Quantum Algorithms for Fluid Simulations

Abstract

This chapter describes results of a recent investigation aiming to assess the potential of quantum computing and suitably designed algorithms for future computational fluid dynamics applications. For quantum computers becoming available in the near future, it can be expected that applications of quantum computing follow the quantum coprocessor model, where selected parts of the computational task for which efficient quantum algorithms exist are executed on the quantum hardware. For example, in computational fluid dynamics algorithm, this hybrid quantum/classical approach is discussed, and in particular it is shown how the approximate quantum Fourier transform (AQFT) can be used in the Poisson solvers of the considered method for the incompressible-flow Navier-Stokes equations. The analysis shows that despite the inevitable errors introduced by applying AQFT, the method produces meaningful results for three-dimensional example problems. A second example of a quantum algorithm for flow simulations is then described. This method based on kinetic modeling of the flow was developed to reduce the information transfer between quantum and classical hardware in the quantum coprocessor model. It is shown that this quantum algorithm can be executed fully on quantum hardware during a simulation. The conclusion summarizes further challenges for algorithm developments and future work.

Quantum Algorithms for Nonlinear Equations in Fluid Mechanics

“…Important developments for a wider range of applications include quantum algorithms for linear systems [9,10] and the Poisson equation [11]. Applications to computational science and engineering problems beyond quantum chemistry have only recently begun to appear [4][5][6][12][13][14]. Despite this research effort, progress in defining suitable engineering applications for quantum computers has been limited.…”

“…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.…”

Finding Solutions of the Navier‐Stokes Equations through Quantum Computing—Recent Progress, a Generalization, and Next Steps Forward

“…It has gone from a research topic which in 1995 was regarded by many to be of (at best) academic interest, to a thriving, wellfunded, multi-disciplinary research juggernaut racing to create a technology from which a scalable, robust, universal quantum the past few years a number of papers have appeared which also consider quantum algorithms for the NS equations. [28][29][30][31][32] That so little has been done is surprising, given that the US aerospace industry in 2019 was a nearly one trillion (US) dollar industry, [33] while weather-forecasting directly impacts the logistics of businesses, militaries, and disaster-relief agencies worldwide.…”

“…Note that these new quantum algorithms open up a large new application area for quantum computing (quantum simulation of classical nonlinear continuum systems and fields) with substantial scientific and economic impact, including the trillion-dollar aerospace industry, weatherforecasting, fiber-optics communication, and engineered-plasma technologies. Note that the quantum NS and quantum PDE algorithms are independent of the previously cited works, [24][25][26][27][28][29][30][31][32] and of a quantum algorithm for elliptic PDEs. [35]…”

Solving Burgers’ equation with quantum computing

“…The quantum lattice-gas model was shown to provide a significant advantage over a classical lattice-gas model. Steijl [21,22] used a hybrid quantum/classical approach to solve Poisson's equation which arises (inter alia) in incompressible NS flows. References [4,5,16,19] also examined fluid flow problems through quantum algorithms.…”