A large spectrum of problems in classical physics and engineering, such as turbulence, is governed by nonlinear differential equations, which typically require high-performance computing to be solved. Over the past decade, however, the growth of classical computing power has slowed down because the miniaturisation of chips has been approaching the atomic scale. This is marking an end to Moore's law, which calls for a new computing paradigm: Quantum computing is a prime candidate. In this paper, we offer a perspective on the current challenges that need to be overcome in order to use quantum computing for the simulation of nonlinear dynamics. We review and discuss progress in the development of both quantum algorithms for nonlinear equations and quantum hardware. We propose pairings between quantum algorithms for nonlinear equations and quantum hardware concepts. These avenues open new opportunities for the simulation of nonlinear systems and turbulence.
Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of
N-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in
log(N) and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of
N interacting quantum harmonic oscillators with cost polylogarithmic in
N. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.
Abstract Emerging as a new field, quantum computation has reinvented the fundamentals of Computer Science and knowledge theory in a manner consistent with quantum physics. The fact that quantum computation has superior features and new events than classical computation provides benefits in proving mathematical theories. With advances in technology, the nonlinear partial differential equations are used in almost every area, and many difficulties have been overcome by the solutions of these equations. In particular, the complex solutions of KdV and Burgers equations have been shown to be used in modeling a simple turbulence flow. In this study, Burger-like equation with complex solutions is defined in Hilbert space and solved with an example. In addition, these solutions were analyzed. Thanks to the Quantum Burgers-Like equation, the nonlinear differential equation is solved by linearizing. The pattern changes of time made the result linear. This means that the Quantum Burgers-Like equation can be used to smoothen the sonic processing.