Deep learning-based quantum algorithms for solving nonlinear partial differential equations

TL;DR

This paper investigates integrating quantum subroutines into deep learning methods for solving high-dimensional nonlinear partial differential equations, exploring potential speedups and hybrid architectures for future quantum computing applications.

Contribution

It introduces hybrid quantum-classical architectures, analyzes quantum-accelerated Monte Carlo methods, and discusses quantum algorithms for neural network training to enhance PDE solving.

Findings

Hybrid architectures perform comparably or worse than classical ones in simulations.

Quantum-accelerated Monte Carlo methods can potentially speed up loss estimation.

Trade-offs exist when using quantum methods for gradient estimation.

Abstract

Partial differential equations frequently appear in the natural sciences and related disciplines. Solving them is often challenging, particularly in high dimensions, due to the "curse of dimensionality". In this work, we explore the potential for enhancing a classical deep learning-based method for solving high-dimensional nonlinear partial differential equations with suitable quantum subroutines. First, with near-term noisy intermediate-scale quantum computers in mind, we construct architectures employing variational quantum circuits and classical neural networks in conjunction. While the hybrid architectures show equal or worse performance than their fully classical counterparts in simulations, they may still be of use in very high-dimensional cases or if the problem is of a quantum mechanical nature. Next, we identify the bottlenecks imposed by Monte Carlo sampling and the training of the neural networks. We find that quantum-accelerated Monte Carlo methods offer the potential to speed up the estimation of the loss function. In addition, we identify and analyse the trade-offs when using quantum-accelerated Monte Carlo methods to estimate the gradients with different methods, including a recently developed backpropagation-free forward gradient method. Finally, we discuss the usage of a suitable quantum algorithm for accelerating the training of feed-forward neural networks. Hence, this work provides different avenues with the potential for polynomial speedups for deep learning-based methods for nonlinear partial differential equations.