pith. sign in

arxiv: 2603.09789 · v2 · submitted 2026-03-10 · 💻 cs.LG · cs.AI· quant-ph

A Hybrid Quantum-Classical Framework for Financial Volatility Forecasting Based on Quantum Circuit Born Machines

Yixiong Chen This is my paper

Pith reviewed 2026-05-15 13:16 UTC · model grok-4.3

classification 💻 cs.LG cs.AIquant-ph
keywords volatility forecastingquantum circuit born machineLSTMhybrid quantum-classicalDrop-Priorfinancial time serieshigh-frequency data
0
0 comments X

The pith

Hybrid LSTM-QCBM model outperforms classical LSTM in volatility forecasting while allowing full classical inference after training

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

The paper proposes integrating a Long Short-Term Memory network with a Quantum Circuit Born Machine to forecast financial volatility from high-frequency market data. The QCBM serves as a learnable generative prior to capture complex distributions that guide the LSTM's predictions. By applying a stochastic Drop-Prior mechanism during training, the LSTM distills knowledge from the quantum component. This setup allows the model to achieve superior performance on metrics like MSE, RMSE, and QLIKE for SSE Composite and CSI 300 indices compared to a standard LSTM. The approach demonstrates quantum-assisted training that results in a classically efficient model at deployment time.

Core claim

By training an LSTM with a QCBM prior under stochastic Drop-Prior, the resulting model captures complex market distributions and outperforms pure classical LSTM forecasting, yet runs entirely classically at inference without loss of the gained accuracy.

What carries the argument

The stochastic Drop-Prior mechanism, which randomly drops the quantum prior during training to force the LSTM to internalize the distributional knowledge.

Load-bearing premise

That the QCBM successfully captures complex market distributions and that the stochastic Drop-Prior mechanism transfers this knowledge to the LSTM without substantial performance loss when the quantum module is removed at inference time.

What would settle it

Training the same LSTM architecture with a classical generative prior instead of QCBM and checking whether the accuracy advantage over standard LSTM disappears on the same SSE and CSI 300 datasets.

Figures

Figures reproduced from arXiv: 2603.09789 by Yixiong Chen.

Figure 1
Figure 1. Figure 1: Quantum Circuit Born Machine Structure adjacent qubit pairs through RXX gates, and finally applies another set of Rz and Rx rotation gates. The entire circuit repeats this layered structure L times, and finally measures all qubits to obtain a classical bit string. The parameters of the circuit, including the rotation angles θ and the RXX gate parameters ϕ, are trainable. By optimizing these parameters, the… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagram of the LSTM-QCBM hybrid model architecture. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of RMSE convergence curves of the models on the test set. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Histogram comparison of the probability distributions generated by the QCBM at [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative probability coverage of the Top- [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
read the original abstract

Accurate financial volatility forecasting is crucial but challenged by the non-linear, highly correlated nature of market data. Recently, quantum computing has emerged as a promising paradigm for solving complex high-dimensional sampling problems. To harness this, we propose a novel hybrid framework combining the temporal representation power of classical neural networks with the distribution-learning capabilities of quantum models. Specifically, we integrate a Long Short-Term Memory (LSTM) network with a Quantum Circuit Born Machine (QCBM). The LSTM extracts dynamic features, while the QCBM acts as a learnable generative prior modeling complex market distributions to guide forecasting. Evaluated on 5-minute high-frequency data from the SSE Composite and CSI 300 indices, our model significantly outperforms a classical LSTM baseline across MSE, RMSE, and QLIKE metrics. Furthermore, by introducing a stochastic ``Drop-Prior" mechanism during training, the LSTM implicitly distills structured information from the quantum prior. This establishes a pragmatic paradigm of ``quantum-assisted training with classical-efficient inference", whereby the model retains its quantum-enhanced accuracy even when the quantum module is entirely disabled during deployment. This demonstrates a practical pathway for leveraging quantum computing to enhance classical models without real-time quantum inference latency.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The paper proposes a hybrid LSTM-QCBM framework for financial volatility forecasting on 5-minute high-frequency data from the SSE Composite and CSI 300 indices. The QCBM serves as a learnable generative prior that models complex market distributions and guides the LSTM via a stochastic Drop-Prior mechanism during training; at inference the quantum module is removed entirely, yielding a classical model that retains the reported performance gains. The central empirical claim is that this architecture significantly outperforms a classical LSTM baseline on MSE, RMSE, and QLIKE.

Significance. If the performance gains can be shown to arise specifically from the quantum prior rather than generic hybrid-training effects, the work would supply a concrete, deployable template for quantum-assisted training of classical time-series models. The separation of quantum computation to the training phase only removes a major practical obstacle (real-time quantum latency) and could be transferred to other domains that benefit from generative priors, such as risk modeling or synthetic data generation.

major comments (3)
  1. Abstract and Results: the claim of significant outperformance on MSE, RMSE, and QLIKE is presented without error bars, number of independent runs, data-split protocol, or any statistical significance test, rendering the headline result impossible to evaluate from the supplied information.
  2. Experimental design (implicit in the workflow description): no ablation is reported that replaces the QCBM with a classical generative model (VAE, GMM, or flow) trained under the identical stochastic Drop-Prior schedule. Without this control it is impossible to determine whether the observed gains stem from quantum-specific distribution capture or from the mere presence of an auxiliary generative signal during training.
  3. Methods: the manuscript supplies no quantitative verification that the QCBM actually captures the target market distribution (e.g., MMD, Wasserstein distance, or held-out log-likelihood on returns). The premise that the quantum module “models complex market distributions” therefore remains an untested assumption rather than a demonstrated fact.
minor comments (2)
  1. The Drop-Prior mechanism is described only at a high level; a formal definition, pseudocode, or explicit loss term would improve reproducibility.
  2. Notation for the hybrid loss and the stochastic dropping probability is not introduced until the methods section; early use of these symbols in the abstract creates unnecessary ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the detailed review. We appreciate the opportunity to clarify and strengthen our manuscript. Below we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: Abstract and Results: the claim of significant outperformance on MSE, RMSE, and QLIKE is presented without error bars, number of independent runs, data-split protocol, or any statistical significance test, rendering the headline result impossible to evaluate from the supplied information.

    Authors: We fully agree with this observation. The original manuscript indeed omitted these statistical details. In the revised version, we will report results with standard deviations from at least 5 independent runs with different random seeds, explicitly describe the temporal data splitting protocol used to avoid look-ahead bias, and include p-values from appropriate statistical tests (e.g., Diebold-Mariano test for forecast accuracy) to substantiate the significance of the improvements. revision: yes

  2. Referee: Experimental design (implicit in the workflow description): no ablation is reported that replaces the QCBM with a classical generative model (VAE, GMM, or flow) trained under the identical stochastic Drop-Prior schedule. Without this control it is impossible to determine whether the observed gains stem from quantum-specific distribution capture or from the mere presence of an auxiliary generative signal during training.

    Authors: This is a valid concern. To address it, we will include an ablation study in the revised manuscript where we replace the QCBM with a classical Gaussian Mixture Model (GMM) as the generative prior, trained under the same stochastic Drop-Prior mechanism. This will help isolate whether the performance gains are due to the quantum model's distribution learning capabilities or the hybrid training approach in general. We note that training more complex classical models like VAEs or normalizing flows with the exact same integration would require additional computational resources, but the GMM ablation should provide initial evidence. revision: partial

  3. Referee: Methods: the manuscript supplies no quantitative verification that the QCBM actually captures the target market distribution (e.g., MMD, Wasserstein distance, or held-out log-likelihood on returns). The premise that the quantum module “models complex market distributions” therefore remains an untested assumption rather than a demonstrated fact.

    Authors: We acknowledge that the manuscript lacks explicit quantitative evaluation of the QCBM's generative performance. We will add in the Methods and Results sections metrics such as the Maximum Mean Discrepancy (MMD) and Wasserstein-1 distance between samples generated by the trained QCBM and the empirical distribution of the financial returns. Additionally, we will report the log-likelihood on a held-out test set of returns to demonstrate that the QCBM effectively models the complex market distributions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents an empirical hybrid framework combining LSTM with QCBM and a stochastic Drop-Prior training mechanism. No equations, derivations, or first-principles results are described that reduce a claimed prediction or performance metric to a fitted parameter or self-defined input by construction. The central claims rest on experimental comparisons (MSE/RMSE/QLIKE on SSE and CSI 300 data) rather than any tautological renaming, ansatz smuggling, or self-citation load-bearing step. The workflow is self-contained against external benchmarks and does not invoke uniqueness theorems or prior self-work to force the outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient technical detail to enumerate free parameters, axioms, or invented entities; standard LSTM and QCBM components are assumed but not specified.

pith-pipeline@v0.9.0 · 5504 in / 969 out tokens · 43427 ms · 2026-05-15T13:16:05.125400+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · 1 internal anchor

  1. [1]

    Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics, 31(3):307–327, 1986

    Tim Bollerslev. Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics, 31(3):307–327, 1986

    work page 1986

  2. [2]

    From garch to neu- ral network for volatility forecast

    Pengfei Zhao, Haoren Zhu, Wilfred Siu Hung Ng, and Dik Lun Lee. From garch to neu- ral network for volatility forecast. InProceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 16998–17006, 2024

    work page 2024

  3. [3]

    Neu- ral network–based financial volatility forecasting: A systematic review.ACM Computing Surveys (CSUR), 55(1):1–30, 2022

    Weijia Ge, Pedram Lalbakhsh, Liudmila Isai, Artem Lenskiy, and Hanna Suominen. Neu- ral network–based financial volatility forecasting: A systematic review.ACM Computing Surveys (CSUR), 55(1):1–30, 2022

    work page 2022

  4. [4]

    Prediction of realized volatility and implied volatility indices using ai and machine learning: A review.International Review of Financial Analysis, 93:103221, 2024

    Eirik S Gunnarsson, Herman R Isern, Aris Kaloudis, Michael Risstad, Bjarte Vigdel, and Sjur Westgaard. Prediction of realized volatility and implied volatility indices using ai and machine learning: A review.International Review of Financial Analysis, 93:103221, 2024

    work page 2024

  5. [5]

    Financial time series forecasting with deep learning: A systematic literature review: 2005–2019.Applied Soft Computing, 90:106181, 2020

    Omer Berat Sezer, Mehmet Ugur Gudelek, and Ahmet Murat Ozbayoglu. Financial time series forecasting with deep learning: A systematic literature review: 2005–2019.Applied Soft Computing, 90:106181, 2020

    work page 2005

  6. [6]

    Long short-term memory.Neural Computation, 9(8):1735–1780, 1997

    Sepp Hochreiter and J¨ urgen Schmidhuber. Long short-term memory.Neural Computation, 9(8):1735–1780, 1997

    work page 1997

  7. [7]

    Deep learning with long short-term memory networks for financial market predictions.European Journal of Operational Research, 270(2):654–669, 2018

    Thomas Fischer and Christopher Krauss. Deep learning with long short-term memory networks for financial market predictions.European Journal of Operational Research, 270(2):654–669, 2018

    work page 2018

  8. [8]

    Cnnpred: Cnn-based stock market prediction using a diverse set of variables.Expert Systems with Applications, 129:273–285, 2019

    Ehsan Hoseinzade and Saman Haratizadeh. Cnnpred: Cnn-based stock market prediction using a diverse set of variables.Expert Systems with Applications, 129:273–285, 2019

    work page 2019

  9. [9]

    Time series forecasting using a deep belief network with restricted boltzmann machines.Neuro- computing, 137:47–56, 2014

    Takashi Kuremoto, Shinsuke Kimura, Kunikazu Kobayashi, and Masanao Obayashi. Time series forecasting using a deep belief network with restricted boltzmann machines.Neuro- computing, 137:47–56, 2014

    work page 2014

  10. [10]

    Attention is all you need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. InAdvances in Neural Information Processing Systems, volume 30, 2017

    work page 2017

  11. [11]

    Multi- transformer: A new neural network-based architecture for forecasting s&p volatility.Math- ematics, 9(15):1794, 2021

    Eduardo Ramos-P´ erez, Pablo J Alonso-Gonz´ alez, and Jos´ e Javier N´ u˜ nez-Vel´ azquez. Multi- transformer: A new neural network-based architecture for forecasting s&p volatility.Math- ematics, 9(15):1794, 2021. 16

    work page 2021

  12. [12]

    Forecasting the volatility of stock price index: A hybrid model integrating lstm with multiple garch-type models.Expert Systems with Applications, 103:25–37, 2018

    Ha Young Kim and Chang Hyun Won. Forecasting the volatility of stock price index: A hybrid model integrating lstm with multiple garch-type models.Expert Systems with Applications, 103:25–37, 2018

    work page 2018

  13. [13]

    Twitter mood predicts the stock market

    Johan Bollen, Huina Mao, and Xiaojun Zeng. Twitter mood predicts the stock market. Journal of Computational Science, 2(1):1–8, 2011

    work page 2011

  14. [14]

    Listening to chaotic whispers: A deep learning framework for news-oriented stock trend prediction

    Ziniu Hu, Weiqing Liu, Jiang Bian, Xuanzhe Liu, and Tie-Yan Liu. Listening to chaotic whispers: A deep learning framework for news-oriented stock trend prediction. InPro- ceedings of the Eleventh ACM International Conference on Web Search and Data Mining, pages 261–269, 2018

    work page 2018

  15. [15]

    Quantum machine learning.Nature, 549(7671):195–202, 2017

    Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning.Nature, 549(7671):195–202, 2017

    work page 2017

  16. [16]

    Quantum convolutional neural networks

    Iris Cong, Soonwon Choi, and Mikhail D Lukin. Quantum convolutional neural networks. Nature Physics, 15(12):1273–1278, 2019

    work page 2019

  17. [17]

    arXiv:2012.03755

    Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, and Alexis Toumi. Founda- tions for near-term quantum natural language processing.arXiv preprint arXiv:2012.03755, 2020

    work page arXiv 2012

  18. [18]

    Quantum deep learning for time series prediction

    Devinder Kaur and Manoj Kumar. Quantum deep learning for time series prediction. In2021 International Conference on Artificial Intelligence and Smart Systems (ICAISS), pages 58–63. IEEE, 2021

    work page 2021

  19. [19]

    Quantum vs. classical machine learning: A benchmark study for financial prediction,

    R Ahmad, M Kashif, N Innan, and M Shafique. Quantum vs. classical machine learning: A benchmark study for financial prediction.arXiv preprint arXiv:2601.03802, 2026

    work page arXiv 2026

  20. [20]

    Advancing financial prediction through quantum machine learning.International Journal of Intelligent Data and Machine Learning, 2(02):1–7, 2025

    A Patel. Advancing financial prediction through quantum machine learning.International Journal of Intelligent Data and Machine Learning, 2(02):1–7, 2025

    work page 2025

  21. [21]

    The potential of quantum techniques for stock price prediction

    N Srivastava, G Belekar, and N Shahakar. The potential of quantum techniques for stock price prediction. In2023 IEEE International Conference on Recent Advances in Systems Science and Engineering (RASSE), pages 1–7. IEEE, 2023

    work page 2023

  22. [22]

    Quantum temporal convolutional neural networks for cross-sectional equity return prediction: A comparative benchmark study

    C S Chen, X Zhang, R Fu, Q Xie, and F Zhang. Quantum temporal convolutional neural networks for cross-sectional equity return prediction: A comparative benchmark study. arXiv preprint arXiv:2512.06630, 2025

    work page arXiv 2025

  23. [23]

    BLS-QLSTM: a novel hybrid quantum neural network for stock index forecasting.Humanities and Social Sciences Communications, 12(1):1–15, 2025

    L Su, D Li, and D Qiu. BLS-QLSTM: a novel hybrid quantum neural network for stock index forecasting.Humanities and Social Sciences Communications, 12(1):1–15, 2025

    work page 2025

  24. [24]

    A hybrid quantum-classical model for stock price prediction using quantum-enhanced long short-term memory.Entropy, 26(11):954, 2024

    K Kea, D Kim, C Huot, T K Kim, and Y Han. A hybrid quantum-classical model for stock price prediction using quantum-enhanced long short-term memory.Entropy, 26(11):954, 2024

    work page 2024

  25. [25]

    A hybrid quantum–classical LSTM approach for pre- dicting the stock market

    S Arora, D Kumar, and S Gupta. A hybrid quantum–classical LSTM approach for pre- dicting the stock market. InQuantum Machine Learning in Industrial Automation, pages 75–98. Springer Nature Switzerland, 2025

    work page 2025

  26. [26]

    QADQN: Quantum attention deep Q-network for financial market prediction

    S Dutta, N Innan, A Marchisio, S B Yahia, and M Shafique. QADQN: Quantum attention deep Q-network for financial market prediction. In2024 IEEE International Conference on Quantum Computing and Engineering (QCE), volume 2, pages 341–346. IEEE, 2024

    work page 2024

  27. [27]

    Contextual quantum neural networks for stock price prediction.Scientific Reports, 2026

    S Mourya, H Leipold, and B Adhikari. Contextual quantum neural networks for stock price prediction.Scientific Reports, 2026. To appear. 17

    work page 2026

  28. [28]

    Hybrid quantum-classical ensemble learning for S&P 500 directional pre- diction.arXiv preprint arXiv:2512.15738, 2025

    A I Weinberg. Hybrid quantum-classical ensemble learning for S&P 500 directional pre- diction.arXiv preprint arXiv:2512.15738, 2025

    work page arXiv 2025

  29. [29]

    HQNN-FSP: A hybrid classical- quantum neural network for regression-based financial stock market prediction.arXiv preprint arXiv:2503.15403, 2025

    P K Choudhary, N Innan, M Shafique, and R Singh. HQNN-FSP: A hybrid classical- quantum neural network for regression-based financial stock market prediction.arXiv preprint arXiv:2503.15403, 2025

    work page arXiv 2025

  30. [30]

    Noisy intermediate-scale quantum (nisq) algorithms.Reviews of Modern Physics, 94(1):015004, 2022

    Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S Kottmann, Tim Menke, Wai-Keong Mok, Soon Sim, Leong-Chuan Kwek, and Al´ an Aspuru-Guzik. Noisy intermediate-scale quantum (nisq) algorithms.Reviews of Modern Physics, 94(1):015004, 2022

    work page 2022

  31. [31]

    Variational quantum algorithms.Nature Reviews Physics, 3(9):625–644, 2021

    M Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms.Nature Reviews Physics, 3(9):625–644, 2021

    work page 2021

  32. [32]

    Supervised learning with quantum-enhanced feature spaces.Nature, 567(7747):209–212, 2019

    Vojtˇ ech Havl´ ıˇ cek, Antonio D C´ orcoles, Kristan Temme, Aram W Harrow, Abhinav Kan- dala, Jerry M Chow, and Jay M Gambetta. Supervised learning with quantum-enhanced feature spaces.Nature, 567(7747):209–212, 2019

    work page 2019

  33. [33]

    Quantum generative adversarial learning.Physical review letters, 121(4):040502, 2018

    Seth Lloyd and Christian Weedbrook. Quantum generative adversarial learning.Physical review letters, 121(4):040502, 2018

    work page 2018

  34. [34]

    Quantum circuit learning.Physical Review A, 98(3):032309, 2018

    Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning.Physical Review A, 98(3):032309, 2018

    work page 2018

  35. [35]

    Eval- uating analytic gradients on quantum hardware.Physical Review A, 99(3):032331, 2019

    Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Eval- uating analytic gradients on quantum hardware.Physical Review A, 99(3):032331, 2019

    work page 2019

  36. [36]

    Quantum embeddings for machine learning,

    Seth Lloyd, Maria Schuld, Aullon Ijaz, Josh Izaac, and Nathan Killoran. Quantum embed- dings for machine learning.arXiv preprint arXiv:2001.03622, 2020

    work page arXiv 2001

  37. [37]

    Effect of data encoding on the expressive power of variational quantum-machine-learning models.Physical Review A, 103(3):032430, 2021

    Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. Effect of data encoding on the expressive power of variational quantum-machine-learning models.Physical Review A, 103(3):032430, 2021

    work page 2021

  38. [38]

    Encoding-dependent generalization bounds for parametrized quantum circuits.Quantum, 5:582, 2021

    Matthias C Caro, Elies Gil-Fuster, Johannes Jakob Meyer, Jens Eisert, and Ryan Sweke. Encoding-dependent generalization bounds for parametrized quantum circuits.Quantum, 5:582, 2021

    work page 2021

  39. [39]

    Barren plateaus in quantum neural network training landscapes.Nature Commu- nications, 9(1):4812, 2018

    Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes.Nature Commu- nications, 9(1):4812, 2018

    work page 2018

  40. [40]

    Noise-induced barren plateaus in variational quantum algorithms

    Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. Noise-induced barren plateaus in variational quantum algorithms. Nature Communications, 12(1):6961, 2021

    work page 2021

  41. [41]

    Generative modeling with multi-mode quantum approxi- mate optimization algorithms.NPJ Quantum Information, 5(1):45, 2019

    Marcello Benedetti, D Garcia-Pintos, O Perdomo, Vicente Leyton-Ortega, Yunseong Nam, and Alejandro Perdomo-Ortiz. Generative modeling with multi-mode quantum approxi- mate optimization algorithms.NPJ Quantum Information, 5(1):45, 2019

    work page 2019

  42. [42]

    Differentiable learning of quantum circuit born machines

    Jin-Guo Liu and Lei Wang. Differentiable learning of quantum circuit born machines. Physical Review A, 98(6):062324, 2018. 18

    work page 2018

  43. [43]

    Expressive power of parametrized quantum circuits.Physical Review Research, 2(3):033125, 2020

    Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, and Dacheng Tao. Expressive power of parametrized quantum circuits.Physical Review Research, 2(3):033125, 2020

    work page 2020

  44. [44]

    Quantum versus classical generative modelling in finance.Quantum Science and Technol- ogy, 6(2):024013, 2021

    Brian Coyle, Maxwell Henderson, Justin C Le, Niraj Kumar, M Marco, and Elham Kashefi. Quantum versus classical generative modelling in finance.Quantum Science and Technol- ogy, 6(2):024013, 2021

    work page 2021

  45. [45]

    Deep learning and the information bottleneck principle

    Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In2015 IEEE Information Theory Workshop (ITW), pages 1–5. IEEE, 2015

    work page 2015

  46. [46]

    Distilling the Knowledge in a Neural Network

    Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  47. [47]

    A new learning paradigm: Learning using privileged information.Neural networks, 22(5-6):544–557, 2009

    Vladimir Vapnik and Akshay Vashist. A new learning paradigm: Learning using privileged information.Neural networks, 22(5-6):544–557, 2009

    work page 2009

  48. [48]

    Learning using privileged information: similarity control and knowledge transfer.Journal of machine learning research, 16(1):2023–2049, 2015

    Vladimir Vapnik and Rauf Izmailov. Learning using privileged information: similarity control and knowledge transfer.Journal of machine learning research, 16(1):2023–2049, 2015

    work page 2023

  49. [49]

    A learning algorithm for boltzmann machines.Cognitive science, 9(1):147–169, 1985

    David H Ackley, Geoffrey E Hinton, and Terrence J Sejnowski. A learning algorithm for boltzmann machines.Cognitive science, 9(1):147–169, 1985. 19

    work page 1985