  We propose a physics-informed quantum algorithm to solve nonlinear and
multidimensional differential equations (DEs) in a quantum latent space. We
suggest a strategy for building quantum models as state overlaps, where
exponentially large sets of independent basis functions are used for implicitly
representing solutions. By measuring the overlaps between states which are
representations of DE terms, we construct a loss that does not require
independent sequential function evaluations on grid points. In this sense, the
solver evaluates the loss in an intrinsically parallel way, utilizing a global
type of the model. When the loss is trained variationally, our approach can be
related to the differentiable quantum circuit protocol, which does not scale
with the training grid size. Specifically, using the proposed model definition
and feature map encoding, we represent function- and derivative-based terms of
a differential equation as corresponding quantum states. Importantly, we
propose an efficient way for encoding nonlinearity, for some bases requiring
only an additive linear increase of the system size $\mathcal{O}(N + p)$ in the
degree of nonlinearity $p$. By utilizing basis mapping, we show how the
proposed model can be evaluated explicitly. This allows to implement arbitrary
functions of independent variables, treat problems with various initial and
boundary conditions, and include data and regularization terms in the
physics-informed machine learning setting. On the technical side, we present
toolboxes for exponential Chebyshev and Fourier basis sets, developing tools
for automatic differentiation and multiplication, implementing nonlinearity,
and describing multivariate extensions. The approach is compatible with, and
tested on, a range of problems including linear, nonlinear and multidimensional
differential equations.
