Finding flows of a Navier–Stokes fluid through quantum computing

Abstract

There is great interest in using quantum computers to efficiently simulate a quantum system’s dynamics as existing classical computers cannot do this. Little attention, however, has been given to quantum simulation of a classical nonlinear continuum system such as a viscous fluid even though this too is hard for classical computers. Such fluids obey the Navier–Stokes nonlinear partial differential equations, whose solution is essential to the aerospace industry, weather forecasting, plasma magneto-hydrodynamics, and astrophysics. Here we present a quantum algorithm for solving the Navier–Stokes equations. We test the algorithm by using it to find the steady-state inviscid, compressible flow through a convergent-divergent nozzle when a shockwave is (is not) present. We find excellent agreement between numerical simulation results and the exact solution, including shockwave capture when present. Finally, we compare the algorithm’s computational cost to deterministic and random classical algorithms and show that a significant speed-up is possible. Our work points to a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry, weather-forecasting, and engineered-plasma technologies.

Solving nonlinear differential equations with differentiable quantum circuits

“…The specific example we choose to start with is the flow through a convergent-divergent nozzle, being a paradigmatic task in the aerospace industry [Fig. 9(a)] [2,142]. The Navier-Stokes equations can be rewritten for the inviscid fluid in quasi-1D approximation.…”

“…The Navier-Stokes equations can be rewritten for the inviscid fluid in quasi-1D approximation. They read [2,142]…”

“…Another hybrid quantum-classical approach is presented in Ref. [142], where computational fluid dynamics applications are considered. This relies heavily on classical computation, with the quantum processor being delegated to perform function averaging (using amplitude estimation as subroutine).…”

Quantum Algorithms for Nonlinear Equations in Fluid Mechanics

“…However, the computational complexity of this work involves exponential dependency on the time interval used in the time integration. A small number of more recent works have addressed nonlinear differential equations and typically algorithms for very specific problems were obtained [8]. Therefore, much research work is needed into quantum algorithms for a wider range of nonlinear problems.…”

“…The quantum circuits for obtaining the non-linear product terms are new developments and form the main contribution of this work. In recent years, a small number of works have considered quantum computing applications to fluid mechanics [2][3][4][5][6][7][8]. A brief review of this previous work will be presented in Section 2 and will provide context to the proposed approach.…”