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.