Achieving the limits of automorphism gates

Quantum Physics arXiv:2609.19250 (quant-ph) [Submitted on 16 Sep 2026] Title:Achieving the limits of automorphism gates Authors:Jin Ming Koh, Shayan Majidy, Aranya Chakraborty, Anqi Gong, Shi Jie Samuel Tan, Norman Y. Yao View a PDF of the paper titled Achieving the limits of automorphism gates, by Jin Ming Koh and 5 other authors View PDF HTML (experimental) Abstract:Universal fault-tolerant quantum computing combines versatile but expensive operations with specialized but cheap ones. Its efficiency depends on how much computation can be pushed onto the cheap operations and on the size of the code needed to do so. Automorphism gates provide such cheap operations using only physical single-qubit Clifford gates and qubit permutations. Yet no general theory characterizes their maximum logical power or the minimum code size needed to attain it. We develop such a theory. For stabilizer codes encoding $k\geq3$ logical qubits, we show that the largest logical group attainable by automorphisms is generated by all addressable $S$ and $\mathrm{CX}$ gates, and we construct codes attaining it. While this group contains exponentially fewer gates than the full Clifford group, adding one suitable non-Clifford gate yields universality. We further classify the largest logical groups attainable using qubit permutations, physical single-qubit Cliffords, or both across general stabilizer and CSS codes, and derive refined bounds for self-dual CSS subclasses. Achieving the maximum-size logical group through automorphisms requires $n=\Theta(2^k)$ physical qubits. By contrast, all addressable diagonal Clifford gates, generated by $S$ and $\mathrm{CZ}$, require only $n=\Theta(k^2)$ physical qubits when implemented using physical single-qubit Cliffords alone. Both bounds are tight. This polynomial qubit cost extends beyond Cliffords to all addressable diagonal gates at any fixed level of the Clifford hierarchy, using physical single-qubit diagonal gates. Thus, for full addressability, the sharpest physical-qubit cost divide lies between diagonal and $\mathrm{CX}$-type gates, not between Clifford and non-Clifford gates. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2609.19250 [quant-ph] (or arXiv:2609.19250v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2609.19250 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Shayan Majidy [view email] [v1] Wed, 16 Sep 2026 18:00:00 UTC (4,871 KB) Full-text links: Access Paper: View a PDF of the paper titled Achieving the limits of automorphism gates, by Jin Ming Koh and 5 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-09 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?)