mlsanityData to 7 Oct 2026

Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series

arXiv 2108.03190 ↗First posted 6 Aug 2021quant-ph cond-mat.dis-nn stat.MLcited by 10

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Abstract (arXiv)

We propose a quantum algorithm for sampling from a solution of stochastic differential equations (SDEs). Using differentiable quantum circuits (DQCs) with a feature map encoding of latent variables, we represent the quantile function for an underlying probability distribution and extract samples as DQC expectation values. Using quantile mechanics we propagate the system in time, thereby allowing for time-series generation. We test the method by simulating the Ornstein-Uhlenbeck process and sampling at times different from the initial point, as required in financial analysis and dataset augmentation. Additionally, we analyse continuous quantum generative adversarial networks (qGANs), and show that they represent quantile functions with a modified (reordered) shape that impedes their efficient time-propagation. Our results shed light on the connection between quantum quantile mechanics (QQM) and qGANs for SDE-based distributions, and point the importance of differential constraints for model training, analogously with the recent success of physics informed neural networks.

Key facts. Cited by 10 papers in the map (data through 7 Oct 2026). Builds on 31 of its 113 references that are in the map. First posted on arXiv 6 Aug 2021.

Builds on

31 of 113 references.

References · 31

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  3. 2020Variational Quantum Algorithmscited by 687
  4. 2020The power of quantum neural networkscited by 377
  5. 2019Parameterized quantum circuits as machine learning modelscited by 330
  6. 2016Quantum Machine Learningcited by 325
  7. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  8. 2020A Review on Generative Adversarial Networks: Algorithms, Theory, and Applicationscited by 209
  9. 2021Neural SDEs as Infinite-Dimensional GANscited by 159
  10. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  11. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  12. 2018Quantum generative adversarial networkscited by 128
19 more
  1. 2019Transfer learning in hybrid classical-quantum neural networkscited by 113
  2. 2018Differentiable Learning of Quantum Circuit Born Machinecited by 90
  3. 2020Experimental Quantum Generative Adversarial Networks for Image Generationcited by 83
  4. 2021Federated Quantum Machine Learningcited by 82
  5. 2021Encoding-dependent generalization bounds for parametrized quantum circuitscited by 81
  6. 2019The Born Supremacy: Quantum Advantage and Training of an Ising Born Machinecited by 75
  7. 2020Quantum Convolutional Neural Networks for High Energy Physics Data Analysiscited by 64
  8. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  9. 2018A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equationscited by 23
  10. 2018Learning and Inference on Generative Adversarial Quantum Circuitscited by 22
  11. 2020Quantum versus Classical Generative Modelling in Financecited by 21
  12. 2018The Expressive Power of Parameterized Quantum Circuitscited by 18
  13. 2019Neural Networks with Cheap Differential Operatorscited by 18
  14. 2018Adversarial quantum circuit learning for pure state approximationcited by 16
  15. 2021Learning quantum data with the quantum Earth Mover's distancecited by 13
  16. 2020Noise robustness and experimental demonstration of a quantum generative adversarial network for continuous distributionscited by 11
  17. 2021Dual-Parameterized Quantum Circuit GAN Model in High Energy Physicscited by 9
  18. 2019Data-driven PDE discovery with evolutionary approachcited by 7
  19. 2019Efficient Online Quantum Generative Adversarial Learning Algorithms with Applicationscited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 5; 2023: 1; 2024: 0; 2025: 0; 2026: 1.

10 papers.

  1. 2021Variational Quantum Reinforcement Learning via Evolutionary Optimizationcited by 50
  2. 2021Quantum Architecture Search via Continual Reinforcement Learningcited by 22
  3. 2023What can we learn from quantum convolutional neural networks?cited by 6
  4. 2021Quantum Model-Discoverycited by 4
  5. 2022Protocols for classically training quantum generative models on probability distributionscited by 4
  6. 2022Financial Risk Management on a Neutral Atom Quantum Processorcited by 4
  7. 2022Quantum Extremal Learningcited by 3
  8. 2022Integral Transforms in a Physics-Informed (Quantum) Neural Network setting: Applications & Use-Casescited by 1
  9. 2022Harmonic (Quantum) Neural Networkscited by 1
  10. 2026Accelerating physics-informed neural networks for full waveform inversion using a hybrid quantum-classical finite-basis architecturecited by 0