Consider a four-qubit circuit with the following structure:
where the boxes can be arbitrary two-qubit unitary operations, and the time- evolution proceeds from left to right.
That not every four-qubit unitary can be decomposed in such a way follows from a simple parameter-counting argument: $(2^4)^2\gg 3(2^2)^2$. However, consider the case in which we have a fixed input state, and want to generate a target output. From the point of view of the number of free parameters, this seems feasible: $2(2^4)< 3(2^2)^2$. Is it always possible?
In other words, given a fixed input $|\psi_0\rangle$ and a target $|\psi_t\rangle$, can we always find two-qubit unitaries $U_{12},U_{23},U_{34}$ (acting nontrivially only on the respective subspaces) such that $|\psi_t\rangle = (U_{12}\otimes U_{34})U_{23} |\psi_0\rangle$ ?