, and D. Dong, Trainability enhance- ment of parameterized quantum circuits via reduced-domain parameter initialization, arXiv preprint arXiv:2302.06858 (2023)

    2023 arXiv

  26. [33]

    C.-Y . Park, M. Kang, and J. Huh, Hardware-efficient ansatz without barren plateaus in any depth, arXiv preprint arXiv:2403.04844 (2024)

    2024 arXiv

  27. [34]

    Shi and Y

    X. Shi and Y . Shang, Avoiding barren plateaus via gaussian mixture model, arXiv preprint arXiv:2402.13501 (2024)

    2024 arXiv

  28. [35]

    Bengio, J

    Y . Bengio, J. Louradour, R. Collobert, and J. Weston, Curricu- lum learning, in Proceedings of the 26th annual international conference on machine learning (2009) pp. 41–48

    2009

  29. [36]

    Q. H. Tran, Y . Endo, and H. Oshima, Quantum curriculum learning, arXiv preprint arXiv:2407.02419 (2024)

    2024

  30. [37]

    Shrivastava, A

    A. Shrivastava, A. Gupta, and R. Girshick, Training region- based object detectors with online hard example mining, inPro- ceedings of the IEEE conference on computer vision and pat- tern recognition (2016) pp. 761–769

    2016

  31. [38]

    X. Wang, Y . Chen, and W. Zhu, A survey on curriculum learn- ing, IEEE Trans. Patt. Anal. Mach. Int. 44, 4555 (2021)

    2021

  32. [39]

    Soviany, R

    P. Soviany, R. T. Ionescu, P. Rota, and N. Sebe, Curriculum learning: A survey, Int. J. Comp. Vis. 130, 1526 (2022)

    2022

  33. [40]

    Tudor Ionescu, B

    R. Tudor Ionescu, B. Alexe, M. Leordeanu, M. Popescu, D. P. Papadopoulos, and V . Ferrari, How hard can it be? Estimating the difficulty of visual search in an image, inProceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2016) pp. 2157–2166

    2016

  34. [41]

    S. Guo, W. Huang, H. Zhang, C. Zhuang, D. Dong, M. R. Scott, and D. Huang, Curriculumnet: Weakly supervised learn- ing from large-scale web images, in Proceedings of the Euro- pean conference on computer vision (ECCV) (2018) pp. 135– 150

    2018

  35. [42]

    Jiang, D

    L. Jiang, D. Meng, T. Mitamura, and A. G. Hauptmann, Easy samples first: Self-paced reranking for zero-example multime- dia search, in Proceedings of the 22nd ACM international con- ference on Multimedia (2014) pp. 547–556

    2014

  36. [43]

    E. A. Platanios, O. Stretcu, G. Neubig, B. Poczos, and T. Mitchell, Competence-based curriculum learning for neural machine translation, in Proceedings of the 2019 Conference of the North American Chapter of the Association for Computa- tional Linguistics: Human Language Techn...

    2019

  37. [44]

    Matiisen, A

    T. Matiisen, A. Oliver, T. Cohen, and J. Schulman, Teacher– student curriculum learning, IEEE Trans. Neur. Net. learn. sys. 31, 3732 (2019)

    2019

  38. [45]

    Zhang, G

    X. Zhang, G. Kumar, H. Khayrallah, K. Murray, J. Gwinnup, M. J. Martindale, P. McNamee, K. Duh, and M. Carpuat, An empirical exploration of curriculum learning for neural machine translation, arXiv preprint arXiv:1811.00739 (2018)

    2018 arXiv

  39. [46]

    P ´erez-Salinas, A

    A. P ´erez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Data re-uploading for a universal quantum classifier, Quantum 4, 226 (2020)

    2020

  40. [47]

    Schuld, R

    M. Schuld, R. Sweke, and J. J. Meyer, Effect of data encod- ing on the expressive power of variational quantum-machine- learning models, Phys. Rev. A 103, 032430 (2021)

    2021

  41. [48]

    Penha and C

    G. Penha and C. Hauff, Curriculum learning strategies for IR: An empirical study on conversation response ranking, in Ad- vances in Information Retrieval (2020) pp. 699–713

    2020

  42. [49]

    M. C. Caro, H.-Y . Huang, M. Cerezo, K. Sharma, A. Sorn- borger, L. Cincio, and P. J. Coles, Generalization in quantum machine learning from few training data, Nature Comm. 13, 4919 (2022)

    2022

  43. [50]

    Y . Wu, B. Wu, J. Wang, and X. Yuan, Quantum phase recogni- tion via quantum kernel methods, Quantum 7, 981 (2023)

    2023

  44. [51]

    Y .-J. Liu, A. Smith, M. Knap, and F. Pollmann, Model- independent learning of quantum phases of matter with quan- tum convolutional neural networks, Phys. Rev. Lett. 130, 220603 (2023)

    2023

  45. [52]

    Gil-Fuster, J

    E. Gil-Fuster, J. Eisert, and C. Bravo-Prieto, Understanding quantum machine learning also requires rethinking generaliza- tion, Nature Comm. 15, 2277 (2024)

    2024

  46. [53]

    Umeano, A

    C. Umeano, A. E. Paine, V . E. Elfving, and O. Kyriienko, What can we learn from quantum convolutional neural networks?, arXiv preprint arXiv:2308.16664 (2023)

    2023 arXiv

  47. [54]

    Zapletal, N

    P. Zapletal, N. A. McMahon, and M. J. Hartmann, Error- tolerant quantum convolutional neural networks for symmetry- protected topological phases, Phys. Rev. Res.6, 033111 (2024)

    2024

  48. [55]

    Carrasquilla and R

    J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nature Phys. 13, 431–434 (2017)

    2017

  49. [56]

    Sachdev, Quantum phases of matter (Cambridge University Press, Massachusetts, 2023)

    S. Sachdev, Quantum phases of matter (Cambridge University Press, Massachusetts, 2023)

    2023

  50. [58]

    Schuch, D

    N. Schuch, D. P ´erez-Garc´ıa, and I. Cirac, Classifying quantum phases using matrix product states and projected entangled pair states, Phys. Rev. B 84, 165139 (2011)

    2011

  51. [59]

    Elben, J

    A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, and B. Vermersch, Many-body topological invariants from random- ized measurements in synthetic quantum matter, Science ad- vances 6, eaaz3666 (2020)

    2020

  52. [60]

    I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neu- ral networks, Nature Phys. 15, 1273 (2019)

    2019

  53. [61]

    Recio-Armengol and C

    E. Recio-Armengol and C. Bravo-Prieto, Learning complexity gradually in QML models repository. 11

  54. [62]

    Bergholm, J

    V . Bergholm, J. Izaac, M. Schuld, C. Gogolin, S. Ahmed, V . Ajith, M. S. Alam, G. Alonso-Linaje, B. AkashNarayanan, A. Asadi, et al. , Pennylane: Automatic differentiation of hybrid quantum-classical computations, arXiv preprint arXiv:1811.04968 (2018)

    2018 arXiv

  55. [63]

    D. P. Kingma, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014)

    2014 arXiv

  56. [64]

    Aguilar, S

    G. Aguilar, S. Cichy, J. Eisert, and L. Bittel, Full classification of Pauli Lie algebras, arXiv preprint arXiv:2408.00081 (2024)

    2024 arXiv

  57. [65]

    Gross, K

    D. Gross, K. M. R. Audenaert, and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys. 48, 052104 (2007)

    2007

  58. [66]

    M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for variational quantum computing, arXiv preprint arXiv:2308.01432 (2023)

    2023

  59. [67]

    Larocca, P

    M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo, Diagnosing barren plateaus with tools from quantum optimal control, Quantum 6, 824 (2022)

    2022

  60. [68]

    Cincio, Y

    L. Cincio, Y . Subas ¸ı, A. T. Sornborger, and P. J. Coles, Learn- ing the quantum algorithm for state overlap, New Journal of Physics 20, 113022 (2018)

    2018

  61. [69]

    K. Beer, D. Bondarenko, T. Farrelly, T. J. Osborne, R. Salz- mann, D. Scheiermann, and R. Wolf, Training deep quantum neural networks, Nature Comm. 11, 808 (2020)

    2020

  62. [70]

    Cincio, K

    L. Cincio, K. Rudinger, M. Sarovar, and P. J. Coles, Machine learning of noise-resilient quantum circuits, PRX Quantum 2, 010324 (2021)

    2021

  63. [71]

    Z. Yu, X. Zhao, B. Zhao, and X. Wang, Optimal quantum dataset for learning a unitary transformation, Phys. Rev. Ap- plied 19, 034017 (2023)

    2023

  64. [72]

    D. F. Locher, L. Cardarelli, and M. M ¨uller, Quantum error cor- rection with quantum autoencoders, Quantum 7, 942 (2023)

    2023

  65. [73]

    Shalev-Shwartz and S

    S. Shalev-Shwartz and S. Ben-David, Understanding machine learning: From theory to algorithms (Cambridge university press, 2014). Appendix A: Proof of Proposition 1 In this section, we re-state and prove Proposition 1. Proposition 1 (Mitigation of barren plateaus) . Let {|ψi⟩ ...

    2014

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