Provably Efficient Quantum Algorithms for Solving Nonlinear Differential Equations Using Multiple Bosonic Modes Coupled with Qubits

Quantum Physics arXiv:2511.09939 (quant-ph) [Submitted on 13 Nov 2025] Title:Provably Efficient Quantum Algorithms for Solving Nonlinear Differential Equations Using Multiple Bosonic Modes Coupled with Qubits Authors:Yu Gan, Hirad Alipanah, Jinglei Cheng, Zeguan Wu, Guangyi Li, Juan José Mendoza-Arenas, Peyman Givi, Mujeeb R. Malik, Brian J. McDermott, Junyu Liu View a PDF of the paper titled Provably Efficient Quantum Algorithms for Solving Nonlinear Differential Equations Using Multiple Bosonic Modes Coupled with Qubits, by Yu Gan and 9 other authors View PDF HTML (experimental) Abstract:Quantum computers have long been expected to efficiently solve complex classical differential equations. Most digital, fault-tolerant approaches use Carleman linearization to map nonlinear systems to linear ones and then apply quantum linear-system solvers. However, provable speedups typically require digital truncation and full fault tolerance, rendering such linearization approaches challenging to implement on current hardware. Here we present an analog, continuous-variable algorithm based on coupled bosonic modes with qubit-based adaptive measurements that avoids Hilbert-space digitization. This method encodes classical fields as coherent states and, via Kraus-channel composition derived from the Koopman-von Neumann (KvN) formalism, maps nonlinear evolution to linear dynamics. Unlike many analog schemes, the algorithm is provably efficient: advancing a first-order, $L$-grid point, $d$-dimensional, order-$K$ spatial-derivative, degree-$r$ polynomial-nonlinearity, strongly dissipative partial differential equations (PDEs) for $T$ time steps costs $\mathcal{O}\left(T(\log L + d r \log K)\right)$. The capability of the scheme is demonstrated by using it to simulate the one-dimensional Burgers' equation and two-dimensional Fisher-KPP equation. The resilience of the method to photon loss is shown under strong-dissipation conditions and an analytic counterterm is derived that systematically cancels the dominant, experimentally calibrated noise. This work establishes a continuous-variable framework for simulating nonlinear systems and identifies a viable pathway toward practical quantum speedup on near-term analog hardware. Subjects: Quantum Physics (quant-ph); Computational Engineering, Finance, and Science (cs.CE) Cite as: arXiv:2511.09939 [quant-ph] (or arXiv:2511.09939v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2511.09939 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Yu Gan [view email] [v1] Thu, 13 Nov 2025 04:09:32 UTC (2,591 KB) Full-text links: Access Paper: View a PDF of the paper titled Provably Efficient Quantum Algorithms for Solving Nonlinear Differential Equations Using Multiple Bosonic Modes Coupled with Qubits, by Yu Gan and 9 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-11 Change to browse by: cs cs.CE 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?)