Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations
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Quantum Physics arXiv:2605.27408 (quant-ph) [Submitted on 12 May 2026] Title:Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations Authors:Chanyoung Kim, Myeonghwan Seong, Yujin Kim, Daniel K. Park, Youngjoon Hong View a PDF of the paper titled Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations, by Chanyoung Kim and 4 other authors View PDF HTML (experimental) Abstract:Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning. Comments: Subjects: Quantum Physics (quant-ph); Machine Learning (cs.LG); Numerical Analysis (math.NA) Cite as: arXiv:2605.27408 [quant-ph] (or arXiv:2605.27408v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.27408 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Chayoung Kim [view email] [v1] Tue, 12 May 2026 10:30:14 UTC (9,270 KB) Full-text links: Access Paper: View a PDF of the paper titled Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations, by Chanyoung Kim and 4 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: cs cs.LG cs.NA math 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?) 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?)