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arXiv:0803.3396 · Europhysics Letters (EPL)

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

Xinhua Peng, Dieter Suter

Published Nov 1, 2008• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/84/40006

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Comment on "NMR Experiment Factors Numbers with Gauss Sums"

Julia A. Wei Xiaohua Jones

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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.
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.
Recent experiments have shown that truncated Gauss sums allow us to find the factors of an integer N . This method relies on the fact that for a factor the absolute value of the Gauss sum is unity. However, for every integer N there exist integers which are not factors, but where the Gauss sum reaches a value which is arbitrarily close to unity. In order to distinguish such ghost factors from real factors we need to amplify this difference. We show that a proper choice of the truncation parameter of the Gauss sum suppresses the ghost factors below a threshold value. We derive the scaling law of the truncation parameter on the number to be factored. Moreover, we show that this scaling law is also necessary for the success of our factorization scheme, even if we relax the threshold or allow limited error tolerance.
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.
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.
A challenge in the Gauss-sum factorization scheme is the presence of ghost factors, nonfactors that behave similarly to actual factors of an integer, which might lead to the misidentification of nonfactors as factors or vice versa, especially in the presence of noise. We investigate type II ghost factors, which are the class of ghost factors that cannot be suppressed with techniques previously laid out in the literature. The presence of type II ghost factors and the coherence time of the qubit set an upper limit for the total experiment time, and hence the largest factorizable number with this scheme. Discernibility is a figure of merit introduced to characterize this behavior. We introduce preprocessing as a strategy to increase the discernibility of a system, and demonstrate the technique with a transmon qubit. This can bring the total experiment time of the system closer to its decoherence limit, and increase the largest factorizable number.