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

The famous Goldbach conjecture states that any even natural number N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> greater than 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> can be written as the sum of two prime numbers p^{\text{(I)}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>p</mml:mi> <mml:mtext mathvariant="normal">(I)</mml:mtext> </mml:msup> </mml:math> and p^{\text{(II)}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>p</mml:mi> <mml:mtext mathvariant="normal">(II)</mml:mtext> </mml:msup> </mml:math> . In this article we propose a quantum analogue device that solves the following problem: given a small prime p^{\text{(I)}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>p</mml:mi> <mml:mtext mathvariant="normal">(I)</mml:mtext> </mml:msup> </mml:math> , identify a member N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> of a \mathcal{N} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒩</mml:mi> </mml:mstyle> </mml:math> -strong set even numbers for which N-p^{\text{(I)}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mtext mathvariant="normal">(I)</mml:mtext> </mml:msup> </mml:mrow> </mml:math> is also a prime. A table of suitable large primes p^{\text{(II)}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>p</mml:mi> <mml:mtext mathvariant="normal">(II)</mml:mtext> </mml:msup> </mml:math> is assumed to be known a priori. The device realizes the Grover quantum search protocol and as such ensures a \sqrt{\mathcal{N}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msqrt> <mml:mstyle mathvariant="script"> <mml:mi>𝒩</mml:mi> </mml:mstyle> </mml:msqrt> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>p</mml:mi> <mml:mtext mathvariant="normal">(I)</mml:mtext> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>223</mml:mn> </mml:mrow> </mml:math> , it took our quantum algorithm 5 steps to identify the number N=4× 10^{18}+14 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>18</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>14</mml:mn> </mml:mrow> </mml:math> as featuring a Goldbach partition involving 223 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>223</mml:mn> </mml:math> and another prime, namely p^{\text{(II)}}=4× 10^{18}-239 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>p</mml:mi> <mml:mtext mathvariant="normal">(II)</mml:mtext> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>18</mml:mn> </mml:msup> <mml:mo>−</mml:mo> <mml:mn>239</mml:mn> </mml:mrow> </mml:math> . Currently, our algorithm limits the number of evens to be tested simultaneously to \mathcal{N} \sim \ln(N) <