Factorization in a single run with an optical interferometer

Abstract

We will describe a new factorization algorithm based on the reproduction of continuous exponential sums, using the interference pattern produced by polychromatic light interacting with an interferometer with variable optical paths. We will describe two possible interferometers: a generalized symmetric Michelson interferometer and a liquid crystal grating. Such an algorithm allows, for the first time, to find all the factors of a number N in a single run without precalculating the ratio N/l, where l are all the possible trial factors. It also allows to solve the problem of ghost factors and to factorize different numbers using the same output interference pattern.

Factorization algorithm based on the periodicity measurement of a continuous truncated exponential sum

“…We want to point out that the values o ξ in Eq. (20) of the observable O ξ , defined in Section 3.4 for a general analog approach, correspond now to the wavelengths…”

“…(22) as a function of the rescaled variable ξ N which, using the condition in Eq. (20), is given by:…”

“…[20][21][22] We will also show that such an algorithm can be implemented with an optical analog computer, based on a multi-path Michelson interferometer and a spectrometer.…”

Factorization of numbers with truncated Gauss sums at rational arguments

“…Several experimental implementations have been suggested (3,4,6,7). In the meantime Gauss sum factorization has been demonstrated experimentally in various systems ranging from NMR techniques (8)(9)(10), cold atoms (11)(12)(13), ultra short laser pulses (14,15) to classical light interferometry (16,17). The largest number factored so far had 17 digits.…”

“…Most of the experiments performed so far in the realm of Gauss sum factorization had to check every trial factor ℓ individually. However, there already exists an experiment (16,17) with classical light, where the Gauss sum A (M ) N (ℓ) is estimated simultaneously for every trial factor ℓ. Here the number ℓ is encoded in the wavelength of the light and we deduce the factor of N from the spectrum of the light.…”