Krylov-Lie Algebras for Variational Quantum Algorithms: Geometric, Depth-Aware Insights into Expressivity and Trainability

Quantum Physics arXiv:2607.02626 (quant-ph) [Submitted on 2 Jul 2026] Title:Krylov-Lie Algebras for Variational Quantum Algorithms: Geometric, Depth-Aware Insights into Expressivity and Trainability Authors:Anžej Margeta-Cacace View a PDF of the paper titled Krylov-Lie Algebras for Variational Quantum Algorithms: Geometric, Depth-Aware Insights into Expressivity and Trainability, by An\v{z}ej Margeta-Cacace View PDF HTML (experimental) Abstract:Variational quantum algorithms (VQAs) are a leading approach to near-term quantum computation, but their utility is limited by barren plateaus and other pathologies in their loss landscapes. Existing landscape theories based on dynamical Lie algebras, Jordan-algebraic Wishart systems, approximate t-designs, and Haar-random circuits are foundational, but they often neglect the finite-depth geometry of realistic ansätze and are therefore poorly suited to the shallow-depth regime, where VQAs are poor approximators of 2-designs and trainability is most feasible. This thesis introduces Krylov algebras, algebraic structures induced by the Krylov span of a finite generator set acting on one or more seed vectors, as a framework for VQA landscape theory. We show that VQA reachable manifolds can be approximated in a numerically robust, geometrically faithful way by Krylov-Lie algebras and groups, and that these structures induce canonical invariant measures for computing expectation values and variances under general sampling measures. In particular, we derive weighted non-Haar variance formulas that recover the usual Lie-algebraic Haar formulas as a special case while isolating non-Haar effects into explicit correction terms. We also show that the common heuristic that sufficiently deep circuit ensembles must converge to Haar fails in general without additional hypotheses, identify concrete obstructions to naive Haar convergence, and recover convergence under natural necessary and sufficient ergodic conditions. Lastly, our formulas further imply that non-Haar contributions may mitigate barren plateaus by reweighting the visible sectors of the loss landscape, suggesting that VQAs may be more trainable than recent literature has posited. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) MSC classes: 81Q93 (Primary) 81R05, 22E30, 65F60 (Secondary) Cite as: arXiv:2607.02626 [quant-ph] (or arXiv:2607.02626v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2607.02626 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Anžej Margeta-Cacace [view email] [v1] Thu, 2 Jul 2026 12:39:30 UTC (115 KB) Full-text links: Access Paper: View a PDF of the paper titled Krylov-Lie Algebras for Variational Quantum Algorithms: Geometric, Depth-Aware Insights into Expressivity and Trainability, by An\v{z}ej Margeta-CacaceView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-07 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?)