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To answer your first question, whenever you run a QPE circuit on, let's say, $n$ ancilla qubits, you will be left with the binary representation of the phase $\theta$, encoded in the ancilla qubits. Measuring these ancillary qubits provides you an estimate of the phase $\theta$ (up to n-bit precision). Note, the probabilities of the states are there to ...

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The IQPE procedure tries to reduce the extra number of ancilla qubits needed to measure the phase and replaces that with the extra number of iterations that we perform to measure our phase. Imagine you have the phase as $\theta = 0.\theta_{0}\theta_{1}\theta_{2}\theta_{3}$. So, what IQPE does is that in the first iteration, it tries to identify the value of $... 1 Even I have implemented Iterative QPE and the basic QPE in qiskit using approximately the same approach. The point where my simulations also started producing random results was as the phase estimation got beyond 4 qubit precision. The question of how we mitigate this error is actually a bit wrong to ask at this point because we also need to start looking at ... 2 The original Amplitude Estimation algorithm (Brassard et al., 2002) uses Phase Estimation. So, it is OK to notice the similarity between them. Other approaches, however, have emerged in recent years that do not use PE for Amplitude Estimation like this and this. 2 To find the differences between them, you only need to know the aim of the problems. The aim of AA is to find the answers from unstructured data(or more directly, amplify the probability of the right answer). The aim of PE is to find the phase, more specifically the$\phi$in the book of Nielsen: Suppose a unitary operator$U$has an eigenvector$\mid u\...

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When we make a measurement we project the state of the qubit to the z-basis (i.e 0 or 1) so in general it is not possible to measure a global phase. That said, Measuring such global phases is an important subroutine in many quantum algorithms such as Shor's algorithm. This can be done using the Quantum phase estimation algorithm. For details, check https://...

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