Quantum Simulation of the Factorization Problem

TL;DR

This paper proposes a quantum simulator approach to factorize numbers by deriving a Hamiltonian that encodes the problem, predicting prime distribution, and solving the spectrum exactly without external assumptions.

Contribution

It introduces a novel Hamiltonian for quantum simulation of factorization, linking quantum spectra to prime counting functions without relying on traditional analytic methods.

Findings

Exact spectrum solution for the quantum system.

Prediction of prime counting function matching Riemann's R(x).

Derivation of a Hamiltonian based solely on primes below √N.

Abstract

Feynman's prescription for a quantum simulator was to find a hamitonian for a system that could serve as a computer. P\'olya and Hilbert conjecture was to demonstrate Riemann's hypothesis through the spectral decomposition of hermitian operators. Here we study the problem of decomposing a number into its prime factors, $N=xy$, using such a simulator. First, we derive the hamiltonian of the physical system that simulate a new arithmetic function, formulated for the factorization problem, that represents the energy of the computer. This function rests alone on the primes below $\sqrt N$. We exactly solve the spectrum of the quantum system without resorting to any external ad-hoc conditions, also showing that it obtains, for $x\ll \sqrt{N}$, a prediction of the prime counting function that is almost identical to Riemann's $R(x)$ function. It has no counterpart in analytic number theory and its derivation is a consequence of the quantum theory of the simulator alone.