Peng et. al. Describe a circuit for entanglement cloning (i.e. entanglement broadcasting) which on its own looks fairly straightforward:

Confusingly however, the gates $U_1$ and $U_2$ are not logical gates per sey. Rather, they act on the creation operators for their respective modes. In the supplimentary material for the paper, U is defined in the following way:

Apparently, Peng has a way to find the resulting state post-selected on the condition that all modes have exactly one photon:

Obviously I'm not asking anyone to do the math and show why this is true. What I am asking is how one would go about doing this? I've never seen a quantum circuit that has these kinds of "gates" thrown in like this before. Is there any way to systematically unpack this circuit into something that qiskit would be able to handle?