A mathematical technique that decomposes a function into its constituent frequencies.

## Definition
![[Pasted image 20260730181449.png]]

The transformation is a process that takes a periodic vector as an input and outputs
the period of that vector.

![[Pasted image 20260730181941.png]]

## Construction

We will now model DFT as a quantum circuit.
For this we only look at N = $2^n$, DFT applied on n qubits.

![[Pasted image 20260730182117.png]]

Let us look at an example DFT for n = 3:
![[Pasted image 20260730183136.png]]
![[Pasted image 20260730183145.png]]


The more general approach works as follows:

![[Pasted image 20260730185053.png]]

![[Pasted image 20260730185144.png]]


