  The numerical solution of partial differential equations by discretization
techniques is ubiquitous in computational physics. In this work we benchmark
this approach in the quantum realm by solving the heat equation for a square
plate subject to fixed temperatures at the edges and random heat sources and
sinks within the domain. The hybrid classical-quantum approach consists in the
solution on a quantum computer of the coupled linear system of equations that
result from the discretization step. Owing to the limitations in the number of
qubits and their connectivity, we use the Gauss-Seidel method to divide the
full system of linear equations into subsystems, which are solved iteratively
in block fashion. Each of the linear subsystems were solved using 2000Q and
Advantage quantum computers developed by D-Wave Systems Inc. By comparing
classical numerical and quantum solutions, we observe that the errors and chain
break fraction are, on average, greater on the 2000Q system. Unlike the
classical Gauss-Seidel method, the errors of the quantum solutions level off
after a few iterations of our algorithm. This is partly a result of the span of
the real number line available from the mapping of the chosen size of the set
of qubit states. We verified this by using techniques to progressively shrink
the range mapped by the set of qubit states at each iteration (increasing
floating-point accuracy). As a result, no leveling off is observed. However, an
increase in qubits does not translate to an overall lower error. This is
believed to be indicative of the increasing length of chains required for the
mapping to real numbers and the ensuing limitations of hardware.
