# What is the difference between amplitude amplification, amplitude estimation, and phase estimation?

I'm confused about the difference among Amplitude amplification (AA) , phase estimation (PE), and Amplitude Estimation.

I thought I understood AA and PE somewhat but when I heard the amplitude estimation and the circuit looked so similar to phase estimation.

I think I missed something here. Sorry, still new to the field.

To find the differences between them, you only need to know the aim of the problems.

• The aim of AA is to find the answers from unstructured data(or more directly, amplify the probability of the right answer).

• The aim of PE is to find the phase, more specifically the $$\phi$$ in the book of Nielsen:

Suppose a unitary operator $$U$$ has an eigenvector $$\mid u\rangle$$ with eigenvalue $$e^{2\pi i\phi}$$, where the value of $$\phi$$ is unknown.

Then amplitude estimation is the problem of estimating $$a= \langle \Psi_1\mid\Psi_1\rangle$$, the probability that a measurement of $$\mid\Psi\rangle$$ yields a good state.

These are three different tasks. To solve one problem, you will need an algorithm to solve it, while these algorithms might have some parts in common, i.e., use some algorithm as a subroutine to solve other problems(tasks). For example, Quantum Fourier Transform(QFT) is the subroutine for the problem of Shor's algorithm. Another example is the QFT can be a subroutine for PE.

To sum up, to differ from those things you mentioned, you only need to know what you want to solve, what's your problem while the way to solve them(the algorithms) might have something in common and even one is another's subroutine and should not bother you.

• Thank you @narip Very clear Jun 4, 2021 at 20:32
• Hey @JohnParker, remember that you can also accept an answer if it helped you solve your doubts. This will help others that may have the same question in the future. Jun 10, 2021 at 22:38
• @epelaaez I forgot. Thanks. Jun 14, 2021 at 17:27

The original Amplitude Estimation algorithm (Brassard et al., 2002) uses Phase Estimation. So, it is OK to notice the similarity between them.

Other approaches, however, have emerged in recent years that do not use PE for Amplitude Estimation like this and this.