Solving the Hele–Shaw flow using the Harrow–Hassidim–Lloyd algorithm on superconducting devices: A study of efficiency and challenges

The development of quantum processors for practical fluid flow problems is a promising yet distant goal. Recent advances in quantum linear solvers have highlighted their potential for classical fluid dynamics. In this study, we evaluate the Harrow–Hassidim–Lloyd (HHL) quantum linear systems algorithm (QLSA) for solving the idealized Hele–Shaw flow. Our focus is on the accuracy and computational cost of the HHL solver, which we find to be sensitive to the condition number, scaling exponentially with problem size. This emphasizes the need for preconditioning to enhance the practical use of QLSAs in fluid flow applications. Moreover, we perform shots-based simulations on quantum simulators and test the HHL solver on superconducting quantum devices, where noise, large circuit depths, and gate errors limit performance. Error suppression and mitigation techniques improve accuracy, suggesting that such fluid flow problems can benchmark noise mitigation efforts. Our findings provide a foundation for future, more complex application of QLSAs in fluid flow simulations.