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Achieving quantum advantage in a search for a violations of the Goldbach conjecture, with driven atoms in tailored potentials

Year: 2025

Type: article

Abstract: The famous Goldbach conjecture states that any even natural number N N greater than 2 2 can be written as the sum of two prime numbers p^{\text{(I)}} p (I) and p^{\text{(II)}} p (II) . In this article we propose a quantum analogue device that solves the following problem: given a small prime p^{\text{(I)}} p (I) , identify a member N N of a \mathcal{N} 𝒩 -strong set even numbers for which N-p^{\text{(I)}} N − p (I) is also a prime. A table of suitable large primes p^{\text{(II)}} p (II) is assumed to be known a priori. The device realizes the Grover quantum search protocol and as such ensures a \sqrt{\mathcal{N}} 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 83, 2033 (2013)]. For a given small prime number p^{\text{(I)}}=223 p (I) = 223 , it took our quantum algorithm 5 steps to identify the number N=4× 10^{18}+14 N = 4 × 10 18 + 14 as featuring a Goldbach partition involving 223 223 and another prime, namely p^{\text{(II)}}=4× 10^{18}-239 p (II) = 4 × 10 18 − 239 . Currently, our algorithm limits the number of evens to be tested simultaneously to \mathcal{N} \sim \ln(N) <

Source: SciPost Physics Core

DOI: 10.21468/scipostphyscore.8.4.074

Authors: Oleksandr Marchukov, Andrea Trombettoni, Giuseppe Mussardo, Maxim Olshanii