An HHL-Based Quantum-Classical Solver for the Incompressible Navier-Stokes Equations with Approximate QST
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Quantum Physics arXiv:2603.18222 (quant-ph) [Submitted on 18 Mar 2026] Title:An HHL-Based Quantum-Classical Solver for the Incompressible Navier-Stokes Equations with Approximate QST Authors:Moshe Inger, Steven Frankel View a PDF of the paper titled An HHL-Based Quantum-Classical Solver for the Incompressible Navier-Stokes Equations with Approximate QST, by Moshe Inger and Steven Frankel View PDF HTML (experimental) Abstract:In computational fluid dynamics (CFD), the numerical integration of the Navier-Stokes equations is frequently constrained by the Poisson equation to determine the pressure. Discretization of this equation often results in the need to solve a system of linear algebraic equations. This step typically represents the primary computational bottleneck. Quantum linear system algorithms such as Harrow-Hassidim-Lloyd (HHL) offer the potential for exponential speedups for solving sparse linear systems, such as those that arise from the discretized Poisson equation. In this work, we successfully couple HHL to a discretized formulation of the incompressible Navier-Stokes equations and demonstrate both accurate lid-driven cavity flow simulations as a fully integrated benchmark problem, and accurate flow of the Taylor-Green vortex. To address the readout limitation, we utilize a recent novel quantum state tomography (QST) approach based on Chebyshev polynomials, which enables approximate statevector extraction without full state reconstruction. Together, these results clarify the algorithmic structure required for quantum CFD, explicitly confront the measurement bottleneck, and establish benchmark problems for future quantum fluid simulations. We implement the solver using IBM's Qiskit framework and validate the hybrid quantum-classical simulation against standard classical numerical methods. Our results demonstrate that the hybrid solver successfully captures the global vortex dynamics of the lid-driven cavity problem and the Taylor-Green vortex, offering a robust pathway for integrating quantum subroutines into more practical higher-Reynolds number CFD workflows. Comments: Subjects: Quantum Physics (quant-ph); Computational Physics (physics.comp-ph); Fluid Dynamics (physics.flu-dyn) Cite as: arXiv:2603.18222 [quant-ph] (or arXiv:2603.18222v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.18222 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Moshe Inger [view email] [v1] Wed, 18 Mar 2026 19:13:06 UTC (1,076 KB) Full-text links: Access Paper: View a PDF of the paper titled An HHL-Based Quantum-Classical Solver for the Incompressible Navier-Stokes Equations with Approximate QST, by Moshe Inger and Steven FrankelView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 Change to browse by: physics physics.comp-ph physics.flu-dyn References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)