  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.
