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

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• Physik

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2007
• Nombre de lectures: 19
• Langue: English
• Poids de l'ouvrage: 1 Mo

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FACTORIZATION OF NUMBERS
WITH PHYSICAL SYSTEMS
Dissertation
zur Erlangung des Doktorgrades
Dr. rer. nat.
der Fakulta¨t fu¨r Naturwissenschaften
der Universita¨t Ulm
vorgelegt von
WOLFGANG MERKEL
Institut fu¨r Quantenphysik
Universita¨t Ulm
Leiter: Prof. Dr. Wolfgang P. Schleich
Ulm 2007Amtierender Dekan: Prof. Dr. Klaus-Dieter Spindler
Erstgutachter: Prof. Dr. Wolfgang P. Schleich
Zweitgutachter: Prof. Dr. Ferdinand Schmidt-Kaler
Tag der Promotion: 6. Juni 2007Contents
List of Figures vii
1 Introduction & Motivation 1
1.1 A glimpse on Shor’s algorithm . . . . . . . . . . . . . . . . . . . 4
1.2 Our approach to the factorizationof numbers . . . . . . . . . . 7
1.3 Relation to other work . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Organizationof the present work . . . . . . . . . . . . . . . . . 8
2 Factorization with Gauss sums 9
2.1 An alternative representationof Gauss sums . . . . . . . . . . . 10
2.1.1 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 . . . . . . . . . . . . 13m
2.2 Continuoussampling of Gauss sums . . . . . . . . . . . . . . . . 14
2.2.1 Discriminationof factors from non-factors . . . . . . . . 14
2.2.2 Scaling property . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Discrete samplingof Gausssums . . . . . . . . . . . . . . . . . 17
2.3.1 Analytical expressionsfor the standard Gauss sumG(ℓ,N) 17
22.3.2 Factorization with|G(ℓ,N)| . . . . . . . . . . . . . . . . 18
2.3.3 Factorization with real and imaginarypart ofG(ℓ,N) . . 21
2.4 A second type of Gauss sum . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Factorization based on complete Gauss sums . . . . . . . 22
2.4.2 Factorization based on truncated Gauss sums . . . . . . 25
2.4.3 Complexity of the factorizationscheme . . . . . . . . . . 33
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
iii3 Gauss sums arising from a chirped two-photon transition 37
3.1 Chirped laser pulses . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Description of chirped pulses . . . . . . . . . . . . . . . 38
3.1.2 Applicationsof chirped pulses . . . . . . . . . . . . . . . 39
3.2 Chirped pulse excitation in a three-state ladder . . . . . . . . . 40
3.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Time evolution . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Perturbationtheory in the weakfield limit . . . . . . . . 43
3.2.4 Interference of quantumpaths . . . . . . . . . . . . . . . 45
3.2.5 Origin of quadratic phases . . . . . . . . . . . . . . . . . 45
3.2.6 Interference in time-frequency phase space . . . . . . . . 48
3.3 Chirped pulse excitation in a multistate ladder . . . . . . . . . . 50
3.3.1 Exact expression for the total excitation probability am-
plitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 Asymptotic expansionof weight factors . . . . . . . . . . 54
3.4 Candidate systems for an experimentalrealization . . . . . . . . 54
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Gauss sums arising from laser-driven one-photon transitions 59
4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Floquet ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Excitation probabilityin the weakfield limit . . . . . . . 63
4.2.2 Excitation probabilityfor a chirped pulse . . . . . . . . . 64
4.2.3 Choice of modulationindex . . . . . . . . . . . . . . . . 65
4.2.4 Choice of phase . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Pulse train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 Excitation probabilityin the weakfield limit . . . . . . . 68
4.3.2 Purely quadraticphases for a specific choice of parameters 68
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Factorization with physical systems 71
5.1 Chirped multistate ladder . . . . . . . . . . . . . . . . . . . . . 72
5.1.1 Continuoussampling of the signal . . . . . . . . . . . . . 73
5.1.2 Scaling property . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Factorizationbased on Floquet ladder . . . . . . . . . . . . . . . 74
5.2.1 Factorization with different types of sums . . . . . . . . 76
5.2.2 Fluorescence signalat discrete arguments . . . . . . . . 78
5.3 Factorizationwith pulse train . . . . . . . . . . . . . . . . . . . 79
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
iv6 NMR-Realizations of Gausssums 85
6.1 NMR experiment factorsN = 157573 . . . . . . . . . . . . . . . 86
6.2 A second NMR experiment . . . . . . . . . . . . . . . . . . . . . 92
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Conclusions and Outlook 95
A Evaluation of Gausssums 97
B Gauss sum for broad weight factor distributions 99
C Pocket factorizer 101
D Another type of Gausssum 103
E Asymptotic expansion of the complementary error function 107
E.1 Integration path fora> 0: decaying regime . . . . . . . . . . . 110
E.2 Integration path fora< 0: oscillatory regime . . . . . . . . . . 110
E.3 Asymptotic expressions . . . . . . . . . . . . . . . . . . . . . . . 111
Bibliography 115
List of Publications 129
vviList of Figures
2.1 FactorizationofN = 33 . . . . . . . . . . . . . . . . . . . . . . . 15e2.2 FactorizationofN = 65 by scaling the signal forN = 33 . . . . 16
2.3 FactorizationofN = 39 and 41 . . . . . . . . . . . . . . . . . . 20
2.4 FactorizationofN = 40 and 42 . . . . . . . . . . . . . . . . . . 21
2.5 Factorizationwith real and imaginarypart ofG(ℓ,N) . . . . . . 23
(ℓ−1)
2.6 FactorizationofN = 1911withA (ℓ) . . . . . . . . . . . . . 26
N
2.7 Truncating the summationrange in the Gauss sum . . . . . . . 27
2.8 Countingghost factors . . . . . . . . . . . . . . . . . . . . . . . 29
2.9 Ghost factors in the complex plane . . . . . . . . . . . . . . . . 31
2.10 Suppressionof ghost factors . . . . . . . . . . . . . . . . . . . . 34
2.11 Optimal upper bound . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Three-state ladder system . . . . . . . . . . . . . . . . . . . . . 41
3.2 Chirping a two-photon excitation in a three-state ladder . . . . 46
3.3 Origin of Gauss sums in phase space . . . . . . . . . . . . . . . 49
3.4 Extended ladder system . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Chirpingatwo-photonexcitationthroughanequidistantmanifold 53
3.6 Exact vs. approximated weight factors . . . . . . . . . . . . . . 55
4.1 Floquet ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Weight factorsw (φ) . . . . . . . . . . . . . . . . . . . . . . . . 66m
4.3 Pulse train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 FactorizingN = 15 . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Scaling property. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
(±π/2)′5.3 Floquet ladder: factorizingN = 21 withS . . . . . . . . . 77′N
(0)′5.4 Floquet ladder: factorizingN = 21 withS . . . . . . . . . . . 78
′N
vii(π/3)′5.5 Floquet ladder: factorizingN = 21 withS . . . . . . . . . . 79′N
′5.6 Floquet ladder: factorizingN = 105 . . . . . . . . . . . . . . . 80
5.7 Pulse train: factorizationofN = 1911 . . . . . . . . . . . . . . 820
6.1 M. Mehring’s NMR-experiment: pulse sequence . . . . . . . . . 86
6.2 M. Mehring’s NMR-experiment factorizesN = 157573 . . . . . . 90
6.3 Probingthe limits . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.4 D. Suter’s NMR-experiment: pulse sequence . . . . . . . . . . . 93
6.5 D. Suter’s NMR-experiment forN = 1911 compared to theory . 93
C.1 Pocket factorizer . . . . . . . . . . . . . . . . . . . . . . . . . . 102
E.1 Integration paths in complex plane . . . . . . . . . . . . . . . . 109
E.2 Approximationof erfc(ζ ) . . . . . . . . . . . . . . . . . . . . . 113m
viiiCHAPTER1
Introduction & Motivation
Dw vxqulvh zh zloo dwwdfn wkh vpdoo Jdoolf yloodjh.
In history it has always been a major interest to keep the content of a mes-
sage secret when exchanged between two parties [Sin00, Kip97]. It is interest-
ing to analyze the history of cryptography in connection with the progress in
physics and technology. In course of time the competition between the latter
two counterparts has increased in intensity.
Originally,the applicationofcryptography waspredominantly usedfor mil-
itaryorpoliticalpurposes. AlreadyGaiusJuliusCaesar(102-44B.C.)employed
asimplecryptographicschemetohidethecontentofhismessages. Thisscheme
is based on a simple substitution within the alphabet: Each letter of the plain
textisreplacedbytheonewhichoriginatesfromaconstantshiftoftheoriginal
letter. With this knowledge and Caesar’s original shift of three letters the plain
1text can be decrypted by simple means .
Oncenewtechnologieshadbeendevelopedtheyhadimmediateapplication
incryptography. InWorldWarIItheGermanarmyusedtheelectro-mechanical
crypto-machine “Enigma” for encrypting top secret messages. One reason for
the victory of the allied forces was certainly the ability of the cryptographs at
Bletchley Park to decrypt the German