# phase estimation

## 1. problem

- given \(U\) with eigenvector \(\ket{u}\) and eigenvalue \(e^{2\pi i \phi}\), find \(\phi\).

## 2. pre-reqs

## 3. procedure

- Let's assume that \(\phi\) has the binary representation \(0.\phi_1\phi_2...\phi_n\)
- Produce the state

\[(\ket{0} + e^{2\pi i 0.\phi_n \ket{1}}) (\ket{0} + e^{2\pi i 0.\phi_{n-1}\phi_n \ket{1}}) (\ket{0} + e^{2\pi i 0.\phi_1\phi_2...\phi_n \ket{1}})\]

- This can be done using similar techniques to the circuit for the quantum fourier transform
- Take a look at that state. It's exactly the state that results from applying the QFT to \(\phi\). Then, the inverse QFT can be used to recover \(\phi\) exactly!
- But in practice, \(\phi\) can not be exactly represented using \(n\) bits
- But, if we use enough bits, we can bound the error and the probability of error