# Quantum Fourier Transform on two qubits: Non orthogonal outputs

We know the QFT gives us a new orthogonal basis from the original one, however, when I apply it on two qubits, I am not getting the output vectors orthogonal.

$$|out(k)\rangle = \Sigma^{N-1}_{j=0} e^{\frac{2\pi ij.k}{N}}|j\rangle$$

Where $$j.k$$ is the bitwise 'AND' and then summed up.

Applying this on the basis:

$$|00\rangle , |01\rangle , |10\rangle , |11\rangle$$

I get the following vectors (ignore normalization factor):

$$|00\rangle + |01\rangle + |10\rangle + |11\rangle$$

$$|00\rangle + i|01\rangle + |10\rangle + i|11\rangle$$

$$|00\rangle + |01\rangle - |10\rangle - |11\rangle$$

$$|00\rangle + i|01\rangle + |10\rangle -i|11\rangle$$

However these are not orthogonal, as you can see, the dot product does not equal zero for the first and second vector.

Why is this the case?

• where does that definition of the QFT come from? Sep 17 '18 at 13:54
• @DaftWullie, I found it in Nielsen and Chuang, page 217 of the 2002 edition. Sep 17 '18 at 13:55

You are mis-quoting the definition of the QFT. You simply take the product of the decimal values of $j$ and $k$, and don't use their binary representations.