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 ($k≥ 3,\ k \neq 4$). In the process, we show how the constructions of topological quantum memory and gates for $k=2,4$ connect naturally to those for $k≥ 3,\ k \neq 4$, 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. ; Comment: 4.5 pages, 3 figures, published in PRA