Quantum algorithms for nonlinear equations in fluid mechanics

Steijl, R. (2022) Quantum algorithms for nonlinear equations in fluid mechanics. In: Zhao, Y. (ed.) Quantum Computing and Communications. IntechOpen: London. ISBN 9781839681332 (doi: 10.5772/intechopen.95023)

Abstract

In recent years, significant progress has been made in the development of quantum algorithms for linear ordinary differential equations as well as linear partial differential equations. There has not been similar progress in the development of quantum algorithms for nonlinear differential equations. In the present work, the focus is on nonlinear partial differential equations arising as governing equations in fluid mechanics. First, the key challenges related to nonlinear equations in the context of quantum computing are discussed. Then, as the main contribution of this work, quantum circuits are presented that represent the nonlinear convection terms in the Navier–Stokes equations. The quantum algorithms introduced use encoding in the computational basis, and employ arithmetic based on the Quantum Fourier Transform. Furthermore, a floating-point type data representation is used instead of the fixed-point representation typically employed in quantum algorithms. A complexity analysis shows that even with the limited number of qubits available on current and near-term quantum computers (<100), nonlinear product terms can be computed with good accuracy. The importance of including sub-normal numbers in the floating-point quantum arithmetic is demonstrated for a representative example problem. Further development steps required to embed the introduced algorithms into larger-scale algorithms are discussed.

• Item Type: Book Sections
• Status: Published
• Glasgow Author(s) Enlighten ID: Steijl, Dr Rene
• Authors: Steijl, R.
• College/School: College of Science and Engineering > School of Engineering > Autonomous Systems and Connectivity
• Publisher: IntechOpen
• ISBN: 9781839681332
• Published Online: 10 December 2020
• Copyright Holders: Copyright © 2020 The Author
• First Published: First published in Quantum Computing and Communications
• Publisher Policy: Reproduced under a Creative Commons licence