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Solving partial differential equations on near-term quantum computers

In this work, we obtain the numerical temperature field to a thermally developing fluid flow inside parallel plates problem with a quantum computing method. The physical problem deals with the heat transfer of a steady state, hydrodinamically developed and thermally developing fluid flow inside two parallel plates channel subjected to a prescribed constant heat flux. Its solution is formulated numerically with Finite Differences method, where a sequence of linear systems must be solved in order to determine the complete temperature field. Such linear systems are written as discrete unconstrained optimization problems with floating points being approximated using binary variables and solved using near-term quantum heuristics. Due to the exponential cost of simulating quantum algorithms, a reduced number of qubits had to be used in the simulations, causing a loss of precision in the results. However, this work advances the state of the art of solutions of differential equations with noisy quantum devices and could be used for useful applications when quantum computers with thousands of qubits become available.