Potential quantum advantage for simulation of fluid dynamics

Numerical simulation of turbulent fluid dynamics needs to either parametrize 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 speedup can be achieved to simulate fluid dynamics using quantum computing. Specifically, we provide a lattice Boltzmann formulation of fluid dynamics for which we give evidence that low-order Carleman linearization is much more accurate than previously believed for these systems. This is achieved via a combination of reformulating the Navier-Stokes nonlinearity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>(</a:mo><a:mi mathvariant="bold-italic">u</a:mi><a:mo>·</a:mo><a:mrow><a:mi mathvariant="bold">∇</a:mi><a:mi mathvariant="bold-italic">u</a:mi></a:mrow><a:mo>)</a:mo></a:mrow></a:math> to lattice-Boltzmann nonlinearity <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mo>(</e:mo><e:msup><e:mrow><e:mi mathvariant="bold-italic">u</e:mi></e:mrow><e:mn>2</e:mn></e:msup><e:mo>)</e:mo></e:math> and accurately linearizing the dynamical equations, which effectively trades 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 with polynomial scaling in the best known classical algorithms. In this paper, we suggest that a 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.