Optimized topological quantum compilation of three-qubit controlled gates in the Fibonacci anyon model: A controlled-injection approach

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 class of controlled three-qubit gates and conducts a numerical simulation of the topological $i\mathrm{Toffoli}$ gate to validate the approach. In addition, we propose an optimization method for single-qubit gate approximation using algebraic relations and numerical methods, including distributed computing.