  A method, termed controlled-injection, is proposed for compiling three-qubit
controlled gates within the non-abelian Fibonacci anyon model. Building on
single-qubit compilation techniques with three Fibonacci anyons, the approach
showcases enhanced accuracy and reduced braid length compared to the
conventional decomposition method for the controlled three-qubit gates. This
method necessitates only four two-qubit gates for decomposition, a notable
reduction from the conventional five. In conjunction, the study introduces a
novel class of controlled three-qubit gates and conducts a numerical simulation
of the topological iToffoli gate to validate the approach. In addition, we
propose an optimization method for single-qubit gate approximation using novel
algebraic relations and numerical methods, including distributed computing.
