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

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