Recent advances in quantum computing and their increased availability has led
to a growing interest in possible applications. Among those is the solution of
partial differential equations (PDEs) for, e.g., material or flow simulation.
Currently, the most promising route to useful deployment of quantum processors
in the short to near term are so-called hybrid variational quantum algorithms
(VQAs). Thus, variational methods for PDEs have been proposed as a candidate
for quantum advantage in the noisy intermediate scale quantum (NISQ) era. In
this work, we conduct an extensive study of utilizing VQAs on real quantum
devices to solve the simplest prototype of a PDE -- the Poisson equation.
Although results on noiseless simulators for small problem sizes may seem
deceivingly promising, the performance on quantum computers is very poor. We
argue that direct resolution of PDEs via an amplitude encoding of the solution
is not a good use case within reach of today's quantum devices -- especially
when considering large system sizes and more complicated non-linear PDEs that
are required in order to be competitive with classical high-end solvers.Comment: 19 pages, 18 figure