Systematic Computation of Braid Generator Matrix in Topological Quantum Computing

TL;DR

This paper introduces a systematic numerical method for computing elementary braid matrices in topological quantum computing, applicable to all anyon models, enabling complex quantum circuit simulations and state preparations.

Contribution

The paper presents a novel, general numerical approach for braid matrix computation in TQC, including new braiding moves and algorithms for systems with many anyons and complex fusion patterns.

Findings

Validated by simulating an approximate CNOT gate

Successfully simulated a GHZ state with five qubits using Fibonacci anyons

Method is effective across various anyonic models

Abstract

We provide a comprehensive systematic method for the numerical computation of elementary braid operations in topological quantum computation (TQC). This {procedure} is systematically applicable to all anyon models, including $SU(2)_k$. Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.