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

Abstract

Efficient simulation of a quantum system's dynamics is expected to be an important application area for quantum computers as existing classical computers cannot do this. However, quantum systems are not unique in being hard to simulate. For example, classical nonlinear continuum systems and fields are governed by nonlinear partial differential equations whose solution is also hard for classical computers. Solving such equations is essential for many economically important industries/applications such as the aerospace industry, weather-forecasting, fiber-optics communication, and plasma magneto-hydrodynamics. This raises the question: can a quantum computer speed-up solving these equations? In this Review, a new quantum algorithm is described for solving nonlinear partial differential equations for which the answer is yes. First, a new quantum algorithm is discussed for solving the Navier-Stokes nonlinear partial differential equations which govern the flow of a viscous fluid. Its construction, verification, and computational cost are described, and it is shown that a significant quantum speed-up is possible. Its generalization to a quantum algorithm for solving nonlinear partial differential equations is described. The Review closes with a discussion of next steps forward. These new quantum algorithms open up a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry.

Review and perspectives in quantum computing for partial differential equations in structural mechanics

“…What is even more surprising is that other disciplines away from quantum physics, yet heavily relying on numerical calculus (fluid mechanics, finance, etc) already applied quantum algorithms to their own cost-intensive problems. For instance, several works in fluid mechanics field used quantum subroutines to solve both the lattice Boltzmann (Mezzacapo et al, 2015;Todorova and Steijl, 2020;Budinski, 2021a) and the Navier-Stokes (Steijl and Barakos, 2018;Gaitan, 2020;Budinski, 2021b;Gaitan, 2021) equations. The hope is that structual mechanics will also explore the use of quantum algorithms to support expensive simulations, such as those involving material nonlinearities and large structural deformations.…”

“…Specifically, this work is aimed at the applied mechanics community and thus limits its focus to equations in structural mechanics. However, it must be pointed out that the field of quantum algorithms for PDEs spans many other branches of science, such as fluid mechanics (Mezzacapo et al, 2015;Steijl and Barakos, 2018;Todorova and Steijl, 2020;Gaitan, 2020Gaitan, , 2021Budinski, 2021a,b), finance (An et al, 2020;Chakrabarti et al, 2021;Fontanela et al, 2021), model discovery (Heim et al, 2021) and cosmology (Mocz and Szasz, 2021), which may also benefit from specialized reviews.…”

Observability of fidelity decay at the Lyapunov rate in few-qubit quantum simulations

“…Second, their estimate of the border to localization was imprecise and has been improved to form bound (2). Third and lastly, they did not recognize the early time regime which is crucial for quantifying the maximum allowable noise in bound (3). These details are necessary in the context of few-qubit quantum simulation.…”

“…Because chaotic systems are typically modeled through numerical computation, they are one of the primary application areas for scientific computing. Examples from classical physics include simulations of molecular dynamics [1], fluid dynamics [2,3], plasma physics [4,5], Monte Carlo methods, and gravitational N-body problems. Examples from quantum physics include nonequilibrium condensed matter physics [6], chemistry [7], nuclear physics [8,9], and lattice gauge theory [10].…”

Quantum Signal Processing for simulating cold plasma waves

“…In recent years, there has been a significant development of QC applications to physics problems, including simulations of classical systems. Various methods have been proposed to solve the wave equation 23 , Poisson's equation 24,25 , Maxwell's equations 26,27 , and first-order linear hyperbolic systems 28 , the Navier-Stokes equation 29,30 , the Boltzmann equation 31 , and to simulate advection-diffusion processes 32 . It was also shown that most of quantum algorithms can be considered as special cases of a more general algorithm called Quantum Singular Value Transform 33 .…”