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Fortschritte der Physik

Factorization of numbers with physical systems

W. Merkel, Ilya Sh. Averbukh, B. Girard, G. G. Paulus, Wolfgang P. Schleich

Published Aug 1, 2006• 35 citations
Publisher

Abstract

Abstract The periodicity properties of Gauss sums allow us to factor integer numbers. We show that the excitation probability amplitudes of appropriate quantum systems interacting with specific laser fields are determined by Gauss sums. The resulting probabilities are experimentally accessible by measuring the fluorescence from this level. In particular, we discuss a two‐photon transition in a ladder system driven by a chirped laser pulse. In addition, we consider two realizations of laser driven one‐photon transitions. For each quantum system we demonstrate the power of this factorization scheme using numerical examples.

DOI
10.1002/prop.200610315

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Abstract The periodicity properties of Gauss sums allow us to factor integer numbers. We show that the excitation probability amplitudes of appropriate quantum systems interacting with specific laser fields are determined by Gauss sums. The resulting probabilities are experimentally accessible by measuring the fluorescence from this level. In particular, we discuss a two‐photon transition in a ladder system driven by a chirped laser pulse. In addition, we consider two realizations of laser driven one‐photon transitions. For each quantum system we demonstrate the power of this factorization scheme using numerical examples.
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.
The security of codes, for example in credit card and government information, relies on the fact that the factorization of a large integer N is a rather costly process on a classical digital computer. Such a security is endangered by Shor's algorithm which employs entangled quantum systems to find, with a polynomial number of resources, the period of a function which is connected with the factors of N. We can surely expect a possible future realization of such a method for large numbers, but so far the period of Shor's function has been only computed for the number 15. Inspired by Shor's idea, our work aims to methods of factorization based on the periodicity measurement of a given continuous periodic factoring function which is physically implementable using an analogue computer. In particular, we have focused on both the theoretical and the experimental analysis of Gauss sums with continuous arguments leading to a new factorization algorithm. The procedure allows, for the first time, to factor several numbers by measuring the periodicity of Gauss sums performing first-order factoring'' interference processes. We experimentally implemented this idea by exploiting polychromatic optical interference in the visible range with a multi-path interferometer, and achieved the factorization of seven digit numbers. The physical principle behind this factoring interference procedure can be potentially exploited also on entangled systems, as multi-photon entangled states, in order to achieve a polynomial scaling in the number of resources.