Using Quantum Fourier Transform in adding two 2-bit numbers

Question

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?

Answer 1

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.