# Controlled controlled adder gates involved

Let's say I have a circuit that given in the figure

As we can see that this circuit consists of $$2$$-Toffoli gates and $$4$$ C-NOT gates, and to construct this entire circuit using only single qubit gates I would require at least $$12$$ C-NOT gates $$6$$ for each of the step UMA and MAJ. That makes it a total of at least $$16$$ C-NOT Gates. Now my question is if I modify these circuits by putting two controls for each of them such that each of them operates if both controls are (say) $$1$$. Then I have $$2$$ CCCC-NOT gates and $$4$$ CCC-NOT gates. Now for this, I need at least $$(8\times 4 + 10\times 2=52)$$ C-NOT gates.

This is using an article that says to stimulate a $$n$$ qubit Toffoli gate we need at least $$2n$$ C-NOT gates.

Is my reasoning for my circuit correct? Can somebody help?

• You are not doing any simplifications. For example, see answer about automatic groups. – AHusain Jun 8 '19 at 21:35
• Please clarify the question. You want to know the minimum number of CNOTs to simulate an $n$ qubit Toffoli? Or you want to rewrite your circuit using only CNOT? Or something else? – bRost03 Jun 10 '19 at 2:34