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Europhysics Letters (EPL)

Factoring numbers with interfering random waves

S. J. Weber, B. Chatel, B. Girard

Published Jul 29, 2008• 22 citations
Publisher

Abstract

We report on a new implementation of the factorisation of numbers using Gauss sums which improves tremendously the efficiency to eliminate all "ghost" factors. We show that by choosing randomly the terms in the Gauss sum, the required number of terms varies as ln N instead of . As an illustration, we present experimental results obtained by interfering thirty ultrashort laser pulses where we factorise 1340333404807. This new approach is totally general and can be implemented for all the experiments based on the Gauss sum.

DOI
10.1209/0295-5075/83/34008

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We report on a new implementation of the factorisation of numbers using Gauss sums which improves tremendously the efficiency to eliminate all "ghost" factors. We show that by choosing randomly the terms in the Gauss sum, the required number of terms varies as ln N instead of . As an illustration, we present experimental results obtained by interfering thirty ultrashort laser pulses where we factorise 1340333404807. This new approach is totally general and can be implemented for all the experiments based on the Gauss sum.
We propose three implementations of the Gauss sum factorization schemes discussed in part I of this series (Wölk et al 2011 New J. Phys. 13 103007): (i) a two-photon transition in a multi-level ladder system induced by a chirped laser pulse, (ii) a chirped one-photon transition in a two-level atom with a periodically modulated excited state and (iii) a linearly chirped one-photon transition driven by a sequence of ultrashort pulses. For each of these quantum systems, we show that the excitation probability amplitude is given by an appropriate Gauss sum. We provide rules on how to encode the number N to be factored in our system and how to identify the factors of N in the fluorescence signal of the excited state.
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.
We present the experimental proof of principle of a new factorization algorithm based on the implementation of generalized continuous truncated Gauss sums using a generalized Michelson interferometer with variable interfering optical paths. Respect to the past Gauss sums realizations, such algorithm allow us to check all the trial factors l of a number N at the same time in a single run, avoiding the pre-calculation of the ratio N/l and it is generalizable to higher order j. Most important, this procedure allows, for the first time, to factorize different numbers in a single run, despite the previous Gauss sums realization, in which was necessary to run the experiment for each trial factors.
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.