NMR Experiment Factors Numbers with Gauss Sums

Abstract

We factor the number 157573 using an NMR implementation of Gauss sums. Although the current implementation is classical and scales exponentially, we believe that an extension to the quantum regime using entangled states is possible.

Factorization of Numbers with Physical Systems

“…Yet another advantage over the schemes basing on S N relies on the fact that all terms enter the sum A We conclude that the pulse train approach and the implementation of the Gauss sum A (M ) N offers itself as the most favorable system for an experimental realization. Recently, Prof. Mehring (Universität Stuttgart) translated this scheme into the language of NMR, which led to the first experimental demonstration of factorization based on Gauss sums [Meh07]. We discuss the key ideas of this experimental realization in chapter 6.…”

“…This approach allows to factor relatively large numbers on the basis of a simple pulse sequence which contains only few RF-pulses [Meh07]. Unlike the Shor algorithm our scheme does not feature an exponential speed-up in the required operations.…”

“…In order to achieve a clearer representation and to distill the periodic dependence of the entries in the first register we project 5 the second register and arrive at…”

Factorization of numbers with physical systems

“…Recently, Prof. Mehring (Universität Stuttgart) translated this scheme into the language of NMR, which led to the first experimental demonstration of factorization based on Gauss sums [Meh07]. We discuss the key ideas of this experimental realization in chapter 6.…”

“…In order to achieve a clearer representation and to distill the periodic dependence of the entries in the first register we project 5 where the order r appears as a period in the entry of the first register. But how can we gain information on r from |Ψ 4 ?…”