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Achieving quantum advantage in a search for a minimal Goldbach partition with driven atoms in tailored potentials
by Oleksandr V. Marchukov, Andrea Trombettoni, Giuseppe Mussardo, Maxim Olshanii
Submission summary
| Authors (as registered SciPost users): | Maxim Olshanii |
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| Preprint Link: | https://arxiv.org/abs/2404.00517v2 (pdf) |
| Date submitted: | 2024-07-31 05:21 |
| Submitted by: | Olshanii, Maxim |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
The famous Goldbach conjecture states that any even natural number $N$ greater than $2$ can be written as the sum of two prime numbers $p$ and $p'$, with $p \, , p'$ referred to as a Goldbach pair. In this article we present a quantum analogue protocol for detecting -- given a even number $N$ -- the existence of a so-called minimal Goldbach partition $N=p+p'$ with $p\equiv p_{\rm min}(N)$ being the so-called minimal Goldbach prime, i.e. the least possible value for $p$ among all the Goldbach pairs of $N$. The proposed protocol is effectively a quantum Grover algorithm with a modified final stage. Assuming that an approximate smooth upper bound $\mathcal{N}(N)$ for the number of primes less than or equal to $ p_{\rm min}(N)$ is known, our protocol will identify if the set of $\mathcal{N}(N)$ lowest primes contains the minimal Goldbach prime in approximately $\sqrt{\mathcal{N}(N)}$ steps, against the corresponding classical value $\mathcal{N}(N)$. In the larger context of a search for violations of Goldbach's conjecture, the quantum advantage provided by our scheme appears to be potentially convenient. E.g., referring to the current state-of-art numerical search for violations of the Goldbach conjecture among all even numbers up to $N_{\text{max}} = 4\times 10^{18}$ [T. O. e Silva, S. Herzog, and S. Pardi, Mathematics of Computation 83, 2033 (2013)], a quantum realization of the search would deliver a quantum advantage factor of $\sqrt{\mathcal{N}(N_{\text{max}})} \approx 37$ and it will require a Hilbert space spanning $\mathcal{N}(N_{\text{max}}) \approx 1376$ basis states.
Current status:
Reports on this Submission
Strengths
1- Relevance. In their work, the authors investigate the possibility of using a modified Grover algorithm to test Goldbach's conjecture, which states that every natural number greater than 2 can be written as the sum of two prime numbers. The authors' idea, which seems well-based, is that, thanks to the use of a quantum algorithm, the search process would be extremely faster. This, given the ever-increasing relevance that prime numbers have in various fields of modern science, should justify the publication of their paper.
Weaknesses
1- From a scientific point of view, I am puzzled by the fact that the authors provide no evidence, neither analytical nor numerical, that their modified version of Grover's algorithm continues to correctly converge to the searched state with the same rate. Perhaps the authors consider it sufficient that the algorithm is similar to a well-known one to ensure that everything goes smoothly. However, the change in the algorithm is not trivial and doubts about the convergence may arise. The authors should make an effort to make their claims more solid.
2- Even less do the authors take the trouble to support with analytical and/or numerical evidence the fact that the dynamics of the systems they considered (equations 6 and 7) correctly simulates their algorithm.
This is not a trivial point since the authors make several assumptions, and it would be interesting to test them numerically. To provide an example, the authors require that some quantities must be much smaller than 1 (non enumerated equations). Great! Is 10^-1 enough? Or do we need to reach 10^-10? And how are these values affected by the other parameters of the system? These seem to me to be interesting questions that are not answered within their paper.
3- The paper seems to have been rushed out to publication. This feeling comes from several small problems. For example, on page 2, the authors quickly summarize the Grover algorithm in its classic version. After equation 4 they write: "The sought key |w> be read out using the state measurement in the {|n>} \otimes |g > basis". What is the state |g >? It has never been defined before. Moreover, what would be its role in a Grover algorithm? After a few pages, we see that the state |g > is nothing more than one of the two internal states of a two-level system introduced to realize a system whose dynamics should simulate the modified Grover algorithm that they introduce. OK, but why insert it pages before in a cryptic sentence that only adds confusion?
4- Of course, if this were the only problem, it would be a small oversight that happens to everyone. The paper presents a huge amount of typos in various mathematical expressions, especially those in the text. They range from "|m_{max}>>" with the double >> to an artistic "|n′<\otimes|e>", etc. Before resubmitting the paper, the authors should do a thorough job of cleaning up these errors.
5- I would also suggest extending the introduction of the paper to better frame the problem and make it accessible to a wider audience of readers.
Report
Even if the topic is relevant and the results seems to be of wide interest, I cannot recommend publishing the paper in this version. However, once that the problems of the paper will be fixed, after a thorough rewriting, an improved version of their work will deserve a publication.
Requested changes
1- Provide a proof that the modified algorithm that they introduced, converges at the right state with a rate comparable with the original Grover's one.
2- Provide a test of the stability of the dynamics in the system introduced in equations 6 and 7 for different choices of parameters
3- Subject the text of the article to a careful rewriting that better clarifies the various points and removes the numerous typos
4- Extend the introduction to make the paper more accessible to a wider audience
Recommendation
Ask for major revision