Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing
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Quantum Physics arXiv:2603.27004 (quant-ph) [Submitted on 27 Mar 2026] Title:Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing Authors:Ahmed Adel Mahmoud, Gabrielle Tournaire, Sven Bachmann, Steven Rayan View a PDF of the paper titled Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing, by Ahmed Adel Mahmoud and 3 other authors View PDF HTML (experimental) Abstract:Fault-tolerant measurement-based quantum computing (MBQC) provides a compelling framework for fault-tolerant quantum computation, in which quantum information is processed through single-qubit measurements on a three-dimensional entangled resource known as cluster state. To date, this resource has been predominantly studied on Euclidean lattices, most notably in the Raussendorf-Harrington-Goyal (RHG) construction, which underlies topological fault tolerance in MBQC. In this work, we introduce the hyperbolic cluster state, a generalization of the three-dimensional cluster state to negatively curved geometries, obtained via the foliation of periodic hyperbolic lattices. We present an explicit construction of hyperbolic cluster states and investigate their fault-tolerant properties under a realistic circuit-level depolarizing noise model. Using large-scale numerical simulations, we perform memory experiments to characterize their logical error rates and decoding performance. Our results demonstrate that hyperbolic cluster states exhibit a fault-tolerance threshold comparable to that of the Euclidean RHG cluster state, while simultaneously supporting a constant encoding rate in the thermodynamic limit. This represents a substantial improvement in qubit overhead relative to conventional cluster-state constructions. These findings establish hyperbolic geometry as a powerful and experimentally relevant resource for scalable, fault-tolerant MBQC and open new avenues for leveraging negative curvature in quantum information processing. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Algebraic Topology (math.AT) Cite as: arXiv:2603.27004 [quant-ph] (or arXiv:2603.27004v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2603.27004 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Ahmed Adel Mahmoud [view email] [v1] Fri, 27 Mar 2026 21:28:52 UTC (1,212 KB) Full-text links: Access Paper: View a PDF of the paper titled Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing, by Ahmed Adel Mahmoud and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-03 Change to browse by: math math-ph math.AG math.AT 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?)