# Quantum simulation of the integer factorization problem: Bell states in a Penning trap

## TL;DR

This paper proposes a quantum simulation approach using Bell states in a Penning trap to factor integers, linking quantum energy spectra to number theory, and demonstrates its potential to outperform current quantum computers in factoring large numbers.

## Contribution

It introduces a novel quantum simulation method for integer factorization using entangled particles in a Penning trap, connecting quantum physics with number theory and complexity theory.

## Key findings

- Quantum energy spectra relate to prime factors of N.
- Only o(log√N)^3 energy measurements needed for factorization.
- Experimental feasibility of large number factoring with Penning traps.

## Abstract

The arithmetic problem of factoring an integer $N$ can be translated into the physics of a quantum device, a result that supports P\'olya's and Hilbert's conjecture to prove Riemann's hypothesis. The energies of this system, being univocally related to the factors of $N$, are the eigenvalues of a bounded Hamiltonian. Here we solve the quantum conditions and show that the histogram of the discrete energies, provided by the spectrum of the system, should be interpreted in number theory as the relative probability for a prime to be a factor candidate of $N$. This is equivalent to a quantum sieve that is demonstrated to require only $ o(\log \sqrt N)^3$ energy measurements to solve the problem, recovering Shor's complexity result. Hence, the outcome can be seen as a probability map that a pair of primes solve the given factorization problem. Furthermore, we show that a possible embodiment of this quantum simulator corresponds to two entangled particles in a Penning trap. The possibility to build the simulator experimentally is studied in detail. The results show that factoring numbers, many orders of magnitude larger than those computed with experimentally available quantum computers, is achievable using typical parameters in Penning traps.