Factoring Using Shor's Quantum Algorithm

Transcription

1 Factorng Usng Shor's uantum Algorthm Frank Rou Emertus Professor of Chemstry CSB SJU Ths tutoral presents a toy calculaton dealng wth quantum factorzaton usng Shor's algorthm. Before begnnng that task, tradtonal classcal factorzaton s revewed wth the eample of fndng the prme factors of 5. As shown below the key s to fnd the perod of a modulo 5, where a s chosen randomly. a N 5 f ( ) mod a N f ( ) Seeng that the perod of f() s two, the net step s to use the Eucldan algorthm by calculatng the greatest common denomnator of two functons nvolvng the perod and a, and the number to be factored, N. perod gcd a perod N 3 gcd a perod N 5 Factorng 5 by ths method s trval because t s the product of two small prme numbers. However, f N s the product of two large prmes ths method s mpractcal because fndng the perodcty of f() would not be possble by nspecton as t s above. If f() were plotted t would appear to be random nose wth no recognzable perodc structure. Shor recognzed that the dscrete Fourer transform (or, quantum Fourer transform) provded an effcent method for fndng the perod of f() when N s the product of two etremely large prme numbers. So the contrbuton that quantum mechancs may eventually make to code breakng s effcently fndng the perodcty of f(). We proceed by gnorng the fact that we already know that the perod of f() s and demonstrate how t s determned usng a quantum (dscrete) Fourer transform. After the regsters are loaded wth and f() usng a quantum computer, they est n the followng superposton. 3 ( ) 3 f The rearrangement on the rght collects terms on the f() values. Now the values appear n pars wth ther f() partners. Note that the perod of s dscernable n each par of values. After ( > + >) the net par s offset by. The Fourer transform on the -regster removes the offset, clearly revealng the perod of. In preparaton for the Fourer transform the superposton s wrtten as a sum of vector tensor products.

2 T The net step s to fnd the perod of f() by performng a quantum Fourer transform (FT) on the nput regster, >. The dentty operaton (do nothng) acts on the second regster. m n FT mn ep π m n I dentty 5 The result of the FT s: epressed n tensor format kronecker FT I ( ) The calculaton at ths pont s summarzed as follows: ( ) FT f Recallng that -regster orgnally was a superposton nvolvng,,, and 3, we see that the FT brought about constructve and destructve nterference because now the -regster (blue) contans only and. Ths gves us a perod of and we can now proceed to the classcal calculaton that was demonstrated earler. The detals of how the FTon the -regster gves rse to nterference s gven n the summary below. Recommended readng: Two nsghtful analyses of quantum mechancs' role n factorng by Davd Mermn appear n the Aprl and October 7 ssues of Physcs Today. The frst s ttled "What has quantum mechancs to do wth factorng?" and the second "Some curous facts about quantum factorng." Chapter 5 n Julan Brown's The uest for the uantum Computer deals n depth wth code breakng and Shor's algorthm.

3 Summary Fgure 5 n "uantum Computaton," by Davd P. DVncenzo, Scence 7, 58 (995) provdes a graphcal llustraton of the steps of Shor's factorzaton algorthm. Load the -regster Calculate f() Fnd perod of f() Ths tutoral deals wth steps and 3 of the algorthm, summarzed mathematcally below. The negatve sgn n the far rght column vector s an accumulated phase due to the quantum Fourer transform. FT f ( ) How a Fourer transform on the -regster can yeld the perodcty of f() whch s on the y-regster s revealed by carryng out the Fourer transform on the ndvdual members of the -regster n the mddle superposton term above. FT FT FT FT FT FT FT FT (A) (B)

4 Usng the results from (B) the FT on the -regster n (A) yelds the followng superposton. (C) Constructve and destructve nterference between the terms of ths superposton leads to the fnal state. (D) The detals of the nterference between the terms n (C) can be seen by epandng them usng vector tensor multplcaton. (E)

5 An algebrac vew of (C) also reveals the -regster nterference. Destructve nterference occurs wthn the terms hghlghted n red ( >) and wthn the terms hghlghted n blue ( 3>)