Quantum algorithms for Young measures: applications to nonlinear partial differential equations
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Quantum Physics arXiv:2604.11825 (quant-ph) [Submitted on 11 Apr 2026] Title:Quantum algorithms for Young measures: applications to nonlinear partial differential equations Authors:Shi Jin, Nana Liu, Maria Lukacova-Medvidova, Yuhuan Yuan View a PDF of the paper titled Quantum algorithms for Young measures: applications to nonlinear partial differential equations, by Shi Jin and 3 other authors View PDF HTML (experimental) Abstract:Many nonlinear PDEs have singular or oscillatory solutions or may exhibit physical instabilities or uncertainties. This requires a suitable concept of physically relevant generalized solutions. Dissipative measure-valued solutions have been an effective analytical tool to characterize PDE behavior in such singular regimes. They have also been used to characterize limits of standard numerical schemes on classical computers. The measure-valued formulation of a nonlinear PDE yields an optimization problem with a linear cost functional and linear constraints, which can be formulated as a linear programming problem. However, this linear programming problem can suffer from the curse of dimensionality. In this article, we propose solving it using quantum linear programming (QLP) algorithms and discuss whether this approach can reduce costs compared to classical algorithms. We show that some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method. For problems where one is interested in the dissipative weak solution, namely the expected values of the Young measure, we show that the QLP algorithms offer no advantage over direct classical solvers. Moreover, for random PDEs, there can be up to polynomial advantage in obtaining the Young measure over direct classical PDE solvers. This is a significant advantage over standard PDE solvers, since the Young measure provides a more detailed description of the solution. We also propose some open questions for future development in this direction. Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) Cite as: arXiv:2604.11825 [quant-ph] (or arXiv:2604.11825v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.11825 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Nana Liu [view email] [v1] Sat, 11 Apr 2026 11:37:24 UTC (862 KB) Full-text links: Access Paper: View a PDF of the paper titled Quantum algorithms for Young measures: applications to nonlinear partial differential equations, by Shi Jin and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 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?) 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?)