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There has been significant progress in the development of quantum algorithmsfor solving linear systems of equations with a growing body of applications toComputational Fluid Dynamics (CFD) and CFD-like problems. This work extendsprevious work by developing a non-linear hybrid quantum-classical CFD solverand using it to generate fully converged solutions. The hybrid solver uses theSIMPLE CFD algorithm, which is common in many industrial CFD codes, and appliesit to the 2-dimensional lid driven cavity test case. A theme of this work isthe classical processing time needed to prepare the quantum circuit with afocus on the decomposition of the CFD matrix into a linear combination ofunitaries (LCU). CFD meshes with up to 65x65 nodes are considered with thelargest producing a LCU containing 32,767 Pauli strings. A new method forrapidly re-computing the coefficients in a LCU is proposed, although thisreduces, rather than eliminates, the classical scaling issues. The quantumlinear equation solver uses the Harrow, Hassidim, Lloyd (HHL) algorithm via astate-vector emulator. Test matrices are sampled from the classical CFD solverto investigate the solution accuracy that can be achieved with HHL. For thesmallest 5x5 and 9x9 CFD meshes, full non-linear hybrid CFD calculations areperformed. The impacts of approximating the LCU and the varying the number ofancilla rotations in the eigenvalue inversion circuit are studied. Preliminarytiming results indicate that the classical computer preparation time needed fora hybrid solver is just as important to the achievement of quantum advantage inCFD as the time on the quantum computer. The reported HHL solutions and LCUdecompositions provide a benchmark for future research. The CFD test matricesused in this study are available upon request.

A Hybrid Quantum-Classical CFD Methodology with Benchmark HHL Solutions