Quantum Fourier addition simplified to Toffoli addition

Quantum addition circuits are considered being of two types: (1) Toffoli-adder which use only classical reversible gates (controlled-not and Toffoli), (2) QFT-adder based on the quantum Fourier transformation. We present a systematic translation QFT circuit into Toffoli-based adder. This result shows that has fundamentally same fault-tolerance cost (e.g., T count) as most cost-efficient Toffoli adder: instead using approximate decompositions from circuit, it is more efficient to merge gates. In order achieve this, we formulated identities for multicontrolled apply algorithmically. The employed techniques can be used automate optimization heuristics.