# How do I calculate the number of uses of a unitary $U$ in iterative phase estimation?

How would one go along to calculate the number of uses of an unitary $$U$$ in Iterative Phase Estimation (IPE) to compare it to the number of uses of $$U$$ in standard Phase Estimation (Qiskit QPE)?

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 $$\theta_{3}$$ by applying the unitary $$U$$ matrix, $$2^{3}$$ times. Why? Each phase kickback imparts a phase of $$\theta$$ to our ancilla. If we applied the matrix $$8$$ times, the phase kicked back is going to be $$2^{3} \theta$$ right?
Now, see what actually $$e^{2 \pi i (2^{3}\theta)}$$ is equal to. $$2^{3}\theta$$ is just $$\theta_{0}\theta_{1}\theta_{2}.\theta_{3}$$ as the bits get shifted 3 times left. $$e^{2 \pi i (\theta_{0}\theta_{1}\theta_{2}.\theta_{3})}$$ can be written as $$e^{2 \pi i (0.\theta_{3})}$$ as the integral part evaluates to 1. Measuring in the X basis produces the bit $$\theta_{3}$$ with very high probability.
Now, we see that in the estimation of the $$i_{th}$$ bit position (0-based), we required $$2^{i}$$ number of unitary applications. Summing this over to all the bits we shall have -
$$\sum_{i=0}^{n-1} 2^{i}$$ which evaluates to $$2^{n} - 1$$ unitary applications for n bit precision.