Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0609174 · Physical Review Letters

NMR Experiment Factors Numbers with Gauss Sums

Michael Mehring, Klaus Müller, Ilya Sh. Averbukh, Wolfgang Merkel, Wolfgang P. Schleich

Published Mar 22, 2007• 85 citations
arXivHTMLPDF

Abstract

We factor the number 157573 using an NMR implementation of Gauss sums. Although the current implementation is classical and scales exponentially, we believe that an extension to the quantum regime using entangled states is possible.
DOI
10.1103/physrevlett.98.120502

Related papers

Comment on "NMR Experiment Factors Numbers with Gauss Sums"

Julia A. Wei Xiaohua Jones

Mehring et al. have recently described an elegant nuclear magnetic resonance (NMR) experiment implementing an algorithm to factor numbers based on the properties of Gauss sums. Similar experiments have also been described by Mahesh et al. In fact these algorithms do not factor numbers directly, but rather check whether a trial integer $\ell$ is a factor of a given integer $N$. Here I show that these NMR schemes cannot be used for factor checking without first implicitly determining whether or not $\ell$ is a factor of $N$.

arXiv:0704.2065•Apr 16, 2007•2 citations

Factorizing numbers with the Gauss sum technique: NMR implementations

T. S. Mahesh, Nageswaran Rajendran, Xinhua Peng +1 more

Several physics-based algorithms for factorizing large numbers were recently presented. A notable recent algorthm by Schleich et al. uses Gauss sums for distinguishing between factors and nonfactors. We demonstrate two NMR techniques that evaluate Gauss sums and thus implement their algorithm. The first one is based on differential excitation of a single spin magnetization by a cascade of rf pulses. The second method is based on spatial averaging and selective refocusing of magnetization for Gauss sums corresponding to factors. All factors of 16 637 and 52 882 363 are successfully obtained.

arXiv:quant-ph/0701205•Physical Review A•Jun 4, 2007•42 citations

NMR implementations of Gauss sums

Jonathan A. Jones

arXiv:0807.1608•Physics Letters A•Jul 17, 2008•14 citations

Factorizing numbers with the Gauss sum technique

Dieter Suter, T. S. Mahesh, Nageswaran Rajendran +1 more

Several physics-based algorithms for factorizing large numbers were recently presented. A notable recent algorthm by Schleich et al. uses Gauss sums for distinguishing between factors and nonfactors. We demonstrate two NMR techniques that evaluate Gauss sums and thus implement their algorithm. The first one is based on differential excitation of a single spin magnetization by a cascade of rf pulses. The second method is based on spatial averaging and selective refocusing of magnetization for Gauss sums corresponding to factors. All factors of 16 637 and 52 882 363 are successfully obtained.

•Jun 4, 2007•25 citations

NMR implementation of exponential sums for integer factorization

M. Štefaňák, W. Merkel, M. Mehring +1 more

Contemporary Physics•May 1, 2008•2 citations

NMR implementation of factoring large numbers with Gauß sums: Suppression of ghost factors

Xinhua Peng, Dieter Suter

Finding the factors of an integer can be achieved by various experimental techniques, based on an algorithm developed by Schleich et al . ( Fortschr. Phys. , 54 (2006) 856), which uses specific properties of Gauß 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.

arXiv:0803.3396•Europhysics Letters (EPL)•Nov 1, 2008•24 citations
Mehring et al. have recently described an elegant nuclear magnetic resonance (NMR) experiment implementing an algorithm to factor numbers based on the properties of Gauss sums. Similar experiments have also been described by Mahesh et al. In fact these algorithms do not factor numbers directly, but rather check whether a trial integer ℓ\ellℓ is a factor of a given integer NNN. Here I show that these NMR schemes cannot be used for factor checking without first implicitly determining whether or not ℓ\ellℓ is a factor of NNN.
Several physics-based algorithms for factorizing large numbers were recently presented. A notable recent algorthm by Schleich et al. uses Gauss sums for distinguishing between factors and nonfactors. We demonstrate two NMR techniques that evaluate Gauss sums and thus implement their algorithm. The first one is based on differential excitation of a single spin magnetization by a cascade of rf pulses. The second method is based on spatial averaging and selective refocusing of magnetization for Gauss sums corresponding to factors. All factors of 16 637 and 52 882 363 are successfully obtained.
Several physics-based algorithms for factorizing large numbers were recently presented. A notable recent algorthm by Schleich et al. uses Gauss sums for distinguishing between factors and nonfactors. We demonstrate two NMR techniques that evaluate Gauss sums and thus implement their algorithm. The first one is based on differential excitation of a single spin magnetization by a cascade of rf pulses. The second method is based on spatial averaging and selective refocusing of magnetization for Gauss sums corresponding to factors. All factors of 16 637 and 52 882 363 are successfully obtained.
Finding the factors of an integer can be achieved by various experimental techniques, based on an algorithm developed by Schleich et al . ( Fortschr. Phys. , 54 (2006) 856), which uses specific properties of Gauß 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.