Quantum Fourier addition simplified to Toffoli addition

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