# Qade: Solving Differential Equations on Quantum Annealers

arXiv: https://arxiv.org/abs/2204.03657 · HTML: https://mlsanity.com/paper/2204.03657/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 7 Apr 2022.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [Neural Networks Trained to Solve Differential Equations Learn General Representations](https://mlsanity.com/paper/1807.00042/) (2018)
- [PyDEns: a Python Framework for Solving Differential Equations with Neural Networks](https://mlsanity.com/paper/1909.11544/) (2019)
- [A Discussion on Solving Partial Differential Equations using Neural Networks](https://mlsanity.com/paper/1904.07200/) (2019)
- [Elvet -- a neural network-based differential equation and variational problem solver](https://mlsanity.com/paper/2103.14575/) (2021)
- [Quantum Optimisation of Complex Systems with a Quantum Annealer](https://mlsanity.com/paper/2105.13945/) (2021)
- [Completely Quantum Neural Networks](https://mlsanity.com/paper/2202.11727/) (2022)
- [dNNsolve: an efficient NN-based PDE solver](https://mlsanity.com/paper/2103.08662/) (2021)
- [ODEN: A Framework to Solve Ordinary Differential Equations using Artificial Neural Networks](https://mlsanity.com/paper/2005.14090/) (2020)

## Built on by (most cited first, 1 of 1)

- [Training Neural Networks with Universal Adiabatic Quantum Computing](https://mlsanity.com/paper/2308.13028/) (2023)

## Abstract (arXiv)

We present a general method, called Qade, for solving differential equations using a quantum annealer. The solution is obtained as a linear combination of a set of basis functions. On current devices, Qade can solve systems of coupled partial differential equations that depend linearly on the solution and its derivatives, with non-linear variable coefficients and arbitrary inhomogeneous terms. We test the method with several examples and find that state-of-the-art quantum annealers can find the solution accurately for problems requiring a small enough function basis. We provide a Python package implementing the method at gitlab.com/jccriado/qade.