Quantum Algorithms for Computational Fluid Dynamics
Abstract
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.
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BibTeX
@misc{cecile2026quantum,
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},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39962},
eprint = {2609.39962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39962},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Algorithms for Computational Fluid Dynamics
AU - Mario Guillaume Cecile
AU - Nis-Luca van Hülst
AU - Tomohiro Hashizume
AU - Pia Siegl
AU - Abhishek Setty
AU - José Diogo da Costa Jesus
AU - Paul Over
AU - Sergio Bengoechea
AU - Muhammad Umer
AU - Spyros Tserkis
AU - Eleftherios Mastorakis
AU - Tristan Kraft
AU - Francisco Cárdenas-López
AU - Leonardo Scandurra
AU - Thomas Rung
AU - Felix Motzoi
AU - Belda Yesil
AU - Barbara Kraus
AU - Martin Kiffner
AU - Dimitris G. Angelakis
AU - Eugene de Villiers
AU - Dieter Jaksch
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39962
UR - https://arxiv.org/abs/2609.39962
AB - 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.
ER -