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arXiv:2212.02921

The ribbon category framework for topological quantum computing

Willie Aboumrad

Published Dec 6, 2022• 0 citations
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Abstract

This expository article supplies the mathematical background underpinning the braid representation calculator introduced in arXiv:2212.00831; those representations describe the sets of logic gates available to a topological quantum computer for processing encoded qubits. Assuming little background in category theory, we first recall the notion of a ribbon fusion category (RFC), collecting most of the necessary definitions. Then we discuss how certain RFCs arise from the representation theory of quantum groups. We explore the braiding in these categories in detail, since it is essential for the quantum computing application.

DOI
10.48550/arxiv.2212.02921

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This expository article supplies the mathematical background underpinning the braid representation calculator introduced in arXiv:2212.00831; those representations describe the sets of logic gates available to a topological quantum computer for processing encoded qubits. Assuming little background in category theory, we first recall the notion of a ribbon fusion category (RFC), collecting most of the necessary definitions. Then we discuss how certain RFCs arise from the representation theory of quantum groups. We explore the braiding in these categories in detail, since it is essential for the quantum computing application.
Quantum computing with anyons: an FFF-matrix and braid calculator
We introduce a pentagon equation solver, available as part of SageMath, and use it to construct braid group representations associated to certain anyon systems. We recall the category-theoretic framework for topological quantum computation to explain how these representations describe the sets of logical gates available to an anyonic quantum computer for information processing. In doing so, we avoid venturing deep into topological or conformal quantum field theory. Instead, we present anyons abstractly as sets of labels together with a collection of data satisfying a number of axioms, including the pentagon and hexagon equations, and explain how these data characterize ribbon fusion categories (RFCs). In the language of RFCs, our solver can produce FFF-matrices for anyon systems corresponding to multiplicity-free fusion rings arising in connection with the representation theory of quantum groups associated to simple Lie algebras with deformation parameter a root of unity.
Topological quantum computation offers a paradigm shift for fault-tolerant quantum computation. It proposes encoding qubits in the mutual states of quasiparticles (called anyons) of 2-dimensional topologically ordered matter. Unitary ribbon fusion categories describe the algebraic theory of anyons. Although the category of an anyonic theory specifies suitable computational spaces, qubits, and quantum gates, it does not precisely formalise the category of topological quantum computation (TQC). We describe the TQC category using the elements of categorical quantum mechanics (CQM). Given the qubit-based graphical calculus accompanying CQM, called the ZX-calculus, we present the details of the ZX-representation of Fibonacci and Ising anyons. Finally, we propose a bicategorical approach to convene anyonic theory and TQC in one (bi)category. In the second part of the thesis, we discuss how the bicategory of module categories formalises boundary types and their excitations. We show theorems that reduce the data to physically meaningful data. A 2-category capable of describing topological quantum computation with point-like and string-like excitations is speculated to be a fusion 2-category. We discuss the available definition in the literature. The paper particularly lacks a rigorous definition for additive 2-categories; in this thesis, we give a definition. We propose the idea of typing higher-ranked matrices in additive 2-categories. Lastly, we overview the complete definition of monoidal 2-categories and recover Kapranov and Voevodsky's definition from the algebraic definition of tricategory by Gurski. This definition and the detailed list of data are the complete ingredients physicists need for quantum computation with higher dimensional excitations.
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.
The aim of this thesis is to develop an abstract categorical setup in order to show that C -colored manifolds (i.e. compact closed manifolds with boundary where each boundary component is colored with an object of a semisimple strongly ribbon category) behaves basically in a similar manner as quantum circuits under the action of a unitary modular functor. There, the set of gates is composed only of braid operations, rotations and Dehn-twists. We first introduce the basic mathematical structure of a quantum circuit. We then provide a complete development of a 2-dimensional CW-complex over an extended surface. Furthermore, we provide a complete development of the categorical framework in order to construct a C -extended unitary modular functor (UMF) acting from the category of C -colored surfaces and morphisms of C -colored surfaces to the category of finite-dimensional vector spaces and linear isomorphisms. We then conclude by giving a complete semantics for topological quantum computation including an abstract version of the inner product, basic data units, basic data transformations, projectors and the notion of topological invariance of the algorithms.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.