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The numerical solution of partial differential equations by discretizationtechniques is ubiquitous in computational physics. In this work we benchmarkthis approach in the quantum realm by solving the heat equation for a squareplate subject to fixed temperatures at the edges and random heat sources andsinks within the domain. The hybrid classical-quantum approach consists in thesolution on a quantum computer of the coupled linear system of equations thatresult from the discretization step. Owing to the limitations in the number ofqubits and their connectivity, we use the Gauss-Seidel method to divide thefull system of linear equations into subsystems, which are solved iterativelyin block fashion. Each of the linear subsystems were solved using 2000Q andAdvantage quantum computers developed by D-Wave Systems Inc. By comparingclassical numerical and quantum solutions, we observe that the errors and chainbreak fraction are, on average, greater on the 2000Q system. Unlike theclassical Gauss-Seidel method, the errors of the quantum solutions level offafter a few iterations of our algorithm. This is partly a result of the span ofthe real number line available from the mapping of the chosen size of the setof qubit states. We verified this by using techniques to progressively shrinkthe range mapped by the set of qubit states at each iteration (increasingfloating-point accuracy). As a result, no leveling off is observed. However, anincrease in qubits does not translate to an overall lower error. This isbelieved to be indicative of the increasing length of chains required for themapping to real numbers and the ensuing limitations of hardware.

A hybrid classical-quantum approach to solve the heat equation using quantum annealers