Quantum Algorithms for Solving Nonlinear Equations
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We proposed several quantum algorithms for solving various nonlinear equations, including the quantum finite volume method for Navier-Stokes equations, quantum Newton's method for nonlinear algebraic equations, and the quantum homotopy perturbation method for dissipative nonlinear ODEs and quadratic nonlinear algebraic equations. Compared with their classical counterparts, the quantum methods can show exponential speedup in the dimension of the equations. We also conducted numerical experiments, from toy models to real-world scenarios. Results showed that quantum computing could still solve the equations correctly under a certain amount of quantum error, which supported the existence of quantum advantages. Our works expand the application scope of quantum computing from linear problems to nonlinear problems, fills some gaps in the research field of quantum algorithms for solving nonlinear problems, and have the prospect of solving vaster types of CFD problems in the future.
