= Hamming-weight obstruction to Fredkin-only Toffoli synthesis
{c}
Every <Fredkin gate> preserves the <Hamming weight> of a <Computational basis>[computational-basis] string, so any network of these gates does too. A <Toffoli gate> maps $110$ to $111$ and hence cannot equal such a network on the same three wires. Fixed ancillas returned unchanged do not remove the obstruction. Extra encoded outputs or input-dependent garbage change the resource model and do not by themselves implement the same clean coherent three-qubit gate.
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