Factorization of numbers with physical systems [Elektronische Ressource] / vorgelegt von Wolfgang Merkel

FACTORIZATION OF NUMBERSWITH PHYSICAL SYSTEMSDissertationzur Erlangung des DoktorgradesDr. rer. nat.der Fakulta¨t fu¨r Naturwissenschaftender Universita¨t Ulmvorgelegt vonWOLFGANG MERKELInstitut fu¨r QuantenphysikUniversita¨t UlmLeiter: Prof. Dr. Wolfgang P. SchleichUlm 2007Amtierender Dekan: Prof. Dr. Klaus-Dieter SpindlerErstgutachter: Prof. Dr. Wolfgang P. SchleichZweitgutachter: Prof. Dr. Ferdinand Schmidt-KalerTag der Promotion: 6. Juni 2007ContentsList of Figures vii1 Introduction & Motivation 11.1 A glimpse on Shor’s algorithm . . . . . . . . . . . . . . . . . . . 41.2 Our approach to the factorizationof numbers . . . . . . . . . . 71.3 Relation to other work . . . . . . . . . . . . . . . . . . . . . . . 71.4 Organizationof the present work . . . . . . . . . . . . . . . . . 82 Factorization with Gauss sums 92.1 An alternative representationof Gauss sums . . . . . . . . . . . 102.1.1 Sequence of individual contributions . . . . . . . . . . . 11(r)2.1.2 Factorization made possible by the shape functionI . 12m(r)2.1.3 Factorization made possible byW . . . . . . . . . . . . 13m2.2 Continuoussampling of Gauss sums . . . . . . . . . . . . . . . . 142.2.1 Discriminationof factors from non-factors . . . . . . . . 142.2.2 Scaling property . . . . . . . . . . . . . . . . . . . . . . 142.3 Discrete samplingof Gausssums . . . . . . . . . . . . . . . . . 172.3.1 Analytical expressionsfor the standard Gauss sumG(ℓ,N) 1722.3.