Motivation Sho s Algoith It appas that th univs in which w liv is govnd by quantu chanics Quantu infoation thoy givs us a nw avnu to study & tst quantu chanics Why do w want to build a quantu coput? Pt Sho AT&T Labs Why build a classical coput? Why build a quantu coput? QC Thy a abl to pfo calculations any ods of agnitud fast than can b don with pncil and pap. Thy should b abl to pfo calculations any ods of agnitud fast than can b don on a classical coput. Sho-typ Algoiths Factoing Disct log Ablian stabiliz Spd-up: Eponntial? Quantu Algoiths Quantu Counting Quantu Siulations Gov-typ Algoiths Saching Mad stat Miniu Mdian Spd-up: quadatic Ovviw Sho s factoing algoith Phas stiation algoith Quantu Foui tansfo adaad gat Contolld- gat Equivalnc of factoing and od finding Solving od finding using PE Suay
Disct Foui Tansfo Disct Foui Tansfo Givn a squnc of copl nubs,,, K Th DFT poducs anoth squnc, y, y, K y wh y πi / y πi / It is not had to show that th tansfo y tuns th oiginal squnc. Ecis: Vify th foula fo D Disct Foui Tansfo If w lt and y b -by- vctos, thn y wh D 9 8 M 8 and L D O D y 8 M 9 8 L O Disct Foui Tansfo Suppos Thn π i {, } y By inspction, D D Ecis: Vify th foula fo y Quantu Foui Tansfo Th quantu Foui tansfo is a DFT of th aplituds of a quantu stat. Suppos w hav so stat, ψ + + K+ Th quantu Foui tansfo poducs th stat χ y + K+ y y D Quantu Foui Tansfo Th QFT is unitay can b iplntd vy fficintly An apl: S T S S i T iπ /
Quantu Foui Tansfo ψ ψ χ y + + + + + + + S T S y D χ Quantu Foui Tansfo In gnal,to pfo th QFT on n qubits quis O(n ) on and two qubit gats Rfnc: Clv t al. (quant-ph/98) Tansfoing n aplituds with only n opations Th fastst w can do classically is n n owv, QFT dos not allow us to ipov classical Foui tansfos Th is no fficint way to tact th aplituds of th stat χ y Quantu Foui Tansfo Pfoing a QFT dictly followd by a asunt is vy asy In fact, if you wish to asu dictly aft applying th QFT, you only nd n singl qubit otations! R R Ovviw Sho s factoing algoith Phas stiation algoith Quantu Foui tansfo adaad gat Contolld- gat Contolld- gat Equivalnc of factoing and od finding Solving od finding using PE Suay adaad gat + adaad gat + + + +
Contolld- gat Contolld- gat Two-qubit contolld- Multi-qubit contolld- Phas stiation algoith Givn a unitay opato and an ignstat of th opato Th goal of th PE algoith is to find th cosponding ignvalu ˆ i Phas Phas stiation algoith Th PE algoith uss two gists of qubits Th tagt gist, to which can b applid Th ind gist, which will b usd to sto th ignvalu of Phas stiation algoith 8 QFT Quantu cicuit diaga Phas stiation algoith W initially stat with th syst in th stat Pfoing th adaad gats on th ind gist cats th stat Pfoing th sis of contolld- gats givs ˆ ˆ
Phas stiation algoith W can ov th insid th suation And plac with i ˆ iϕ Phas stiation algoith Raanging, thn i ϕ πi if π Applying th quantu Foui tansfo givs Phas stiation algoith Gnally, will not b an intg With high pobability w will obtain th nast intg to Thus, w hav an -bit appoiation to. Pis RSA ncyption ad aft Rivst, Shai and Adlan, who ca up with th sch Basd on th as with which can b calculatd fo and And th difficulty of calculating and fo RSA ncyption is ad publicly availabl, and is usd to ncypt data and a th sct ys which nabl you to dcypt th data To cac th cod, a cod-ba nds to facto Bst cunt cacing thod on a classical coput ub fild siv Rquis p(o(n / log / n)) n is th lngth of A littl nub thoy Sallst a od Modula Aithtic Co-pi a b od Siply ans a b + is any intg and b < gcd( a, ) Gatst Coon Diviso o factos in coon!
A littl nub thoy a od Consid th quation y od y od ( y + )( y ) od ( y + )( y ) A littl nub thoy a od ( y + )( y ) gcd( y +, ) gcd( y, ) gcd( y +, ) gcd( y, ) Tivial solutions gcd can b calculatd vy fficintly Euclid s algoith BC A littl nub thoy a If w can find And th is vn Thn od / gcd( a +, ) / gcd( a, ) Povidd w don t gt tivial solutions y od A littl nub thoy a od What about th ifs and buts?!? Tho: Lt, wh and a pi nubs not qual to. Suppos a is chosn at ando fo th st {a : < a <, gcd(a,) }. Lt b th od of y od. Thn th pobability Pob( is vn and non-tivial) Poof: long, boing and coplicatd A littl nub thoy a od Finding is quivalnt to factoing Why can t w us a classical coput to find? It tas O( n ) opations Ecis: sing th duction of factoing to od-finding, and th fact that is co-pi to, facto Choosing a Consid th opato, a a od od As a and a co-pi, this opato is unitay Can b fficintly iplntd on a quantu coput What about,, 8,, ^ a od
Choosing an initial stat a od Consid th stat, ψ π i a od ψ is an ignstat of, with ignvalu i( ) π Thfo, if w could ppa ψ, w can us th PE algoith to fficintly find, and hnc facto. ψ Choosing an initial stat QFT ψ Thfo, if w could ppa, w can us th PE algoith to fficintly find, and hnc facto. ψ Choosing an initial stat a od Consid th stats, ψ ψ π i a od {, K} is an ignstat of, with ignvalu i( ) Choosing an initial stat QFT Ecis: Show π ψ ψ Choosing an initial stat a od Thfo, using th PE algoith, w can fficintly calculat Wh and a unnown If and a co-pi, thn cancling to an iducibl faction will yild. If and a not co-pi, w ty again. W want to find Suay a od Equivalnt to solving s two qubit gists, initially in th stat Calculat cicuits fo,,.. ^n Apply th phas stiation algoith Rpat if quid