Achieving quantum advantage in a search for a minimal Goldbach partition with driven atoms in tailored potentials
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 socalled minimal Goldbach partition $N=p+p'$ with $p\equiv p_{\rm min}(N)$ being the socalled 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 stateofart 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2024
 DOI:
 10.48550/arXiv.2404.00517
 arXiv:
 arXiv:2404.00517
 Bibcode:
 2024arXiv240400517M
 Keywords:

 Quantum Physics;
 Condensed Matter  Quantum Gases