NMR implementation of Factoring Large Numbers with Gau\ss{}Sums: Suppression of Ghost Factors

TL;DR

This paper explores NMR-based methods for factoring large numbers using Gauß sums, focusing on suppressing ghost factors through modified algorithms tested experimentally in a nuclear spin system.

Contribution

It introduces two techniques to distinguish true factors from ghost factors with minimal increase in sum terms, enhancing the practicality of NMR-based factorization.

Findings

Successful experimental implementation in NMR system

Effective suppression of ghost factors achieved

Modified algorithms maintain constant or slowly increasing sum terms

Abstract

Finding the factors of an integer can be achieved by various experimental techniques, based on an algorithm developed by Schleich et al., which uses specific properties of Gau\ss{}sums. Experimental limitations usually require truncation of these series, but if the truncation parameter is too small, it is no longer possible to distinguish between factors and so-called "ghost" factors. Here, we discuss two techniques for distinguishing between true factors and ghost factors while keeping the number of terms in the sum constant or only slowly increasing. We experimentally test these modified algorithms in a nuclear spin system, using NMR.