Quantum computing with anyons: an $F$-matrix and braid calculator

TL;DR

This paper presents a computational tool integrated into SageMath for solving pentagon equations, enabling the construction of braid group representations for anyon systems in topological quantum computing, without requiring deep field theory knowledge.

Contribution

We develop a pentagon equation solver in SageMath that constructs braid group representations for anyon systems, facilitating topological quantum computation modeling.

Findings

Solver produces $F$-matrices for multiplicity-free fusion rings.

Represents logical gates in anyonic quantum computers.

Connects fusion categories with quantum group representations.

Abstract

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 $F$-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.