Question about the phase kickback in the phase estimation algorithm

Question

I have an issue with the quantum phase estimation algorithm as explained Nielsen and Chuang.

There was a question very similar to mine asked about this 2 years ago, but my question is different... Namely, I can understand the phase kickback operation if t was equal to 1. What is peculiar to me however is how the phase factor sort of "jumps over" the tensors and goes to the specific control bit. Is this just the standard operation of a controlled u within n qubits, and would need to be verified through examining the specific matrix? Thanks in advance

Answer 1

The phase is applied to the overall wave function $|\phi\rangle$, therefore you can factor the phase to any individual qubit.

For example if we have a wave function as a result of a controlled operation, with the first qubit as the control:

$A|11\dots \rangle$, this can be factored as the tensor product $A|1\rangle \otimes |1\rangle \otimes |\dots\rangle$. Therefore we can say the phase has been 'kickbacked' to the control qubit.

The 'jump' is because the controlled operation only occurs when the control qubit is $|1\rangle$, therefore the phase applied by the controlled operation is only applied to wavefunctions where this qubit is $|1\rangle$, so as described above you can factor the phase as 'kickbacked' to this qubit.