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Achieving quantum advantage in a search for a violations of the Goldbach conjecture, with driven atoms in tailored potentials
by Oleksandr V. Marchukov, Andrea Trombettoni, Giuseppe Mussardo, Maxim Olshanii
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Maxim Olshanii |
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| Preprint Link: | https://arxiv.org/abs/2404.00517v3 (pdf) |
| Date submitted: | March 27, 2025, 3:18 a.m. |
| Submitted by: | Maxim Olshanii |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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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 $p^{\text{(I)}}$ and $p^{\text{(II)}}$. In this article we propose a quantum analogue device that solves the following problem: given a small prime $p^{\text{(I)}}$, identify a member $N$ of a $\mathcal{N}$-strong set even numbers for which $N-p^{\text{(I)}}$ is also a prime. A table of suitable large primes $p^{\text{(II)}}$ is assumed to be known a priori. The device realizes the Grover quantum search protocol and as such ensures a $\sqrt{\mathcal{N}}$ quantum advantage. Our numerical example involves a set of 51 even numbers just above the highest even classical-numerically explored so far [T. O. e Silva, S. Herzog, and S. Pardi, Mathematics of Computation {\bf 83}, 2033 (2013)]. For a given small prime number $p^{\text{(I)}}=223$, it took our quantum algorithm 5 steps to identify the number $N=4\times 10^{18}+14$ as featuring a Goldbach partition involving $223$ and another prime, namely $p^{\text{(II)}}=4\times 10^{18}-239$. Currently, our algorithm limits the number of evens to be tested simultaneously to $\mathcal{N} \sim \ln(N)$: larger samples will typically contain more than one even that can be partitioned with the help of a given $p^{\text{(I)}}$, thus leading to a departure from the Grover paradigm.
Author comments upon resubmission
Our referee remarks inspired us to, effectively, rewrite the paper and run a series of computer simulations, for which we are indefinitely grateful.
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We modified the protocol in such a way that it is now isomorphic to the conventional Grover search scheme. Extensive literature exists that studies the efficiency of the latter.
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We underwent an extensive numerical experimentation cycle and found an optimal set of system parameters
2b. On the negative side, our numerical study showed that the requirements to the relative values of the perturbation matrix elements, for both omega- and s-gates, are much more stringent than we expected. While the unperturbed Hamiltonian remains firmly rooted in potentials realized in Donatella Cassettari's lab, the gate perturbations used in our numerics are represented by their idealized versions. More work is needed to bring our proposal in contact with the AMO reality. Our recent AVS Quantum [O. V. Marchukov and M. Olshanii, AVS Quantum Science 7, 013801 (2025)]is a step in this direction.
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The text is completely rewritten.
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Our introduction is substantially extended.
List of changes
Major changes
- We added numerical simulations.
- The protocol is mildly altered: in the most recent version, even numbers and large primes correspond to the quantum eigenstates and small primes are perturbation frequencies. As the result, the protocol is now identical to the Grover one.
- Overall, the text has been very substantially rewritten.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2025-9-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2404.00517v3, delivered 2025-09-07, doi: 10.21468/SciPost.Report.11884
Strengths
1- Insightful implementation of Grover quantum search 2- Sound analysis of the efficiency
Weaknesses
1- Numerical results are not clearly explained
Report
In my view the authors addressed the concerns of the previous referee. The paper can be accepted for publication after the authors have considered the following points:
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The potential in Eq. (4) is supposed to implement an all-to-all coupling between the states of the databased. However, it does not seem to realise the diffusion operator of Grover search $\hat U_s$. In fact, in the model of Eq. (2) the states of the database have different energies. Thus, since (4) is a perturbation, the effective coupling between a given pair of states will also depend on the energy difference. The authors shall comment how this affects the final efficiency of the protocol.
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The numerical results are not clearly explained. It is difficult to extract what the figures show (which quantity is shown? What does it tell about the efficiency of the protocol? etc)- a text in Sec. 3 that guides the reader through the figures and discusses the findings would significantly increase the clarity of the presentation.
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Fig.3: What limits the efficiency of the protocol to 80%?
Requested changes
In addition to addressing the points listed above, I recommend that the authors fix several typos (e.g., "exited"->"excited", "t hey" ->"they", "targrert"->"target", "hight"->"high").
Recommendation
Ask for minor revision
Author: Maxim Olshanii on 2025-09-19 [id 5837]
(in reply to Report 1 on 2025-09-07)To start let us mention that without the referees, this paper would be two time shorter. Their comments were relevant, focused, and inspiring.
We added several sentences of a comment, in the end of the Section 2.5. Additionally, we completely eliminated the misleading notion of "perturbation."
We expanded the Chapter 3 to better describe our numerical results.
We added two small paragraphs to the ens of the Chapter 3.