arXiv:2305.02019 · Physical Review A

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

Lukas Mouton, Florentin Reiter, Ying Chen, Patrick Rebentrost

Published Aug 14, 2024• 8 citations

Abstract

Partial differential equations frequently appear in the natural sciences and related disciplines. 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. In a first approach, we construct a deep-learning architecture based on variational quantum circuits without provable guarantees. In a second approach, tailored towards fault-tolerant quantum computers, find that quantum-accelerated Monte Carlo methods offer the potential to speed up the estimation of the loss function. In addition, we identify and analyze 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.

DOI: 10.1103/physreva.110.022612

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