Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems

Quantum Physics arXiv:2512.21033 (quant-ph) [Submitted on 24 Dec 2025] Title:Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems Authors:Sachin S. Bharadwaj, Balasubramanya Nadiga, Stephan Eidenbenz, Katepalli R. Sreenivasan View a PDF of the paper titled Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems, by Sachin S. Bharadwaj and 2 other authors View PDF HTML (experimental) Abstract:Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present here a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, and nonlinear PDEs. We embed the PDEs in a truncated, high dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods that use a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of scaling of matrix operator norms, condition number, simulation time, and accuracy. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts and query complexity. A physically motivated measure of nonlinearity is connected to a parameter that is similar to the flow Reynolds number $Re_{\textrm{H}}$, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of a one-dimensional Burgers problem. This work shows the potential of the hybrid quantum algorithm for simulating practical and nonlinear phenomena on near-term and fault-tolerant quantum devices. Comments: Subjects: Quantum Physics (quant-ph); Computational Engineering, Finance, and Science (cs.CE); Applied Physics (physics.app-ph); Computational Physics (physics.comp-ph); Fluid Dynamics (physics.flu-dyn) Cite as: arXiv:2512.21033 [quant-ph] (or arXiv:2512.21033v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.21033 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Sachin Satish Bharadwaj [view email] [v1] Wed, 24 Dec 2025 07:56:34 UTC (548 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems, by Sachin S. Bharadwaj and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: cs cs.CE physics physics.app-ph 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?)