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The QPE on IBM platform finds the eigenvalue of a unitary operator, i.e $$U|\phi\rangle=e^{2\pi i\theta}|\phi\rangle$$ and uses the rotation operators as $$U(\theta)=\begin{bmatrix}0 & 1\\ 1 &e^{i\theta}\end{bmatrix}$$ My question is we can write $U|\phi\rangle=e^{2\pi i\theta}|\phi\rangle$ as $$U|\phi\rangle=e^{2e i\theta}|\phi\rangle$$ where, since even then the magnitude of the eigenvalue remains same. Of course that would require more number of measurement qubits. For instance for $\theta=0.5$, we had $4$, so this time we can increase the $n$ and get closer to $\theta=0.5$. My question is can this be done.

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No - the key intuition of QPE is that $e^{2 \pi i \theta} $ holds for $\theta \in [0, 1) $, and the reading out the ancilla provides the binary representation of the fraction.

If you changed the operator to $e^{2 e i \theta}$, $\theta \not \in [0, 1)$ necessarily. Instead, you should just use the typical QPE approach, but then find $\zeta$ where $2 e i \zeta = 2 \pi i \theta \implies \zeta = \frac{\pi}{e}\theta$, and correct for symmetries afterwards.

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