The 2-Category of Topological Quantum Computation

TL;DR

This paper proposes a braided fusion 2-category framework that unifies anyonic theories and topological quantum computing models, clarifying their categorical structures and superselection rules.

Contribution

It introduces a braided fusion 2-category formalism that captures both anyonic theories and topological quantum computing models in a unified way.

Findings

A 2-category formalism unifies anyonic theories and quantum computing models.

Clarifies the distinction between fusion of anyons and tensor products in linear algebra.

Provides insights into superselection rules in topological quantum computation.

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.