Abstract A quantum computer requires the ability to store and manipulate information globally to protect against local noise. Topologically ordered phases1,2 offer two routes: encoding information in the ground-state subspace3 or in anyonic excitations1,4,5. The toric code1 exemplifies the first approach but does not intrinsically support a universal gate set. The latter—topological quantum computation—implements gates by braiding non-Abelian anyons6 around each other. However, the simplest non-Abelian generalizations of the toric code cannot achieve universality by braiding alone7,8,9. Here we demonstrate that anyon fusion, used as a computational primitive, renders these minimally non-Abelian topologically ordered states universal. We prepare a 54-qubit ground state of the quantum double of S3, the smallest non-Abelian group, on the H2 processor of Quantinuum. We encode logical information in the global fusion space of non-Abelian anyons, and by combining braiding with fusion, we realize a universal topological gate set and read-out, which we demonstrate by topologically preparing a magic state. This demonstrates that the S3 topologically ordered state is scalably preparable, yet rich enough to support a universal gate set. More broadly, this work opens up new pathways for harnessing the intrinsic properties of quantum matter to manipulate quantum information. This is a preview of subscription content, access via your institution Access options Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any time Subscribe to this journal Receive 52 print issues and online access $199.00 per year only $3.83 per issue Buy this article - Purchase on SpringerLink - Instant access to the full article PDF. USD 39.95 Prices may be subject to local taxes which are calculated during checkout Similar content being viewed by others Data availability The data generated in this study are available at Zenodo55 (https://doi.org/10.5281/zenodo.18054264) under open access. Code availability The