# How can I understand the notation for the final state of QPE algorithm?

I've been reading about the standard phase estimation from the Qiskit tutorial and got stuck in interpreting the final state representation. What does the state $$|2^t\theta\rangle$$ mean? Is that the binary representation of the most probable state? Thanks for the help!

There are "2" cases for the final result:

1. The outcome imply that $$2^t\theta=Integer$$, in that case, yes, exactly, it will be a state with the binary encoding of $$2^t\theta$$
2. In case $$2^t\theta=NotInteger$$, QPE will result (before measurement) to a state that has very high probability amplitude in the state nearest to the desired angle that you are looking for.

Schematically, before and after QFT (before measurement):

Is mapped after measurement:

Notice that the term:

will be close to 1 when $$x$$ is close to $$2^t\theta$$ , what will end and terms that do not (or just less) cancel each other after all the sum of the terms:

Given a unitary operator $$U$$ the algorithm estimates $$\theta$$ in $$U|\psi \rangle = e^{2\pi i \theta} |\psi\rangle$$. So $$\theta$$ is an angle that determines a complex eigenvalue $$e^{2\pi i \theta}$$ associated with an eigenvector $$|\psi\rangle$$ of $$U$$. If you know $$\theta$$ you know the eigenvalue of $$U$$.

QPE outputs a state $$| 2^t \theta \rangle$$ which is an integer (even if $$\theta$$ is not an integer). Of course, your integer will be represented in binary.

The notation $$| 2^t \theta \rangle$$ is a bit confusing, cause if $$\theta$$ was non-integer, say $$\theta =0.2$$, then we would be looking at $$| 2^t 0.2\rangle$$. One can infer that this state must be interpreted as a "rounded" to the nearest integer. For example, for $$t=2$$ and $$\theta=0.2$$ we would measure (with high probability) $$|1\rangle$$ because $$| 2^2 0.2\rangle = |0.8\rangle \approx |1\rangle$$ or in binary $$|01\rangle$$.

The conclusion is:
In the case when $$\theta$$ is an integer, the measurement on $$t$$ qubits yields the state $$| 2^t \theta \rangle$$ with certainty. Then you just solve for $$\theta$$.
If $$\theta$$ is not an integer then you will measure the closest integer to $$2^t \theta$$ with a probability $$\geq 0.4$$. Every measurement will be a binary encoding of some integer.