Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions
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Quantum Physics arXiv:2604.09957 (quant-ph) [Submitted on 10 Apr 2026] Title:Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions Authors:Prasad Nimantha Madusanka Ukwatta Hewage, Midhun Chakkravarthy, Ruvan Kumara Abeysekara View a PDF of the paper titled Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions, by Prasad Nimantha Madusanka Ukwatta Hewage and 2 other authors View PDF HTML (experimental) Abstract:The barren plateau phenomenon; where cost function gradients vanish exponentially with system size; remains a fundamental obstacle to training variational quantum circuits (VQCs) at scale. We demonstrate, both theoretically and numerically, that embedding partial differential equation (PDE) constraints into the VQC loss function provides a natural and effective mitigation mechanism against barren plateaus. We derive analytical gradient variance lower bounds showing that physics-constrained loss functions composed of local PDE residuals evaluated at spatial collocation points inherit the favorable polynomial scaling of local cost functions, while additionally benefiting from constraint-induced landscape narrowing that concentrates gradient information. Systematic numerical experiments on the one-dimensional heat equation, Burgers' equation, and the Saint-Venant shallow water equations quantify the gradient variance across 4-8 qubits and 1-5 layer depths, comparing global cost, local cost, PDE-constrained, and PDE-constrained with structured ansatz configurations. We find that PDE-constrained circuits exhibit favorable gradient variance scaling with system size, with the physics constraints creating a stabilizing effect that resists exponential gradient vanishing. Entanglement entropy analysis reveals that structured ansatze operate in a sub-maximal entanglement regime consistent with trainability. Convergence experiments confirm that physics-constrained VQCs achieve lower loss values in fewer epochs. These results establish PDE constraints as a principled, physically motivated strategy for designing trainable variational quantum circuits, with direct implications for quantum physics-informed neural networks and variational quantum simulation. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.09957 [quant-ph] (or arXiv:2604.09957v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.09957 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Prasad Nimantha Madusanka Ukwatta Hewage [view email] [v1] Fri, 10 Apr 2026 23:36:59 UTC (84 KB) Full-text links: Access Paper: View a PDF of the paper titled Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions, by Prasad Nimantha Madusanka Ukwatta Hewage and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-04 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?)