# What is Quantum Phase Estimation in Shor's Algorithm?

While I'm studying Algorithm, I couldn't understand what Quantum Phase Estimation is. And I heard there is relation between Phase-Kickback and Quantum Phase Estimation. I wonder what it is. Also, I'm not sure what this is and how it works within the algorithm. I would like to learn from the mathematical point of view and the conceptual part.

Phase estimation is the process by which you are given a controlled-$$U$$ unitary, and a state that you are promised is an eigenvector of $$U$$ with eigenvalue $$e^{2\pi ix/2^t}$$, then you can use a $$t$$-qubit register to affect the change $$|0\rangle^{\otimes t}|u\rangle\mapsto|y\rangle|u\rangle.$$ If $$x$$ is in integer, then the outcome is guaranteed to be $$y=x$$. If $$x$$ is not an integer, $$y$$ is, with high probability, the closest integer to $$x$$.
The phase-kickback is the way that we get something, apparently on the second register (i.e. the $$|u\rangle$$ being acted upon by $$U$$) to change the first register. In effect, this is making use of the difference between global phase (as it would be for $$U$$) and relative phase in a superposition (which is what you actually get when you use controlled-$$U$$).