Potential quantum advantage for simulation of fluid dynamics

Abstract

Numerical simulation of turbulent fluid dynamics needs to either parameterize turbulence-which introduces large uncertainties-or explicitly resolve the smallest scales-which is prohibitively expensive. Here we provide evidence through analytic bounds and numerical studies that a potential quantum exponential speedup can be achieved to simulate the Navier-Stokes equations governing turbulence using quantum computing. Specifically, we provide a formulation of the lattice Boltzmann equation for which we give evidence that low-order Carleman linearization is much more accurate than previously believed for these systems and that for computationally interesting examples. This is achieved via a combination of reformulating the nonlinearity and accurately linearizing the dynamical equations, effectively trading nonlinearity for additional degrees of freedom that add negligible expense in the quantum solver. Based on this we apply a quantum algorithm for simulating the Carleman-linearized lattice Boltzmann equation and provide evidence that its cost scales logarithmically with system size, compared to polynomial scaling in the best known classical algorithms. This work suggests that an exponential quantum advantage may exist for simulating fluid dynamics, paving the way for simulating nonlinear multiscale transport phenomena in a wide range of disciplines using quantum computing.