Braid Matrices and Quantum Gates for Ising Anyons Topological Quantum Computation

TL;DR

This paper explores the mathematical framework and quantum gate construction for topological quantum computation using Ising anyons, focusing on braiding matrices, encoding schemes, and their implications for quantum information processing.

Contribution

It introduces a normalization of braiding matrices based on Temperley-Lieb theory and constructs quantum gates for Ising anyons, providing alternative proofs and comparisons with other anyon models.

Findings

Derived unitary braiding matrices for Ising anyons

Proved a no-entanglement theorem for a specific encoding scheme

Constructed quantum gates equivalent to known models

Abstract

We study various aspects of the topological quantum computation scheme based on the non-Abelian anyons corresponding to fractional quantum hall effect states at filling fraction 5/2 using the Temperley-Lieb recoupling theory. Unitary braiding matrices are obtained by a normalization of the degenerate ground states of a system of anyons, which is equivalent to a modification of the definition of the 3-vertices in the Temperley-Lieb recoupling theory as proposed by Kauffman and Lomonaco. With the braid matrices available, we discuss the problems of encoding of qubit states and construction of quantum gates from the elementary braiding operation matrices for the Ising anyons model. In the encoding scheme where 2 qubits are represented by 8 Ising anyons, we give an alternative proof of the no-entanglement theorem given by Bravyi and compare it to the case of Fibonacci anyons model. In the encoding scheme where 2 qubits are represented by 6 Ising anyons, we construct a set of quantum gates which is equivalent to the construction of Georgiev.