# Using Quantum Fourier Transform in adding two 2-bit numbers

I am trying to use Qiskit to write a code that uses QFT to add 2 numbers. I am referring to this paper: https://iopscience.iop.org/article/10.1088/1742-6596/735/1/012083

I have a few questions: 1) Is the $$R$$ rotation gate in this paper equivalent to the Controlled-U1 gate in Qiskit?

2) Why does the value of $$k$$ change?

3) The program is supposed to add $$1$$ and $$2$$. Number $$1$$ is encoded as $$01$$, represented by the zero vector $$(1,0)$$ and the 1 vector $$(0,1)$$. When does number $$2$$ come into play? It seems that they keep performing operations on number $$1$$ until the end.

4) Number $$1$$ is represented by $$0$$ on $$a_1$$ and $$1$$ on $$a_2$$. And number $$2$$ consists of $$1$$ on $$b_1$$ and $$0$$ on $$b_2$$, is that right?

It appears that you are asking for details about the following circuit from the paper of Cherkas and Chivilikhin, that they describe as implementing the addition of two $$2$$-bit numbers. Although they don't say it, I believe this is (mod 4). The first application of $$R_1$$ is controlled on the basis of $$a_1$$, the second $$R_2$$ is controlled on the basis of $$a_1$$, and the third is controlled on $$a_2$$. When we say we apply $$R_k$$, we are applying either $$R_1$$ or $$R_2$$. Thus the angle of rotation is dependent on $$k$$.
It appears that the first number is $$a=(a_1,a_2)$$ having the most significant bit be $$a_1$$ and the least significant bit $$a_2$$, while the second number is similarly $$b=(b_1,b_2)$$. That is, I think you fourth question is correct, but I'm not sure if your third question is the right understanding.