Quantum algorithm for anisotropic diffusion and convection equations with vector norm scaling
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Quantum Physics arXiv:2603.08799 (quant-ph) [Submitted on 9 Mar 2026] Title:Quantum algorithm for anisotropic diffusion and convection equations with vector norm scaling Authors:Julien Zylberman, Thibault Fredon, Nuno F. Loureiro, Fabrice Debbasch View a PDF of the paper titled Quantum algorithm for anisotropic diffusion and convection equations with vector norm scaling, by Julien Zylberman and 3 other authors View PDF Abstract:In this work, we tackle the resolution of partial differential equations (PDEs) on digital quantum computers. Two fundamental PDEs are addressed: the anisotropic diffusion equation and the anisotropic convection equation. We present a quantum numerical scheme consisting of three steps: quantum state preparation, evolution with diagonal operators, and measurement of observables of interest. The evolution step relies on a high-order centered finite difference and a product formula approximation, also known as Trotterization. We provide novel vector-norm analysis to bound the different sources of error. We prove that the number of time-steps required in the evolution can be reduced by a factor $\Theta (16^n)$ for the diffusion equation, and $\Theta (4^n)$ for the convection equation, where $n$ is the number of qubits per dimension, an exponential reduction compared to the previously established operator-norm analysis. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Numerical Analysis (math.NA) Cite as: arXiv:2603.08799 [quant-ph] (or arXiv:2603.08799v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.08799 Focus to learn more arXiv-issued DOI via DataCite Journal reference: International Conference on Quantum Engineering Sciences and Technologies for Industry and Services,255-263,2025,Springer Related DOI: https://doi.org/10.1007/978-3-032-13855-2_23 Focus to learn more DOI(s) linking to related resources Submission history From: Julien Zylberman [view email] [v1] Mon, 9 Mar 2026 18:02:12 UTC (68 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum algorithm for anisotropic diffusion and convection equations with vector norm scaling, by Julien Zylberman and 3 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-03 Change to browse by: cs cs.NA math math-ph math.MP math.NA 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?)