Introduction to topological quantum computation with non-Abelian anyons

TL;DR

This paper provides a comprehensive introduction to topological quantum computation using Fibonacci anyons, explaining their braiding properties, computer architecture, and demonstrating quantum algorithms for knot invariants.

Contribution

It offers a pedagogical review of Fibonacci anyons for topological quantum computing and demonstrates explicit simulations of quantum algorithms for knot invariants.

Findings

Simulation of a topological quantum computer performing knot polynomial calculations

Demonstration of exact computation of Jones polynomial at specific points using Fibonacci anyons

Educational overview of braiding, matrices, and computer layout for topological quantum computing

Abstract

Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.