Two paradigms for topological quantum computation

TL;DR

This paper explores two frameworks connecting algebraic, topological, and quantum computational aspects of topological quantum computing, providing evidence, examples, and conjectures to support these paradigms.

Contribution

It introduces two new paradigms linking computational power, link invariants, and braid group representations in topological quantum computation, with supporting evidence and conjectures.

Findings

Supported the paradigms with recent results

Provided a comprehensive list of known examples

Formulated two conjectures to strengthen the paradigms

Abstract

We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum computers, computational complexity of link invariants and images of braid group representations. While at least parts of these paradigms are well-known to experts, we provide supporting evidence for them in terms of recent results. We give a fairly comprehensive list of known examples and formulate two conjectures that would further support the paradigms.