I do understand the math behind phase kickback. The math makes sense. For more context, I find this document very helpful.
But I’m struggling a lot to intuitively understand, why the conditional phase change to the second qubit will end up change the phase of the first qubit (the control qubit)?
Can somebody help to explain this in an intuitive way?
Part of the problem people usually have here is a sort-of-classical intuition. Because you're trying to describe the action of a gate such as controlled-$U$, we divide it up as "if the control qubit is something, do something on the target qubit". It makes it sound like the control qubit doesn't change, and it's only the target that changes. This is completely wrong. One semi-intuitive way to see this is to consider the specific example of controlled-phase. This is symmetric - it doesn't matter which qubit you identify as control and which as target. $$ |0\rangle\langle 0|\otimes I+|1\rangle\langle 1|\otimes Z=I\otimes |0\rangle\langle 0|+Z\otimes|1\rangle\langle 1| $$ Given that the gate is not the identity, something must change, and hence both qubits must change! It is a genuinely two-qubit gate that changes the two-qubit state.
