An impossible quantum adder claimed by a journal article?

Question

In Quantum adder of two states that are themselves superpositions, I asked:

I have two states $|a\rangle = \frac{1}{\sqrt N}\sum_{i=0}^{N-1}|i\rangle|a_i\rangle$ and $|b\rangle = \frac{1}{\sqrt N}\sum_{j=0}^ {N-1} |j\rangle|b_j\rangle$, with $i,j,a_i,b_j \in \mathbb{N}$.

I want to add $|a\rangle$ and $|b\rangle$ such that the output I get from an adder circuit is $|c\rangle = \frac{1}{\sqrt N}\sum_{i=0}^{N-1}|i\rangle|a_i+b_i\rangle$. Is this circuit possible?

One answer suggests 'no' to the question - but the journal article (and it's from International Journal of Theoretical Physics - reputable journal) suggests 'yes'. The problem is, the article does not seem to provide a real justification of the algorithm involved, thus the question. Here is the algorithm diagram of the gated article:

The article claims that given $|\psi\rangle = \frac{1}{2^n}\left[\sum_{x=0}^{2^n -1} |x\rangle|a_x\rangle\right]\left[\sum_{y=0}^{2^n -1} |y\rangle|a_y\rangle\right]$ and with $a_x = a_y$ when $x=y$, which makes $|\psi\rangle$ two identical non-entangled copies of the same state, one can compute $A = \sum_{i=0}^{2^n-1}a_i$ in $O(n)$, with the algorithm in the figure.

The AM module simply adds $|k\rangle$ to $|y\rangle$ modulo $2^n$ in the article and nothing more. The C module simply compares value of $|x\rangle$ and $|y\rangle$ and returns two qubits to be all $|0\rangle$ if they are equal. The AS module is a controlled module, which adds $|a_x\rangle$ and $|b_x\rangle$ only if comparator output bits are all $|0\rangle$. At stage $i$ of the figure, what happens is that $\frac{1}{\sqrt 2^n}\left[\sum_{x=0}^{2^n -1} |x\rangle|a_x\rangle\right]$ is summed with itself but along the same index $x$ as to produce $\frac{1}{\sqrt 2^n}\left[\sum_{x=0}^{2^n -1} |x\rangle|2 a_x\rangle\right]$. Which basically is the question I asked before.

For me, this adder does not seem to make any sense. Yet it comes from a reputable journal, and some articles have been using this algorithm. This makes me wonder what is going on.

Here is another from the article:

Answer 1

it's from International Journal of Theoretical Physics - reputable journal

Consider this paper your lesson in how reliable "reputability" is.

---

Look at Figure 5 from the paper:

Specifically look at the left hand side:

This is laughably blatantly wrong. Input $|0\rangle$ for $|b\rangle$ and it's a cloning circuit. Cloning circuits are not possible.

The details of the mistake being made are clarified in figure 6. It shows the authors are silently discarding qubits that are still entangled with the system:

You can't just throw away entangled qubits like this! Tracing out those qubits degrades the system from a pure state to a mixed state, which fundamentally breaks the whole idea behind the paper's results.