Gauss Optics and Gauss Sum on an Optical Phenomena

O. M. Marchenko, Sergei A. Kazantsev, Laurentius Windholz

Vincenzo Tamma, Heyi Zhang, Xuehua He +2 more

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.

Wim van Dam, G. Seroussi

We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction from the discrete logarithm problem to Gauss sum estimation we also give evidence that this problem is hard for classical algorithms. The workings of the quantum algorithm rely on the interaction between the additive characters of the Fourier transform and the multiplicative characters of the Gauss sum.

L. Mandeļ

W P Schleich

We propose two algorithms to factor numbers using Gauss sums and entanglement: (i) in a Shor-like algorithm we encode the standard Gauss sum in one of two entangled states and (ii) in an interference algorithm we create a superposition of Gauss sums in the probability amplitudes of two entangled states.These schemes are rather efficient provided that there exists a fast algorithm that can detect a period of a function hidden in its zeros.