  In the last years, several quantum algorithms that try to address the problem
of partial differential equation solving have been devised. On one side,
"direct" quantum algorithms that aim at encoding the solution of the PDE by
executing one large quantum circuit. On the other side, variational algorithms
that approximate the solution of the PDE by executing several small quantum
circuits and making profit of classical optimisers. In this work we propose an
experimental study of the costs (in terms of gate number and execution time on
a idealised hardware created from realistic gate data) associated with one of
the "direct" quantum algorithm: the wave equation solver devised in [PCS.
Costa, S. Jordan, A. Ostrander, Phys. Rev. A 99, 012323, 2019]. We show that
our implementation of the quantum wave equation solver agrees with the
theoretical big-O complexity of the algorithm. We also explain in great details
the implementation steps and discuss some possibilities of improvements.
Finally, our implementation proves experimentally that some PDE can be solved
on a quantum computer, even if the direct quantum algorithm chosen will require
error-corrected quantum chips, which are not believed to be available in the
short-term.
