I try to make a Quantum Fourier Transformation with N=6:
w=e^(2*pi*7/6)
so we have 6-th roots: 1,w,w^2,-1,-w,-w^2
My matrix is QFT6= 1/sqrt(6)*
(1 1 1 1 1 1 )
(1 w w^2 -1 -w -w^2)
(1 w^2 -w 1 w^2 -w )
(1 -1 1 -1 1 -1 )
(1 -w w^2 1 -w w^2 )
(1 -w^2 -w -1 w^2 w )
the vector that should be transformed is: 1/sqrt(2)* (|1> + |4>)
=1/sqrt(2)*(0 1 0 0 1 0)^T
if I multiply I get:
1/sqrt(3)*(1 0 w^2 0 -w 0)^T
there must be a mistake but I do not find it..
sorry.. w=e^(2*pi*i/6)
that was not the mistake just a type error ;)
the matrix u get
1/sqrt(3)*(1,0,w^2,0'-w,0)is right
To identify the mistake, let's go step by step through your calculations.
You correctly identified the sixth roots of unity as 1, w, w^2, -1, -w, and -w^2, where w = e^(2*pi*7/6).
Next, you defined the matrix QFT6 as 1/sqrt(6) multiplied by the following matrix:
(1 1 1 1 1 1 )
(1 w w^2 -1 -w -w^2)
(1 w^2 -w 1 w^2 -w )
(1 -1 1 -1 1 -1 )
(1 -w w^2 1 -w w^2 )
(1 -w^2 -w -1 w^2 w )
Now, let's calculate the matrix-vector product of QFT6 with the vector (0 1 0 0 1 0)^T:
1/sqrt(6) * (1*0 + 1*1 + 1*0 + 1*0 + 1*1 + 1*0,
1*0 + w*1 + w^2*0 + (-1)*0 + (-w)*1 + (-w^2)*0,
1*0 + w^2*1 + (-w)*0 + 1*0 + w^2*1 + (-w)*0,
1*0 + (-1)*1 + 1*0 + (-1)*0 + 1*1 + (-1)*0,
1*0 + (-w)*1 + w^2*0 + 1*0 + (-w)*1 + w^2*0,
1*0 + (-w^2)*1 + (-w)*0 + (-1)*0 + w^2*1 + w*0)
Simplifying the above expression:
1/sqrt(6) * (1,
w + w^2,
w^2 + w^2,
-1 + 1,
-w + (-w),
-w^2 + w^2)
= (1/sqrt(6),
2w,
2w^2,
0,
0,
0)
So the result of the quantum Fourier transform (QFT) applied to the vector (0 1 0 0 1 0)^T is (1/sqrt(6), 2w, 2w^2, 0, 0, 0).
At this point, it seems there is a discrepancy between your expected result 1/sqrt(3) * (1 0 w^2 0 -w 0)^T and the actual result we calculated.
The mistake might be in your initial vector or in the calculation of the matrix-vector product. Please review your steps and ensure you have followed them correctly.