Introduction to Quantum Information Processing. Overview. A classical randomised algorithm. q 3,3 00 0,0. p 0,0. Lecture 10.
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1 Itroductio to Quatum Iformatio Procssig Lctur Michl Mosca Ovrviw! Classical Radomizd vs. Quatum Computig! Dutsch-Jozsa ad Brsti- Vazirai algorithms! Th quatum Fourir trasform ad phas stimatio A classical radomisd algorithm! Svral computatioal paths ladig to th sam outcom.! Add up th probabilitis. q, Pr( p q,j p, p, p, p,3 q 3,3 j j,
2 A classical radomisd algorithm! Th probabilitis could corrspod to th squar of a probabilit amplitud (du to masurig th quatum sstm at ach timstp a, a,3 b, b 3,3 Pr( j a b,j j, A quatum algorithm! If w do t masur at ach tim stp, ol at th d, th probabilit amplituds first hav a chac to itrfr. a, a,3 b, b 3,3 Pr( j a b,j j, Dcohrc! A quatum sstm that is cotiuall masurd (or itracts with a tral sstm will bhav li a classical radomizd sstm! Partial masurmts will giv a probabilit distributio somwhr i btw th two trms! Error-corrctig cods will allow a quatum sstm itractig with th viromt to maitai cohrc.
3 Quatum Algorithms! Quatum Algorithms should ploit quatum paralllism ad quatum itrfrc.! W hav alrad s th Dutsch, Dutsch-Jozsa, Brsti-Vazirai ad Simo s algorithms. Multi-qubit adamard ( ( Quatum Algorithms! Ths algorithms hav b computig sstiall classical fuctios o quatum suprpositios! This codd iformatio i th phass of th basis stats: masurig basis stats would provid littl usful iformatio! But a simpl quatum trasformatio traslatd th phas iformatio ito iformatio that was masurabl i th computatioal basis 3
4 Quatum Phas Estimatio! Suppos w wish to stimat a umbr giv th quatum stat [, πi! Not that i biar w ca prss. K 3. K K. K Quatum Phas Estimatio π! Sic i for a itgr, w hav πi( πi(.3... πi πi( πi(.3... πi( πi( Quatum Phas Estimatio! If. th w ca do th followig + πi(. + ( 4
5 ! W ca show that πi sful idtit π ( ( i( π i( + + πi( L ( + π... i(.... ( ( i(. + π π... ( i(. L + Quatum Phas Estimatio! So if. th w ca do th followig + π + πi(. i(. R πi/ Quatum Phas Estimatio! So if. 3 th w ca do th followig πi(.3 πi(.3 πi(
6 Quatum Phas Estimatio! Gralizig this twor (ad rvrsig th ordr of th qubits at th d givs us a twor with O( gats that implmts πi Discrt Fourir Trasform! Th discrt Fourir trasform maps vctors of dimsio N b trasformig th th lmtar vctor accordig to πi (,,...,,,,... (, N, N, K, N! Th quatum Fourir trasform maps vctors i a ilbrt spac of dimsio N accordig to πi N N πi (N πi Discrt Fourir Trasform! Thus w hav illustratd how to implmt (th ivrs of th quatum Fourir trasform i a ilbrt spac of dimsio 6
7 Estimatig arbitrar [,! What if is ot cssaril of th form for som itgr? πi z z! Th QFT will map to a suprpositio whr Prob N ~ α 8 N π α O N Quatum Phas Estimatio! For a ral [, πi(8 + πi(4 + πi( ! With high probabilit 8 3 Eigvalu ic-bac! Rcall th tric : ( f( +f( ( ( f( f( ( ( f( f( ( ( 7
8 8! Cosidr a uitar opratio with igvalu ad igvctor Eigvalu ic-bac πi i π π π i i Eigvalu ic-bac Eigvalu ic-bac β + α i β + α π! As a rlativ phas, bcoms masurabl πi
9 Eigvalu ic-bac! If w potiat, w gt multipls of πi Eigvalu ic-bac + + πi Eigvalu ic-bac + πi( + + πi( + M + O πi( πi L 9
10 Phas stimatio πi( πi( πi πi( O M 3 L L M + + L + Eigvalu stimatio Eigvalu stimatio QFT 8 QFT 8 3
11 Eigvalu ic-bac! Giv with igvctor ad igvalu πi w thus hav a algorithm that maps ~ Eigvalu ic-bac! Giv with igvctors ad i rspctiv igvalus π w thus hav a algorithm that maps ~ α ad thrfor α α ~ Eigvalu ic-bac! Masurig th first rgistr of α ~ is quivalt to masurig ~ with probabilit α
12 Eampl! Suppos w hav a group G ad w wish to fid th ordr of a G (I.. th smallst positiv r such that a r! If w ca fficitl do arithmtic i th group, th w ca ralis a uitar oprator a that maps a! Notic that r r I a a! This mas that th igvalus of ar of th form whr is a itgr a i π r
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