The true cost of factoring: Linking magic and number-theoretic complexity in Shor's algorithm
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Quantum Physics arXiv:2605.05347 (quant-ph) [Submitted on 6 May 2026] Title:The true cost of factoring: Linking magic and number-theoretic complexity in Shor's algorithm Authors:Alessio Paviglianiti, Matteo Seclì, Emanuele Tirrito, Vincenzo Savona View a PDF of the paper titled The true cost of factoring: Linking magic and number-theoretic complexity in Shor's algorithm, by Alessio Paviglianiti and Matteo Secl\`i and Emanuele Tirrito and Vincenzo Savona View PDF HTML (experimental) Abstract:The execution cost of quantum algorithms is typically quantified through asymptotic gate counts and qubit register sizes, yet these metrics do not directly capture which genuinely quantum resources, and in what amount, must be created and maintained for the computation to succeed. The systematic quantification of such information-theoretic requirements in quantum computing protocols remains an extremely challenging open problem, despite their direct role in establishing quantum advantage. To address this gap, we investigate the generation of non-stabilizerness (or magic), one of the key resources, in the paradigmatic Shor's factoring algorithm, revealing a deep connection between intrinsic quantum complexity and the computational hardness of the underlying number-theoretic problem. By developing an explicit analytic theory, we demonstrate the fundamental role of magic in the successful execution of the algorithm, and show that Shor's routine maximally exploits the quantum resource in practically relevant regimes. Our findings create a concise conceptual link between the classical algorithmic difficulty of a task and the non-stabilizer price to solve it on quantum hardware, complementing standard circuit-cost analyses with a resource-based metric that is naturally aligned with the real bottlenecks of fault-tolerant quantum computing. Comments: Subjects: Quantum Physics (quant-ph); Computational Physics (physics.comp-ph) Cite as: arXiv:2605.05347 [quant-ph] (or arXiv:2605.05347v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.05347 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Alessio Paviglianiti [view email] [v1] Wed, 6 May 2026 18:16:29 UTC (262 KB) Full-text links: Access Paper: View a PDF of the paper titled The true cost of factoring: Linking magic and number-theoretic complexity in Shor's algorithm, by Alessio Paviglianiti and Matteo Secl\`i and Emanuele Tirrito and Vincenzo SavonaView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: physics physics.comp-ph 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?)