# Prime Factorization of Arbitrary Integers with a Logarithmic Energy Spectrum

## TL;DR

This paper introduces a quantum-based iterative method for prime factorization using bosonic atoms with a logarithmic energy spectrum, enabling efficient extraction of prime factors through energy measurements.

## Contribution

It presents a novel quantum scheme leveraging a logarithmic energy spectrum of bosonic atoms for prime factorization, a significant departure from classical algorithms.

## Key findings

- Probabilistic factor extraction with 50% success rate per measurement
- Utilizes a logarithmic energy spectrum to encode factors
- Proposes a protocol for prime-only energy spectra

## Abstract

We propose an iterative scheme to factor numbers based on the quantum dynamics of an ensemble of interacting bosonic atoms stored in a trap where the single-particle energy spectrum depends logarithmically on the quantum number. When excited by a time-dependent interaction these atoms perform Rabi oscillations between the ground state and an energy state characteristic of the factors. The number to be factored is encoded into the frequency of the sinusoidally modulated interaction. We show that a measurement of the energy of the atoms at a time chosen at random yields the factors with probability one half. We conclude by discussing a protocol to obtain the desired prime factors employing a logarithmic energy spectrum which consists of prime numbers only.