Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes
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Quantum Physics arXiv:2512.13777 (quant-ph) [Submitted on 15 Dec 2025] Title:Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes Authors:Alison Warman, Sakura Schafer-Nameki View a PDF of the paper titled Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes, by Alison Warman and 1 other authors View PDF HTML (experimental) Abstract:We present a purely 2D transversal realization of phase gates at any level of the Clifford hierarchy, and beyond, using non-Abelian surface codes. Our construction encodes a logical qubit in the quantum double $D(G)$ of a non-Abelian group $G$ on a triangular spatial patch. The logical gate is implemented transversally by stacking on the spatial region a symmetry-protected topological (SPT) phase specified by a group 2-cocycle. The Bravyi--König theorem limits the unitary gates implementable by constant-depth quantum circuits on Pauli stabilizer codes in $D$ dimensions to the $D$-th level of the Clifford hierarchy. We bypass this, by constructing transversal unitary gates at arbitrary levels of the Clifford hierarchy purely in 2D, without sacrificing locality or fault tolerance, however at the cost of using the quantum double of a non-Abelian group $G$. Specifically, for $G = D_{4N}$, the dihedral group of order $8N$, we realize the phase gate $T^{1/N} = \mathrm{diag}(1, e^{i\pi/(4N)})$ in the logical $\overline{Z}$ basis. For $8N = 2^n$, this gate lies at the $n$-th level of the Clifford hierarchy and, importantly, has a qubit-only realization: we show that it can be constructed in terms of Clifford-hierarchy stabilizers for a code with $n$ physical qubits on each edge of the lattice. We also discuss code-switching to the $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_2$ toric codes, which can be utilized for the quantum error correction in this setup. Comments: Subjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph) Cite as: arXiv:2512.13777 [quant-ph] (or arXiv:2512.13777v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.13777 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Alison Warman [view email] [v1] Mon, 15 Dec 2025 19:00:00 UTC (39 KB) Full-text links: Access Paper: View a PDF of the paper titled Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes, by Alison Warman and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: cond-mat cond-mat.str-el hep-th 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?) Links to Code Toggle Papers with Code (What is Papers with Code?) 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?)