I'm learning about the iterative phase estimation (IPE) algorithm from the qiskit textbook. Here's a circuit I generated to implement this algorithm on a random single-qubit Hamiltonian. Instead of using the phase gate to perform time evolution as demonstrated in the textbook, I used controlled $U$ gates. From the qiskit document, the qubit $q_1$ remains in the same state $|\psi\rangle$ throughout the algorithm (here it's initialized to $|0\rangle$).
I don't quite understand why this should be the case, given the circuit above. I was thinking about when we perform a measurement in $q_0$, are we kind of 'projecting' the quantum state on $q_1$ onto a subspace associated with the measurement outcome? If we just look at the unitary operations on $q_1$, I still don't know why that state should be the same.
Also, should the phase corrections (controlled by the classical bits) start from the second iteration be the same for the implementation of IPE on any system? It seems like the phase we want to correct after each iteration is independent of the system we simulate, but I'm worried if there's anything else we need to take into account for a large system when we calculate the phase that needs to be corrected.
Thanks so much for the help:)