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.