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I read the chapter on QPE (quantum phase estimation) in Nielsen and noticed that $\delta$ is defined there as follows: $0 \leq \delta \leq 2^{-t}$, see:

5.2.1 Performance and requirements

The above analysis applies to the ideal case, where $\varphi$ can be written exactly with a $t$ bit binary expansion. What happens when this is not the case? It turns out that the procedure we have described will produce a pretty good approximation to $\varphi$ with high probability, as forshadowed by the notation used in 5.22. Showing this requires some careful manipulations.

Let $b$ be the integer in the range $0$ to $2^t-1$ such that $b/2^t=0.b_1\ldots b_t$ is the best $t$ bit approximation to $\varphi$ which is less than $\varphi$. That is, the difference $\delta\equiv \varphi-b/2^t$ between $\varphi$ and $b/2^t$ satisfies $0\leq\delta\leq 2^{-t}$.

When I read the Wikipedia article about QPE I noticed that $\delta$ is defined as follows, see Wikipedia QPE:

We can approximate the value of $\theta \in [0, 1]$ by rounding $2^n \theta$ to the nearest integer. This means that $2^n \theta = a + 2^n \delta,$ where $a$ is the nearest integer to $2^n \theta,$ and the difference $2^n\delta$ satisfies $0 \leqslant |2^n\delta| \leqslant \tfrac{1}{2}$.

Now I wonder why two such different definitions of $\delta$ are used here. I have a guess why this might be, but I am not sure, so I ask here what the goal of the $\delta$ variant defined in Wikipedia is all about.


My guess:

Wikipedia uses this definition of delta at the point of measurement. My guess is that by using this definition of delta in the measurement, the "surrounding" of nearby values is also taken into account. Let's say the ideal measurement would be e.g. $|010\rangle$, with the $|\delta| \leq \frac{1}{2^{n+1}}$ we reach then that the "environment" around $|010\rangle$ is included in positive AND negative direction, the environment could be understood then e.g. in such a way: $|001\rangle$, $|010\rangle$, $|011\rangle$.

This is only an assumption made by me. I would be very happy if someone is willing to explain this in more detail, why two different definitions of $\delta$ can be found.

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