Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing

Quantum Physics [Submitted on 27 Mar 2026] Hyperbolic Cluster States for Fault-Tolerant Measurement-Based Quantum Computing Ahmed Adel Mahmoud, Gabrielle Tournaire, Sven Bachmann, Steven Rayan 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: 19 pages, 6 figures, 1 table 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 Submission history From: Ahmed Adel Mahmoud [view email] [v1] Fri, 27 Mar 2026 21:28:52 UTC (1,212 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 Change to browse by: math math-ph math.AG math.AT math.MP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)