Quantum Chaos and the Spectrum of Factoring

TL;DR

This paper provides numerical evidence supporting a quantum chaos model for the factorization problem, suggesting a deep link between quantum mechanics and number theory, and confirming the quantum simulator hypothesis.

Contribution

It introduces a Hamiltonian formulation of factorization and demonstrates the spectral properties align with quantum chaos predictions, supporting a quantum-mechanical approach to factoring.

Findings

Spectral analysis of random moduli shows GUE distribution

Supports the quantum simulator hypothesis for factorization

Confirms a link between quantum chaos and number theory

Abstract

There exists a Hamiltonian formulation of the factorisation problem which also needs the definition of a factorisation ensemble (a set to which factorable numbers, $N'=x'y'$, having the same trivial factorisation algorithmic complexity, belong). For the primes therein, a function $E$, that may take only discrete values, should be the analogous of the energy from a confined system of charges in a magnetic trap. This is the quantum factoring simulator hypothesis connecting quantum mechanics with number theory. In this work, we report numerical evidence of the existence of this kind of discrete spectrum from the statistical analysis of the values of $E$ in a sample of random OpenSSL n-bits moduli (which may be taken as a part of the factorisation ensemble). Here, we show that the unfolded distance probability of these $E$'s fits to a {\it Gaussian Unitary Ensemble}, consistently as required, if they actually correspond to the quantum energy levels spacing of a magnetically confined system that exhibits chaos. The confirmation of these predictions bears out the quantum simulator hypothesis and, thereby, it points to the existence of a liaison between quantum mechanics and number theory. Shor's polynomial time complexity of the quantum factorisation problem, from pure quantum simulation primitives, was obtained.