Introduction to topological quantum computation with non-Abelian anyons

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibona...

Topological quantum computation may provide a robust approach for encoding and manipulating information utilizing the topological properties of anyonic quasi-particle excitations. We develop an efficient means to map between dense and sparse representations of quantum information (qubits) and a simple construction of multi-qubit gates, for all anyon models from Chern-Simons-Witten SU(2)$_k$ theory that support universal quantum computation by braiding ($k\geq 3,\ k \neq 4$). ...

We present a systematic numerical method to compute the elementary braiding operations for topological quantum computation (TQC). Braiding non-Abelian anyons is a crucial technique in TQC, offering a topologically protected implementation of quantum gates. However, obtaining matrix representations for braid generators can be challenging, especially for systems with numerous anyons or complex fusion patterns. Our proposed method addresses this challenge, allowing for the inclu...

We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of `knot invariants', among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical comp...

Fibonacci string-net condensate, a complex topological state that supports non-Abelian anyon excitations, holds promise for fault-tolerant universal quantum computation. However, its realization by a static-lattice Hamiltonian has remained elusive due to the inherent high-order interactions demanded. Here, we introduce a scalable dynamical string-net preparation (DSNP) approach, suitable even for near-term quantum processors, that can dynamically prepare the state through rec...

This paper explores the interactions between knot theory and quantum computing. On one side, knot theory has been used to create models of quantum computing, and on the other, it is a source of computational problems. Knot theory is often used to introduce topological idea to people without a formal mathematical background, and we are building on this tradition to discuss some of the deeper ideas of quantum computing.

The fusion basis of Fibonacci anyons supports unitary braid representations that can be utilized for universal quantum computation. We show a mapping between the fusion basis of three Fibonacci anyons, $\{|1\rangle, |\tau\rangle\}$, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard $(2,2)$ Young diagrams using the Jones construction. This corre...

The content of this thesis can be broadly summarised into two categories: first, I constructed modified numerical algorithms based on tensor networks to simulate systems of anyons in low dimensions, and second, I used those methods to study the topological phases the anyons form when they braid around one another. In the first phase of my thesis, I extended the anyonic tensor network algorithms, by incorporating U(1) symmetry to give a modified ansatz, Anyon-U(1) tensor netwo...

We discuss how to significantly reduce leakage errors in topological quantum computation by introducing an irrelevant error in phase, using the construction of a CNOT gate in the Fibonacci anyon model as a concrete example. To be specific, we construct a functional braid in a six-anyon Hilbert space that exchanges two neighboring anyons while conserving the encoded quantum information. The leakage error is $\sim$$10^{-10}$ for a braid of $\sim$100 interchanges of anyons. Appl...

We provide an elementary introduction to topological quantum computation based on the Jones representation of the braid group. We first cover the Burau representation and Alexander polynomial. Then we discuss the Jones representation and Jones polynomial and their application to anyonic quantum computation. Finally we outline the approximation of the Jones polynomial and explicit localizations of braid group representations.