  Two quantum algorithms are presented for the numerical solution of a linear
one-dimensional advection-diffusion equation with periodic boundary conditions.
Their accuracy and performance with increasing qubit number are compared
point-by-point with each other. Specifically, we solve the linear partial
differential equation with a Quantum Linear Systems Algorithms (QLSA) based on
the Harrow--Hassidim--Lloyd method and a Variational Quantum Algorithm (VQA),
for resolutions that can be encoded using up to 6 qubits, which corresponds to
$N=64$ grid points on the unit interval. Both algorithms are of hybrid nature,
i.e., they involve a combination of classical and quantum computing building
blocks. The QLSA and VQA are solved as ideal statevector simulations using the
in-house solver QFlowS and open-access Qiskit software, respectively. We
discuss several aspects of both algorithms which are crucial for a successful
performance in both cases. These are the sizes of an additional quantum
register for the quantum phase estimation for the QLSA and the choice of the
algorithm of the minimization of the cost function for the VQA. The latter
algorithm is also implemented in the noisy Qiskit framework including
measurement and decoherence circuit noise. We reflect the current limitations
and suggest some possible routes of future research for the numerical
simulation of classical fluid flows on a quantum computer.
