CS29A, Sping 25: Quantum Infomation & Quantum Computation Wim van Dam Engineeing, Room 59 vandam@cs http://www.cs.ucsb.edu/~vandam/teaching/cs29/
Administivia ext week talk b Matthias Steffen on uclea Magnetic Resonance (MR) quantum computing. Final will be an exam à la last week s Midtem This week: quantum Fouie tansfom, Sho s algoithms fo factoing and discete logaithms, Gove s seach algoithm. Othe questions?
Last Week Que complexit: Seaching a database of size can be done with Θ( ) quantum queies to F. This quadatic speed-up is nice, but what we eall want is an exponential speed-up. o snake oil; we cannot seach blindl a database with O(log ) queies: thee is no staightfowad wa of solving P-complete poblems in polnomial time. We ae able to solve some poblems efficientl with a quantum compute that as fa as we know equie exponential esouces with a classical compute.
Pimes vs Composite umbes The poblem of distinguishing pime numbes fom composite numbes and of esolving the latte into thei pime factos is known to be one of the most impotant and useful in aithmetic. It has engaged the indust and wisdom of ancient and moden geometes to such an extent that it would be supefluous to discuss the poblem at length Futhe, the dignit of the science itself seems to equie that eve possible means be exploed fo the solution of a poblem so elegant and so celebated. Cal Fiedich Gauß, Disquisitiones Aithmeticæ 8
Pimalit Testing Let be an n-bit intege. Question: Is pime? (efficient ~ pol(n) opeations) Efficient pimalit testing: Pobabilistic tests of complexit O(n 3 ): Solova-Stassen [977], Mille-Rabin [976/8] The Agawal-Kaal-Saxena pimalit test [AKS22] is a deteministic algoithm with unning time O(n 6+ε ) (Assuming the Riemann hpothesis, the Mille-Rabin algoithm is deteministic as well.)
Factoing Integes Let be an n-bit intege. Question: What ae the pime factos of? Relevant fo beaking RSA cptogaph Best known classical algoithm: umbe Field Sieve [Pollad 988] 3 2 Time complexit: exp( log() loglog() 3 O()) [FSET.ORG, Septembe 24]: Factoization of a 73 digit numbe.
Sho s Factoing Algoithm [Pete Sho, 994]: Thee exists a quantum algoithm that finds the pime factos of an intege in time O((log ) 3 ) [Chuang et al, 2] Expeimental implementation fo 5. To undestand Sho s algoithm we have to look at: - Quantum Fouie Tansfom - Classical umbe Theo
Quantum Fouie Tansfom Conside the mod numbes {,,2,, }. The Quantum Fouie Tansfom ove is defined fo each x {,,, } b x a e 2πi x / Hence fo each supeposition ove mod : x α x a x x α x e 2πi x / Impotant fact: The QFT can be efficientl implemented in cicuit size pol(log()) fo each.
Some Small Fouie Tansfoms Fo 2,3,4 we have the following tansfomations: H) ( 2 Fou 2 3 / 2πi 2 2 3 e withω ω ω ω ω 3 Fou i i i i 2 Fou 4 ote unitait
Popeties of Fou The definition: Fou : x a e 2πi x / Hence: Fou x e 2πi x / and: Fou x, e 2πi x / x The invese: Fou : a z e 2πi z / z Know ou phase summations
Example What happens if ou appl Fou twice to? z e z e z 2πiz / z 2πiz / a a The (summation) is if z and if z. Hence the outcome state is. Question: What happens if we appl Fou twice to a basis state x with <x<?
Moe About Fouie Taditionall, Fouie tansfoms ae used to detect peiodic signals (depending on thei fequencies). In quantum computing we will use the QFT to detemine the peiodicit of a function F. Alead inteesting b itself, this peiodicit finding suboutine can be used to factoize numbes and calculate discete logaithms ove Z. See late Handouts fo moe technical details and a desciption of efficient cicuits to implement Fou.
Peiodicit Poblem Conside function F:{,, } S Assume that: F has peiod F is bijective on its peiod F (x) F() if and onl if x mod Task: detemine (efficientl ~ pol(log ) ote: This is the kind of global popet that quantum computing is useful fo.
Peiodicit Algoithm () Stat with a unifom supeposition of x values: x x, Calculate the peiodic function F fo these values: x x,f(x) t t + F() Measue the ightmost egiste; assume outcome F(c) with c< [Cf. Handout 3.]
Peiodicit Algoithm (2) this ields the supeposition fo the left egiste: + t c t Appl the Fouie tansfom ove Z, giving: ( ) + j t jt jc t j c t j j ζ ζ j ζ If j multiple of /, then constuctive intefeence If j not a multiple of /, then destuctive intefeence
Peiodicit Algoithm (3) Calculating the j-dependent intefeence: t ζ jt if j is a multiple of ζ ζ j jt ζ j t if j is not a multiple of Hence we have the output state: j ζ jc jt ζ j t k ζ ck k
Peiodicit Algoithm (4) With ve high pobabilit we will measue a multiple of /, whee is the peiod of the function. B epeating the pocedue seveal times, we obtain enough infomation to detemine / and hence. (This is not entiel tivial and equies the usage of the continued factions method, but it can be done.) Being able to find the (hidden) peiod of a function allows us to solve factoing, discete logaithms and othe (pesumed) had poblems.