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

arXiv: https://arxiv.org/abs/2108.03190 · HTML: https://mlsanity.com/paper/2108.03190/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 113 references that are in the map.
- First posted on arXiv 6 Aug 2021.

## Builds on (most cited first, 30 of 31)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [A Review on Generative Adversarial Networks: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2001.06937/) (2020)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Transfer learning in hybrid classical-quantum neural networks](https://mlsanity.com/paper/1912.08278/) (2019)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Experimental Quantum Generative Adversarial Networks for Image Generation](https://mlsanity.com/paper/2010.06201/) (2020)
- [Federated Quantum Machine Learning](https://mlsanity.com/paper/2103.12010/) (2021)
- [Encoding-dependent generalization bounds for parametrized quantum circuits](https://mlsanity.com/paper/2106.03880/) (2021)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [Quantum Convolutional Neural Networks for High Energy Physics Data Analysis](https://mlsanity.com/paper/2012.12177/) (2020)
- [A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations](https://mlsanity.com/paper/1806.02957/) (2018)
- [Learning and Inference on Generative Adversarial Quantum Circuits](https://mlsanity.com/paper/1808.03425/) (2018)
- [Quantum versus Classical Generative Modelling in Finance](https://mlsanity.com/paper/2008.00691/) (2020)
- [The Expressive Power of Parameterized Quantum Circuits](https://mlsanity.com/paper/1810.11922/) (2018)
- [Neural Networks with Cheap Differential Operators](https://mlsanity.com/paper/1912.03579/) (2019)
- [Adversarial quantum circuit learning for pure state approximation](https://mlsanity.com/paper/1806.00463/) (2018)
- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Noise robustness and experimental demonstration of a quantum generative adversarial network for continuous distributions](https://mlsanity.com/paper/2006.01976/) (2020)
- [Dual-Parameterized Quantum Circuit GAN Model in High Energy Physics](https://mlsanity.com/paper/2103.15470/) (2021)
- [Data-driven PDE discovery with evolutionary approach](https://mlsanity.com/paper/1903.08011/) (2019)

## Built on by (most cited first, 10 of 10)

- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [Quantum Architecture Search via Continual Reinforcement Learning](https://mlsanity.com/paper/2112.05779/) (2021)
- [What can we learn from quantum convolutional neural networks?](https://mlsanity.com/paper/2308.16664/) (2023)
- [Quantum Model-Discovery](https://mlsanity.com/paper/2111.06376/) (2021)
- [Protocols for classically training quantum generative models on probability distributions](https://mlsanity.com/paper/2210.13442/) (2022)
- [Financial Risk Management on a Neutral Atom Quantum Processor](https://mlsanity.com/paper/2212.03223/) (2022)
- [Quantum Extremal Learning](https://mlsanity.com/paper/2205.02807/) (2022)
- [Integral Transforms in a Physics-Informed (Quantum) Neural Network setting: Applications & Use-Cases](https://mlsanity.com/paper/2206.14184/) (2022)
- [Harmonic (Quantum) Neural Networks](https://mlsanity.com/paper/2212.07462/) (2022)
- [Accelerating physics-informed neural networks for full waveform inversion using a hybrid quantum-classical finite-basis architecture](https://mlsanity.com/paper/2606.01110/) (2026)

## 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.