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

TL;DR

The paper addresses the problem of testing Goldbach partitions beyond the numerically verified range by formulating a quantum search over an $N$-strong set of even numbers to find $N$ for which $N-p^{(I)}$ is prime, given a precomputed table of large primes $p^{(II)}$. It proposes a quantum analogue device that implements Grover search using driven atoms in tailored one-body potentials, realized via an $\omega$-gate and an $s$-gate in a two-level system, with a concrete Hamiltonian and readout scheme. A numerical demonstration over a set of $\mathcal{N}=51$ evens shows that five Grover iterations can identify a Goldbach partition $N=4\times 10^{18}+14$ with $p^{(I)}=223$ and $p^{(II)}=4\times 10^{18}-239$ with about $80\%$ success, highlighting both the potential speed-up and practical limitations such as off-resonant effects. The work discusses current capabilities to engineer tailored atomic spectra and outlines future improvements to gate control and scaling to larger search spaces, contributing to the broader pursuit of quantum-accelerated number-theoretic tasks with programmable atomic systems.

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.

Figures (3)

• Figure 1: The major ingredients of the protocol. Energy is expressed in the units of $U_{0}$.
• Figure 2: Real part of the various components of the device wavefunction during the the first Grover cycle. The computational space consists of $\mathcal{N} = 51$ states representing 51 even numbers just above $4\times 10^{18}$, the so far largest number tested for violations of the Goldbach conjecture silva2013_2033. The auxiliary space consists all large prime candidates (7 total) for completion a Goldbach partition with any of the small primes $p_{m_{\text{I}}}$ up to $307$. The frequency of the pulse \ref{['V_w']} corresponds to $p_{m_{\text{I}}}=233$. It identifies the 7'th member of the computational space, $N_{n_{\omega}} = 4\times 10^{18}+14$, as a number that satisfies the Goldbach conjecture, with the second prime being $p_{m_{\text{II},\,\omega}} = 4\times 10^{18} - 209$. (a) Our realization of the Grover $\omega$-pulse. For the magnitude of the $2\pi$-pulse, we chose $V^{(\omega)}_{0} = \frac{1}{6} U_{0}$. As by design, the sign in front of the Grover matching entry $|\omega\rangle$ changes by the end of the pulse, while the amplitudes of the remaining members of the computational space remain the same. (b) Our realization of the Grover $s$-pulse. We choose $V^{(s)}_{0} = 100. \,U_{0}$. Observe that indeed, the prefactor in from of the Grover $|s\rangle$-state changes while its absolute value does not. Here, $n'\equiv n-n_{\text{min}}+1$.
• Figure 3: Grover protocol. Results of a computer simulation of the Grover sequence. System structure and the input parameters are the same as at Fig. \ref{['f:w-and_s--pulses']}. The Grover target state, $|n'=7\rangle \equiv |\omega\rangle$ is the even number $N_{n_{\omega}} = 4\times 10^{18}+14$ that is being identified as a sum of two primes, $p_{m_{\text{I}}}=233$ and $p_{m_{\text{II},\,\omega}} = 4\times 10^{18} - 209$. Red dashed line with symbols is the theoretical prediction for the population of the matching entry bin, $|\langle \omega|\psi \rangle|^2 = \cos^2((2n+1)\delta \theta)$, where $n$ is the number of Grover iterations, $\delta\Theta =\arcsin(1/\sqrt{\mathcal{N}})$, and $\mathcal{N} = 51$ is the size of the computational space. It is evident that $n=5$ is the theoretically predicted optimal number of iterations. Indeed, numerically the matching entry state will be detected in a state measurement with a probability of about 80%. Notice that unlike in the idealized scenario, the $m'_{\text{II},\,\omega}=2$ auxiliary state, corresponding to $p_{m_{\text{II},\,\omega}}$, remains weakly populated. We also propagate the procedure for seven more steps to further compare the numerical results with theoretical predictions. Solid red line with symbols: population of the matching entry state in the end of every Grover iteration, plus it's initial value. Solid blue line with symbols: same, but for the rest of the computational space. Solid green line with symbols: same, but for the auxiliary space. Here, $n'\equiv n-n_{\text{min}}+1$ and $m'_{\text{II}}\equiv m_{\text{II}}-(m_{\text{II}})_{\text{min}}+1$.