Oleksandr V. Marchukov, Andrea Trombettoni, Giuseppe Mussardo, Maxim Olshanii
SciPost Phys. Core 8, 074 (2025) ·
published 28 October 2025
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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 83, 2033 (2013)]. For a given small prime number $p^{\text{(I)}}=223$, it took our quantum algorithm 5 steps to identify the number $N=4× 10^{18}+14$ as featuring a Goldbach partition involving $223$ and another prime, namely $p^{\text{(II)}}=4× 10^{18}-239$. 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.
SciPost Phys. 10, 025 (2021) ·
published 3 February 2021
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We employ the Gross-Pitaevskii equation to study acoustic emission generated in a uniform Bose gas by a static impurity. The impurity excites a sound-wave packet, which propagates through the gas. We calculate the shape of this wave packet in the limit of long wave lengths, and argue that it is possible to extract properties of the impurity by observing this shape. We illustrate here this possibility for a Bose gas with a trapped impurity atom -- an example of a relevant experimental setup. Presented results are general for all one-dimensional systems described by the nonlinear Schr\"odinger equation and can also be used in nonatomic systems, e.g., to analyze light propagation in nonlinear optical media. Finally, we calculate the shape of the sound-wave packet for a three-dimensional Bose gas assuming a spherically symmetric perturbation.
