Achieving quantum advantage in a search for a minimal Goldbach partition with driven atoms in tailored potentials

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.