Mario Guillaume Cecile, Nis-Luca van Hülst, Tomohiro Hashizume, Pia Siegl, Abhishek Setty, José Diogo da Costa Jesus, Paul Over, Sergio Bengoechea, Muhammad Umer, Spyros Tserkis, Eleftherios Mastorakis, Tristan Kraft, Francisco Cárdenas-López, Leonardo Scandurra, Thomas Rung, Felix Motzoi, Belda Yesil, Barbara Kraus, Martin Kiffner, Dimitris G. Angelakis, Eugene de Villiers, Dieter Jaksch
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow–Hassidim–Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum–classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.
Cite as BibTeX
@misc{cecile2026quantumalgorithmscomputationalfluid,
title={Quantum Algorithms for Computational Fluid Dynamics},
author={Mario Guillaume Cecile and Nis-Luca van Hülst and Tomohiro Hashizume and Pia Siegl and Abhishek Setty and José Diogo da Costa Jesus and Paul Over and Sergio Bengoechea and Muhammad Umer and Spyros Tserkis and Eleftherios Mastorakis and Tristan Kraft and Francisco Cárdenas-López and Leonardo Scandurra and Thomas Rung and Felix Motzoi and Belda Yesil and Barbara Kraus and Martin Kiffner and Dimitris G. Angelakis and Eugene de Villiers and Dieter Jaksch},
year={2026},
eprint={2609.39962},
archivePrefix={arXiv},
primaryClass={quant-ph},
url={https://arxiv.org/abs/2609.39962},
}