Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalisation method

Quantum Physics arXiv:2512.07919 (quant-ph) [Submitted on 8 Dec 2025] Title:Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalisation method Authors:Shi Jin, Nana Liu View a PDF of the paper titled Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalisation method, by Shi Jin and Nana Liu View PDF HTML (experimental) Abstract:We present a framework for efficient extraction of the viscosity solutions of nonlinear Hamilton-Jacobi equations with convex Hamiltonians. These viscosity solutions play a central role in areas such as front propagation, mean-field games, optimal control, machine learning, and a direct application to the forced Burgers' equation. Our method is based on an entropy penalisation method proposed by Gomes and Valdinoci, which generalises the Cole-Hopf transform from quadratic to general convex Hamiltonians, allowing a reformulation of viscous Hamilton-Jacobi dynamics by a discrete-time linear dynamics which approximates a linear heat-like parabolic equation, and can also extend to continuous-time dynamics. This makes the method suitable for quantum simulation. The validity of these results hold for arbitrary nonlinearity that correspond to convex Hamiltonians, and for arbitrarily long times, thus obviating a chief obstacle in most quantum algorithms for nonlinear partial differential equations. We provide quantum algorithms, both analog and digital, for extracting pointwise values, gradients, minima, and function evaluations at the minimiser of the viscosity solution, without requiring nonlinear updates or full state reconstruction. Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) Cite as: arXiv:2512.07919 [quant-ph] (or arXiv:2512.07919v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.07919 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Nana Liu [view email] [v1] Mon, 8 Dec 2025 14:19:32 UTC (88 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalisation method, by Shi Jin and Nana LiuView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: math math-ph math.MP 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?)