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REVIEW 3 major objections 5 minor 40 references

Classical versus Quantum Models in Machine Learning: Insights from a Finance Application

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Under equal parameter counts, quantum circuit Born machines beat canonical restricted Boltzmann machines on real financial portfolio benchmarks.

desk verdict A useful new finance-inspired benchmark, but the QCBM-vs-RBM expressivity claim is undercut by an optimizer/information confound; worth reviewing with major revisions. read the letter →

arxiv 1908.10778 v2 pith:NENGTRBY submitted 2019-08-28 quant-ph

Javier Alcazar , Vicente Leyton-Ortega , Alejandro Perdomo-Ortiz This is my paper · ORCID
classification quant-ph
keywords quantumcircuitBornmachinesrestrictedBoltzmanngenerativemodelingportfoliooptimizationKullback-Leiblerdivergencenear-termcomputingunsupervisedlearningapplication-inspiredbenchmarks
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 sets out to compare a quantum generative model, the quantum circuit Born machine (QCBM), against a classical workhorse, the restricted Boltzmann machine (RBM), on benchmarks built from real stock-market data rather than synthetic distributions. The authors construct target distributions from a probabilistic version of mean-variance portfolio optimization with cardinality constraints, then train both models with the same number of parameters. Under canonical RBM training, they report that the QCBM reaches substantially lower distribution-fitting error (KL divergence) on typical instances for system sizes $N=4,6,8,10$, and wins nearly every case-by-case comparison. If correct, this is evidence that quantum generative models can be at least as expressive as classical ones on real-world data with equal resources, using circuit layouts that fit near-term ion-trap hardware. The smallest case, $N=4$, narrows when the classical model is given longer Gibbs sampling chains.

What carries the argument

The load-bearing device is an equal-parameter benchmark built from real financial data. A two-layer hardware-efficient circuit -- $X$ and $Z$ single-qubit rotations followed by a fully connected layer of parametrized two-qubit $XX_{ij}(\chi)$ entangling gates -- defines the QCBM's Born-rule distribution, and an RBM with $M=N/2$ hidden units is sized so both models share exactly $N(N+3)/2$ parameters. The target distributions come from a Boltzmann law over efficient-frontier risks: for each of $\binom{N}{\kappa}$ cardinality-constrained portfolios with $\kappa=N/2$, the risk $\sigma^\kappa_{\rho,i}$ at return level $\rho$ is weighted by $p(\sigma^\kappa_{\rho,i})\propto e^{-\sigma^\kappa_{\rho,i}/T}$, with market temperature $T$. Drawing samples from this distribution yields the benchmark that both models must learn, scored by clipped KL divergence. The pairing of a global optimizer (CMA-ES) acting on exact wavefunction probabilities for the quantum model with a local stochastic-gradient routine (PCD-1) for the classical model is the mechanism that produces the reported gap.

What would settle it

Train the RBM with the same CMA-ES optimizer on exact likelihood values, or with persistent contrastive divergence using 10 to 100 Gibbs steps, and check whether its KL divergence drops to the QCBM's level for $N=6,8,10$. The paper's own appendix shows that for $N=4$ the longer chains already close much of the gap, so the decisive observation is whether the same happens at larger system sizes.

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Extended reading notes

Core claim

The paper's central claim is that, at equal parameter counts, QCBMs represent the portfolio-derived target distributions more accurately than RBMs trained in the standard way. Both models are allotted $N(N+3)/2$ trainable parameters: the quantum ansatz uses $2N$ single-qubit rotations and $N(N-1)/2$ fully connected two-qubit $XX$ entangling gates, while the classical model uses $N$ visible units, $N/2$ hidden units, and the associated biases and weights. Training the QCBM with CMA-ES on noiseless statevector Born probabilities and the RBM with persistent contrastive divergence with one Gibbs step, the paper reports bootstrapped median KL divergences consistently lower for the QCBM at $N=6,8,10$, with most scatter points falling below the equality line. The RBM's performance drifts toward the uniform-distribution baseline as $N$ grows, whereas the QCBM stays far from that baseline. The authors read this as an expressivity or trainability advantage of the quantum model by the standard of canonical RBM training, noting that $N=4$ is the case where longer classical Gibbs chains close much of the gap.

Load-bearing premise

The load-bearing premise is that equal parameter counts make the comparison fair even though the quantum model is trained with a global optimizer on exact probabilities and the classical model with a short local sampling routine; if the optimizer difference, rather than the model family, drives the lower error, the central claim collapses.

Editorial extensions

If this is right

  • If the reported gap is genuine, near-term quantum devices with a few tens of qubits could run generative models that match or beat canonical RBMs on real-world benchmark distributions of comparable size.
  • The benchmark construction itself -- Boltzmann distributions over mean-variance efficient frontiers -- is a reusable, application-inspired testbed for comparing other generative models, including variational autoencoders, GANs, and tensor-network Born machines, once parameter counts can be matched.
  • The results indicate that the RBM's weakness on these instances is not simply a shortage of parameters, since it tends toward the uniform baseline even with the same parameter count as the QCBM; the difficulty lies in training or in the expressive limits of the quadratic energy model.
  • For quantitative finance, the pipeline connects classical portfolio optimization to quantum generative modeling, though the paper is explicit that this is an application-inspired benchmark rather than a practical portfolio solver.

Reading between the lines

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

  • Editorial inference: the comparison does not isolate model family from training procedure; the QCBM uses a global optimizer on exact probabilities while the RBM uses a short local sampling routine, and the paper's appendix shows this matters at $N=4$, where longer classical Gibbs chains close most of the gap.
  • Editorial inference: the target distributions are sparse and peaked, concentrated on $\kappa$-of-$N$ bitstrings; a natural test is to run both models on dense or high-entropy targets to see whether the quantum advantage is specific to sparse, peaked distributions.
  • Editorial inference: equal parameter count is only one figure of merit; comparing sample complexity, wall-clock training cost, and hardware execution time could change the practical ranking, as the paper itself lists these as open considerations.
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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

3 major / 5 minor

Summary. The paper proposes a benchmark for comparing quantum circuit Born machines (QCBMs) and restricted Boltzmann machines (RBMs) on a finance-inspired generative modeling task. Target distributions are constructed from S&P500 time-series data by solving cardinality-constrained portfolio optimization problems and assigning Boltzmann probabilities to efficient frontiers. Both models are designed with the same number of trainable parameters (N(N+3)/2 for N qubits/visible units). The QCBM is trained with CMA-ES on exact, noiseless Born probabilities obtained from full statevector simulation, while the RBM is trained with persistent contrastive divergence with K_RBM=1 Gibbs step (PCD-1). The main reported result is that QCBMs achieve lower KL divergence to the target distributions than RBMs on most of the 30 benchmark instances per system size, with the gap persisting for N=6, 8, 10. The paper also includes an appendix showing sensitivity of the RBM results to K_RBM=10 and 100.

Significance. If the central comparison were sound, the paper would provide a valuable real-data benchmark showing that QCBMs can outperform canonical RBM training on equal parameter counts. The construction of target distributions from real market data via quadratic programming is a useful contribution, and the authors are commendably explicit about their assumptions, such as the noiseless QVM, the infinite measurement precision, and the choice of a uniform baseline. However, the central claim is not yet supported because the comparison conflates model family with training protocol: QCBMs receive a global optimizer (CMA-ES) and exact distributional information, while RBMs receive a local, stochastic gradient method (PCD-1). The authors' own Appendix A shows that increasing K_RBM reverses the ranking at N=4, demonstrating that the outcome depends on the classical training protocol. The benchmark itself could be valuable, but the expressivity claim requires a matched-optimizer comparison or a carefully justified notion of 'canonical' training.

major comments (3)
  1. [Section III.A and III.B, Fig. 3] The comparison is not apples-to-apples: QCBMs are trained with CMA-ES, a global evolutionary optimizer, on exact, noiseless Born probabilities from full statevector simulation, while RBMs are trained with PCD-1, a stochastic local-gradient method that uses one Gibbs step from a persistent chain and sampled gradients. Because optimizer strength and information access differ jointly with the model family, the observed KL divergence gap cannot be attributed to model expressivity. A controlled experiment is needed, for example training the RBM by exact likelihood evaluation with a global optimizer (e.g., CMA-ES on the negative log-likelihood) or training the QCBM with a comparable local stochastic method, before the paper's central claim about quantum models being 'superior' is justified.
  2. [Appendix A, Fig. 4] The authors' own sensitivity analysis undermines the robustness of the main claim. Increasing K_RBM from 1 to 10/100 reverses the ranking at N=4: the scatter plots in Fig. 4 show most points above the identity line, meaning the RBM outperforms the QCBM on most instances. This directly contradicts the 'close to 100% win' language in Section IV and shows that the comparison's outcome depends materially on the chosen classical training protocol. The paper does not provide a principled justification for why K_RBM=1 is the canonical or fair choice, so the abstract's claim of 'superior performance on typical instances' is not robust.
  3. [Section IV, Fig. 3(a)] The uniform distribution baseline is used to argue that RBM performance approaches the no-learning limit for larger N. However, since the RBM is trained only with PCD-1 and no hyperparameter search or alternative optimizer is investigated at N=6, 8, and 10, the degradation could reflect insufficient optimization rather than limited expressivity. The paper itself acknowledges this ambiguity in the sentence 'this energy model with quadratic interactions and N/2 hidden units does not have enough expressive power or that there are difficulties in training such models,' but does not resolve it. The central claim that QCBMs are more expressive than RBMs cannot be separated from the confound without additional experiments that control for optimizer quality.
minor comments (5)
  1. [Section IV] The list of expected return levels is written as {0.010, 0.015, 0.020, 0.025, 0.30, 0.35}; presumably the last two entries should be 0.030 and 0.035. Please correct this typo.
  2. [Section III.A, Eq. (4)] The clip value epsilon introduced in the KL divergence definition is never specified. Please state the value used in the simulations, since it affects the loss landscape.
  3. [Section II and Fig. 1] The symbol M is used both for the number of asset subsets in Section II (M = (N choose kappa)) and for the number of hidden units in the RBM in Fig. 1 (M = N/2). Please disambiguate the notation.
  4. [Fig. 4 caption] In the caption, panels (b) and (c) are both labeled 'K_RBM=10'; the second should presumably be 'K_RBM=100'. Please correct.
  5. [Section IV] The text 'In Appendix 4' should read 'In Appendix A'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QCBM-vs-RBM comparison is self-contained, with target distributions derived independently from market data and neither model's output used as input to the other.

full rationale

The paper's central comparison is between QCBM and RBM trained on benchmark target distributions constructed from S&P 500 data via quadratic programming and the Boltzmann distribution of Eq. (3). The targets are defined before any model is trained and depend only on market data, the Markowitz risk model, and the chosen temperature T = sqrt(mean covariance); neither the QCBM nor RBM parameters appear in their construction. The equal-parameter-count design (Fig. 1) is an experimental control, not a fitted quantity, and the reported DKL values are obtained by independent training runs. The cited QCBM work [2] is used as background and algorithmic reference, not as evidence that QCBMs outperform RBMs on these benchmarks; no uniqueness theorem or external authority is invoked to rule out alternatives. The paper's own Appendix A reports that RBM performance improves with more Gibbs steps at N=4, and Section IV explicitly offers 'curse of dimensionality' or training difficulty as alternative explanations, which are benchmark-interpretation concerns rather than circular reasoning. Any issue with unequal training protocols (CMA-ES vs PCD-1) is a potential confound affecting the validity of the comparison, but it is not a case of a prediction being equivalent to its inputs by construction. Therefore the circularity score is 0.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper's central claim rests on several benchmark-design choices (temperature, return levels, cardinality) and a fairness assumption about equal parameter counts. No new physical entities are introduced. The target distributions are generated from independent market data, so the circularity burden is low, but the ad hoc temperature and return-level choices are free parameters of the benchmark.

free parameters (6)
  • Market temperature T = T = sqrt(mean of covariance matrix)
    Defines the target Boltzmann distribution in Eq. 3. Chosen by analogy to market volatility, not derived; shapes the difficulty of every benchmark instance.
  • Return levels rho = Six values, listed as {0.010, 0.015, 0.020, 0.025, 0.30, 0.35} (likely 0.030, 0.035)
    Chosen 'given the historical return values' (Section IV). These determine which portfolios are feasible and the exact target distribution.
  • Cardinality constraint kappa = kappa = N/2
    Controls the number of feasible portfolios in the benchmark. Fixed by hand, not by a principle.
  • RBM hidden units M = M = N/2
    Chosen to make the RBM parameter count equal to the QCBM parameter count. This affects RBM expressivity and is load-bearing for the fairness claim.
  • KL divergence clip epsilon = Not specified
    Introduced in Eq. 4 to avoid singularities when Q(x)=0; value is not reported and could affect the loss landscape.
  • CMA-ES and PCD hyperparameters = Not reported
    Population size, initial sigma, learning rate, epochs, batch size, and Gibbs chain length are not given. These can significantly affect optimization quality and final DKL.
assumptions (6)
  • domain assumption Boltzmann mapping from risk to probability (Eq. 3)
    Assumes exponential weighting of portfolio risk with a temperature parameter produces a meaningful target distribution for ML benchmarking. Justified only by analogy to an ideal gas (ref [23]).
  • ad hoc to paper Temperature defined as sqrt(mean covariance)
    No derivation; this specific functional form is introduced for this paper and directly sets the target probabilities.
  • domain assumption Equal parameter count is a fair measure of model comparison
    Central to the claim. The paper assumes that counting adjustable parameters is the right resource metric, but does not justify why other metrics (optimization effort, time) are less relevant.
  • domain assumption Noiseless quantum simulation and infinite measurement precision
    Stated in Section III A. The QCBM results are obtained from exact statevector Born probabilities, which may not survive on real hardware with noise and finite shots.
  • standard math KL divergence is an appropriate loss and evaluation metric
    KL divergence is standard, but the chosen clip epsilon and the sensitivity to zero-probability events can favor models with smooth output distributions.
  • domain assumption Target distributions are accurately represented by sampled data
    The paper uses 'a high enough' number of samples to approximate the target distribution (Section V), but this is an acknowledged choice that avoids finite-sample generalization.

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Cite this review

Pith. "Pith review of Classical versus Quantum Models in Machine Learning: Insights from a Finance Application." pith.science (2026). https://pith.science/paper/NENGTRBY

@misc{pith2026190810778,
  author       = {Pith},
  title        = {Pith review of: Classical versus Quantum Models in Machine Learning: Insights from a Finance Application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NENGTRBY}},
  note         = {Machine review of arXiv:1908.10778}
}
read the original abstract

Although several models have been proposed towards assisting machine learning (ML) tasks with quantum computers, a direct comparison of the expressive power and efficiency of classical versus quantum models for datasets originating from real-world applications is one of the key milestones towards a quantum ready era. Here, we take a first step towards addressing this challenge by performing a comparison of the widely used classical ML models known as restricted Boltzmann machines (RBMs), against a recently proposed quantum model, now known as quantum circuit Born machines (QCBMs). Both models address the same hard tasks in unsupervised generative modeling, with QCBMs exploiting the probabilistic nature of quantum mechanics and a candidate for near-term quantum computers, as experimentally demonstrated in three different quantum hardware architectures to date. To address the question of the performance of the quantum model on real-world classical data sets, we construct scenarios from a probabilistic version out of the well-known portfolio optimization problem in finance, by using time-series pricing data from asset subsets of the S\&P500 stock market index. It is remarkable to find that, under the same number of resources in terms of parameters for both classical and quantum models, the quantum models seem to have superior performance on typical instances when compared with the canonical training of the RBMs. Our simulations are grounded on a hardware efficient realization of the QCBMs on ion-trap quantum computers, by using their native gate sets, and therefore readily implementable in near-term quantum devices.

Figures

Figures reproduced from arXiv: 1908.10778 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Reference graph

Works this paper leans on

40 extracted references · 30 canonical work pages

  1. [1]

    Perdomo-Ortiz, M

    A. Perdomo-Ortiz, M. Benedetti, J. Realpe-G ´omez, and R. Biswas, Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers , Quantum Science and Technology 3, 030502 (2018)

    work page 2018

  2. [2]

    Benedetti, D

    M. Benedetti, D. Garcia-Pintos, O. Perdomo, V . Leyton-Ortega, Y . Nam, and A. Perdomo-Ortiz, A generative modeling ap- proach for benchmarking and training shallow quantum cir- cuits, npj Quantum Information 5, 45 (2019)

    work page 2019

  3. [3]

    Z.-Y . Han, J. Wang, H. Fan, L. Wang, and P. Zhang, Unsuper- vised generative modeling using matrix product states, Physical Review X 8 (2018)

    work page 2018

  4. [4]

    Liu and L

    J.-G. Liu and L. Wang, Differentiable learning of quantum cir- cuit born machines, Phys. Rev. A 98, 062324 (2018)

    work page 2018

  5. [5]

    The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine

    B. Coyle, D. Mills, V . Danos, and E. Kashefi, The born supremacy: Quantum advantage and training of an ising born machine, arXiv:1904.02214v2 (2019)

    work page Pith review arXiv 2019

  6. [6]

    curse of dimensionality

    For those configurations, a risk of effectively infinity is assigned such that probability is zero. This is the reason why Fig. 2(b) shows only four peaks, instead of six. In Sec. IV we provide more details about the construction of the 30 realizations of target distributions per system size, i.e., number of stock assets. With the intention of having statis...

    work page 2017

  7. [7]

    D. Zhu, N. M. Linke, M. Benedetti, K. A. Landsman, N. H. Nguyen, C. H. Alderete, A. Perdomo-Ortiz, N. Korda, A. Gar- foot, C. Brecque, et al.,Training of quantum circuits on a hybrid quantum computer, arXiv preprint arXiv:1812.08862 (2018)

    arXiv 2018

  8. [8]

    K. E. Hamilton, E. F. Dumitrescu, and R. C. Pooser,Generative model benchmarks for superconducting qubits , arXiv preprint arXiv:1811.09905 (2018)

    arXiv 2018

Show all 40 references
  1. [9]

    Leyton-Ortega, A

    V . Leyton-Ortega, A. Perdomo-Ortiz, and O. Perdomo, Ro- bust implementation of generative modeling with parametrized quantum circuits, arXiv preprint arXiv:1901.08047 (2019)

    2019 arXiv

  2. [10]

    Zoufal, A

    C. Zoufal, A. Lucchi, and S. Woerner, Quantum generative ad- versarial networks for learning and loading random distribu- tions, arXiv:1904.00043v2 (2019)

    2019 arXiv

  3. [11]

    M. H. Amin, E. Andriyash, J. Rolfe, B. Kulchytskyy, and R. Melko, Quantum boltzmann machine (2016), arXiv:1601.02036

    2016 arXiv

  4. [12]

    Y . Cao, G. G. Guerreschi, and A. Aspuru-Guzik,Quantum neu- ron: an elementary building block for machine learning on quantum computers (2017), arXiv:1711.11240

    2017 arXiv

  5. [13]

    E. R. Anschuetz and Y . Cao, Realizing quantum boltz- mann machines through eigenstate thermalization (2019), arXiv:1903.01359

    2019 arXiv

  6. [14]

    Rosenberg, P

    G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, K. Wu, and M. L. de Prado,Solving the optimal trading trajectory prob- lem using a quantum annealer, IEEE Journal of Selected Topics in Signal Processing 10, 1053 (2016)

    2016

  7. [15]

    Marzec, Portfolio Optimization: Applications in Quantum Computing (John Wiley & Sons, Ltd, 2016), chap

    M. Marzec, Portfolio Optimization: Applications in Quantum Computing (John Wiley & Sons, Ltd, 2016), chap. 4, pp. 73–106, ISBN 9781118593486, https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118593486.ch4

    2016 doi

  8. [16]

    Rebentrost, B

    P. Rebentrost, B. Gupt, and T. R. Bromley, Quantum compu- tational finance: Monte carlo pricing of financial derivatives , Phys. Rev. A 98, 022321 (2018)

    2018

  9. [17]

    R. Orus, S. Mugel, and E. Lizaso, Forecasting financial crashes with quantum computing (2018)

    2018

  10. [18]

    Woerner and D

    S. Woerner and D. J. Egger, Quantum risk analysis, npj Quan- tum Information 5, 15 (2019)

    2019

  11. [19]

    R. Ors, S. Mugel, and E. Lizaso, Quantum computing for fi- nance: Overview and prospects, Reviews in Physics 4, 100028 (2019), ISSN 2405-4283

    2019

  12. [20]

    Stamatopoulos, D

    N. Stamatopoulos, D. J. Egger, Y . Sun, C. Zoufal, R. Iten, N. Shen, and S. Wrner, Option pricing using quantum comput- ers (2019)

    2019

  13. [21]

    Y . Ding, L. Lamata, M. Sanz, J. D. Martn-Guerrero, E. Lizaso, S. Mugel, X. Chen, R. Ors, and E. Solano, Towards prediction of financial crashes with a d-wave quantum computer (2019), arXiv:1904.05808

    2019 arXiv

  14. [22]

    Martin, B

    A. Martin, B. Candelas, ngel Rodrguez-Rozas, J. D. Martn- Guerrero, X. Chen, L. Lamata, R. Ors, E. Solano, and M. Sanz, Towards pricing financial derivatives with an ibm quantum computer (2019), arXiv:1904.05803

    2019 arXiv

  15. [23]

    Venturelli and A

    D. Venturelli and A. Kondratyev, Reverse quantum annealing approach to portfolio optimization problems , Quantum Ma- chine Intelligence 1, 17 (2019)

    2019

  16. [24]

    P. S. de Mattos Neto, D. A. Silva, T. A. Ferreira, and G. D. Cavalcanti, Market volatility modeling for short time window , Physica A: Statistical Mechanics and its Applications390, 3444 (2011), ISSN 0378-4371

    2011

  17. [25]

    Markowitz, Portfolio selection, The Journal of Finance 7, 77 (1952), ISSN 00221082, 15406261

    H. Markowitz, Portfolio selection, The Journal of Finance 7, 77 (1952), ISSN 00221082, 15406261

    1952

  18. [26]

    Moral-Escudero, R

    R. Moral-Escudero, R. Ruiz-Torrubiano, and A. Suarez, in2006 IEEE International Conference on Evolutionary Computation (2006), pp. 2382–2388, ISSN 1089-778X

    2006

  19. [27]

    Kresta, Solving cardinality constrained portfolio optimiza- tion problem by binary particle swarm optimization algorithm, Acta academica karviniensia 11, 24 (2011)

    A. Kresta, Solving cardinality constrained portfolio optimiza- tion problem by binary particle swarm optimization algorithm, Acta academica karviniensia 11, 24 (2011)

    2011

  20. [28]

    Farzi, A

    S. Farzi, A. Rayati Shavazi, and A. Rezaei Pandari, Using quantum-behaved particle swarm optimization for portfolio selection problem, International Arab Journal of Information Technology 10 (2013)

    2013

  21. [29]

    Rifki and H

    O. Rifki and H. Ono (2012)

    2012

  22. [30]

    Ruiz-Torrubiano and A

    R. Ruiz-Torrubiano and A. Suarez, Hybrid approaches and di- mensionality reduction for portfolio selection with cardinality constraints, IEEE Computational Intelligence Magazine 5, 92 (2010), ISSN 1556-603X

    2010

  23. [31]

    Rigetti Computing, ForestTM SDK, https://rigetti.com/forest

  24. [32]

    G. E. Hinton, S. Osindero, and Y .-W. Teh, A fast learning al- gorithm for deep belief nets , Neural Comput. 18, 1527 (2006), ISSN 0899-7667

    2006

  25. [33]

    Tieleman, in Proceedings of the 25th International Confer- ence on Machine Learning(ACM, New York, NY , USA, 2008), ICML ’08, pp

    T. Tieleman, in Proceedings of the 25th International Confer- ence on Machine Learning(ACM, New York, NY , USA, 2008), ICML ’08, pp. 1064–1071

    2008

  26. [34]

    net/tutorial/rbm.html

    Theano Development Team, Restricted boltzmann ma- chines (RBM) (2016), URL http://deeplearning. net/tutorial/rbm.html

    2016

  27. [35]

    Hansen, Y

    N. Hansen, Y . Akimoto, and P. Baudis, CMA-ES/pycma on Github, Zenodo, DOI:10.5281/zenodo.2559634 (2019), URL https://doi.org/10.5281/zenodo.2559634

    2019 doi

  28. [36]

    Hansen and A

    N. Hansen and A. Ostermeier, Completely derandomized self- adaptation in evolution strategies , Evolutionary Computation 9, 159 (2001)

    2001

  29. [37]

    X. Gao, Z. Zhang, and L. Duan, A quantum machine learn- ing algorithm based on generative models, Science Advances 4 (2018)

    2018

  30. [38]

    Glasser, R

    I. Glasser, R. Sweke, N. Pancotti, J. Eisert, and J. I. Cirac, Ex- pressive power of tensor-network factorizations for probabilis- tic modeling, with applications from hidden markov models to quantum machine learning, arXiv:1907.03741v2 (2019)

    2019 arXiv

  31. [39]

    Bradley, E

    T.-D. Bradley, E. M. Stoudenmire, and J. Terilla, Model- ing sequences with quantum states: A look under the hood , arXiv:1910.07425 (2019)

    2019 arXiv

  32. [40]

    Perdomo-Ortiz, A

    A. Perdomo-Ortiz, A. Feldman, A. Ozaeta, S. V . Isakov, Z. Zhu, B. O’Gorman, H. G. Katzgraber, A. Diedrich, H. Neven, J. de Kleer, et al., Readiness of quantum optimization machines for industrial applications , Phys. Rev. Applied 12, 014004 (2019)

    2019

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