Let there be a system of three qubits, $a,b,c,d$ where $d$ is an ancilla qubit. We can add more ancilla if required. The operation I want to implement is as follows:
"If $a$ and $b$ are both in the '$1$' state, then swap qubits $b$ and $c$."
Obviously this is not a unitary transformation but it might be possible with the use of ancilla qubits. I got a similar self-controlling circuit to work with Toffoli gates, but the double-control SWAP gate eludes me. My advisor thinks it might be impossible, but neither of us has proven that.
Basically, does anyone have any idea how this operation might be implemented in a quantum circuit using one or more ancilla?