On Quantum Computation, Anyons, and Categories

Andreas R. Blass, Yuri G. Gurevich

We explain the use of category theory in describing certain sorts of anyons. Yoneda's lemma leads to a simplification of that description. For the particular case of Fibonacci anyons, we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.

Prakash Panangaden, Eric Olive Paquette

Seth Lloyd

Andreas R. Blass, Yuri G. Gurevich

We explain the use of category theory in describing certain sorts of anyons . Yoneda’s lemma leads to a simplification of that description. For the particular case of Fibonacci anyons , we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.

Ortiz Marrero, Merino Carlos

Ahmadi, Fatimah Rita

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.