Proof graph
The lower-bound chain from Sections 3-7 and the explicit five-gate construction meet at the exported necessary-and-sufficient theorem.
Five arbitrary two-qubit gates are necessary and sufficient to implement the three-qubit Toffoli gate exactly.
TwoControl.Top100.five_two_qubit_gates_are_necessary_and_sufficient_to_implement_toffoli
The lower-bound chain from Sections 3-7 and the explicit five-gate construction meet at the exported necessary-and-sufficient theorem.