The 2-Category of Topological Quantum Computation

• URL: http://arxiv.org/abs/2505.22171v2
• Date: Thu, 29 May 2025 21:25:27 GMT
• Title: The 2-Category of Topological Quantum Computation
• Authors: Fatimah Rita Ahmadi,
• Abstract summary: It has been widely assumed that the same category formalizes a topological quantum computing model.<n>In this paper, we argue that a categorical formalism that captures and unifies both anyonic theories and a model of topological quantum computing is a braided (fusion) 2-category.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Unitary Ribbon Fusion Categories (URFC) formalize anyonic theories. It has been widely assumed that the same category formalizes a topological quantum computing model. However, in previous work, we addressed and resolved this confusion and demonstrated while the former could be any fusion category, the latter is always a subcategory of Hilb. In this paper, we argue that a categorical formalism that captures and unifies both anyonic theories (the Hardware of quantum computing) and a model of topological quantum computing is a braided (fusion) 2-category. In this 2-category, 0-morphisms describe anyonic types and Hom-categories describe different models of quantum computing. This picture provides an insightful perspective on superselection rules. It presents furthermore a clear distinction between fusion of anyons versus tensor products as defined in linear algebra, between vector spaces of 1-morphisms. The former represents a monoidal product and sum between 0-morphisms and the latter a tensor product and direct sum between 1-morphisms.

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