Unified approach to topological quantum computation with anyons: From qubit encoding to Toffoli gate

TL;DR

This paper presents a unified framework for topological quantum computation using anyons, enabling efficient qubit encoding, multi-qubit gates, and connecting different anyon models within a simple, comprehensive approach.

Contribution

It introduces a unified method for mapping qubit representations and constructing multi-qubit gates across all universal anyon models from Chern-Simons-Witten theory.

Findings

Efficient mapping between dense and sparse qubit representations.

Construction of multi-qubit gates for all universal anyon models.

Unified framework connecting different anyon models.

Abstract

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 ($k\geq 3,\ k \neq 4$). In the process, we show how the constructions of topological quantum memory and gates for $k=2,4$ connect naturally to those for $k\geq 3,\ k \neq 4$, unifying these concepts in a simple framework. Furthermore, we illustrate potential extensions of these ideas to other anyon models outside of Chern-Simons-Witten field theory.