Another well known quantum algorithm for searching.
Takes a function $f : \{0,1\}^n \rightarrow \{0,1\}$ where f(x) = 1 for exactly one x. 
The goal is to find x.

New gates: $V_f$ and FLIP$_*$ 

$\ket{*}$ denotes the superposition over al classical possibilities.

### Oracle $V_f$

![[Pasted image 20260805102238.png]]

We use the Unitary $U_f$ as previously defined to construct $V_f$

![[Pasted image 20260805102605.png]]


### FLIP$_*$

We first need FLIP$_0$ defined as follows:
![[Pasted image 20260805102659.png]]

This is implemented via this circuit:

![[Pasted image 20260805102717.png]]

Z is a Pauli matrix and the empty circles denote negative control wires.
So Z is only applied if all other wires are $\ket{0}$