  Quantum addition circuits are considered being of two types: 1)
Toffolli-adder circuits which use only classical reversible gates (CNOT and
Toffoli), and 2) QFT-adder circuits based on the quantum Fourier
transformation. We present the first 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 novel circuit identities for multi-controlled
gates and apply the identities algorithmically. The employed techniques can be
used to automate quantum circuit optimisation heuristics.
