# An impossible quantum adder claimed by a journal article?

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:

• Is it really "published" if it's not open access? Makes it rather hard to read it to see what's wrong. Aug 30, 2022 at 2:14

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.

• I think from the no-cloning theorem to conclude that the second picture is wrong is not rigorous? Different $|a\rangle$ is orthogonal to each other, while the no-cloning theorem states that we can not clone any unknown states, especially including non-orthogonal state. Aug 30, 2022 at 3:30
• @narip The diagram doesn't state any constraints that the states must be orthogonal. And the paper explicitly passes non-orthogonal states through the circuit as part of its main result. Aug 30, 2022 at 3:58
• Hmm, IJTP also published an article about image encryption with many of the same authors. Any cryptographer will tell you that image encryption is snake oil; it seems to exist only to pad CVs. See e.g. this. Aug 31, 2022 at 2:18
• @CraigGidney I don't think it's a cloning circuit as written. If I take it seriously, then sending in a state $\sum_a\psi_a|a\rangle|0\rangle$ would result in the state $\sum_a\psi_a|a\rangle|a\rangle$, which is not a clone of the original state. I still agree that their circuit for S doesn't work, which you could guess from the fact that S is not a unitary operation and their circuit includes only unitary operators! Aug 31, 2022 at 18:13
• @JahanClaes I'm pretty sure in the paper their goal is to create an independent superposition not an entangled one. It's hard to speak definitively about what the circuit diagram "should" do, because the intended underlying functionality is unphysical. Aug 31, 2022 at 21:02