Properties of QFT

Being similar to the DFT matrix, QFT matrix has all the properties of DFT matrix.

Some properties are listed below

• 
  1) , means is unitary matrix , like every other Quantum Operator.
  


• 
  2)
  


• 
  3) So
  

Now we focus on the powers of .
We have

When m+n = 0(mod d)we have on other cases the summation is is zero.So we have the

let So we have .

Now (this is easy to prove). then

So we have the following results.

Note that so they commute.

Now let us consider a qudit system with d- cannonical states , Consider the quantum Operator acting on any canonical state , we have

Consider the ring of integers with normal addition and multiplication being the ring operations. Then we can view the operator as

where is a canonical state, where negation is the group negation operation. This is equivalent to time reversal in the DFT case.

Simplest matrices are given below.