I am looking to translate interferometric processes onto a quantum circuit model and am running into issues regarding feedback and the reuse of circuit elements. My question is framed in terms of a Sagnac interferometer but applies to translating more general problems onto quantum circuits.
At its heart, a Sagnac interferometer applies some unitary $U$ to a target system if the control is in some state $|0\rangle$ and the opposite unitary $U^\dagger$ to the target system if the control is in state $|1\rangle$. This is easy to implement with a quantum circuit if we have an individual gate implementing each of $U$ and $U^\dagger$; then we only need two control gates. But the crux of Sagnac interferometry is that $U^\dagger$ arises from implementing $U$ backwards: the control qubit governs the direction in which the target system experiences the unitary $U$. Can this be represented by a quantum circuit?
The first step is straightforward: if control is in $|0\rangle$, implement $U$, otherwise ignore. I could envision the next step to be: if control is in $|1\rangle$, reflect backward through the circuit. Is this allowed? Is there a standard method for doing this?
Perhaps there is a way of implementing some controlled gate that says "if the control is in some state, perform a complex conjugation operation on the next gate" can that be done, instead? And, if so, can that be done in arbitrary dimensions?
At the end of the day, Sagnac interferometry needs the two processes to be able to reinterfere with each other, so an ideal solution would allow for that, but we can postpone that problem for now.