b : the two eigenvectors of the Grover iteration Quantum counting algorithm

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1 .5.3. Quau couig algorih How quickly ca we deerie he uber of arge saes i Grover s algorih, i.e. arked saes, r, o a N= daa base search proble, if r is o kow i advace. Classical search ~ ( N ) Quau search ~ ( N ) (Grover ieraio + phase esiaio algorih) applicaios ) Quau search, eve if he uber of soluios is ukow ) NP-coplee SAT (saisfiabiliy) probles (= exisece of a soluio o a search proble) Quau algorih: a, si b : he wo eigevecors of he Grover ieraio ˆ ˆ ˆ ˆ ˆ Q I U I U U i he space spaed by o-soluio sae ad soluio sae. : he agle of roaio deeried by ˆQ Qˆ cos si si cos basis Correspodig eigevalues i e e i( ) ad 36

2 Q Q Q Sep : iiializaio 0 0 log + qubis eough o iplee he Grover ieraioi (N= + expaded search space) qubis esiae o bis of accuracy wih a probabiliy of success. Sep : Walsch-Hadaard rasfor o he firs ad secod regiser ˆ HH Sep 3: Corolled ˆ x x0 y0 y Liear superposiio of C a C b operaio a b a ad b The above circui provides a esiae of ad -. ˆ ( ) CQ ix ix a b x0 Q x e C a e C b 37

3 Sep 4: iverse Fourier rasfor o he firs regiser ˆ F C a C b a b equivale Sep 5: proecive easuree of he firs regiser wih a accuracy of wih probabiliy of success esiae of esiae r hrough How large a error, r, i his esiae? N r rn, 3 6 r r ~ 4 r N Grover ieraios allowable agle error ad reasoable success probabiliy 38

4 Nuber of required ieraios: R~ Agle error: N r ~ 8 Success probabiliy: cos ~. If (o soluio), so ha he algorih produces he esiae wih he probabiliy >5/6. NP-coplee SAT probles Exaple: 3-SAT (saisfiabiliy) proble Fid if he followig Boolea fucio ca be saisfied or o. f x x x x x x x f or zero? Classical brue-force algorih requires ~ poly Grover algorih requires ~ poly (Soluio exiss or o) This is bad ews for quau copuig. The esseial reaso for he difficuly of NP-coplee probles is ha heir search space has esseially o srucure. The bes possible ehod for solvig such a proble is o adop a Grover search ehod. This eas ha NPcoplee probles cao be solved by quau copuers (excep for a square-roo speed up) sice he Grover algorih is Kow o be opiu. 39

5 .6 Quau siulaio Digial quau siulaio: A quau copuer solves a physical (ay body) proble. Aalog Quau siulaio: A arificial ad corollable quau syse siulaes a real physical syse uder sudy. I order o calculae herodyaical properies p of a ay body syse, a eseble quau copuer, i which ay idepede icroscopic quau copuers execue he sae operaio ad produce a acroscopic readou sigal, is ore efficie ha a sadard quau copuer cosisig of a qubi syse..6.. Spi-laice syses Syse Hailoia,, ageic orderig (phase rasiio) Helholz free eergy per spi F k T B l Z k T B l e E : eperaure peauepaa paraeer ee k B T e E E Z e E de : herodyaic pariio fucio : eigesae idex E : eergy desiy of saes 40

6 herodyaic properies (ageizaio, specific-hea, ageic suscepibiliy ) ca be easily calculaed by he free eergy F oce i is kow. A brue-force approach o euerae he eige-eergies is difficul, because he uber of eigesaes grows expoeially wih he uber of spis i he laice. cf. quau Moe Carlo ehods (sig proble: D.P. Ladau ad K. Bider, A Guide o Moe Carlo Siulaio i Saisical Physics (Cabridge Uiv. Press, Cabridge, 000)) E of The followig quau algorihs are based o f Fourier rasfor of E f flow of copuaioal sep: ie ie E e de e Tr e i # of eigesaes f race of he ie evoluio operaor U ˆ g Tr e i g l g l E Z discree ies iverse Fourier rasfor siple iegraio quau copuaio classical l copuaio 4

7 .6. Isig odel J z i ˆ z ˆ z i h ˆ i i z i : diagoal i copuaioal basis exaple: heero-spi ework wih dipolar couplig uder dc ageic field Siple cofiguraio (failure case) Sep : 0 W ˆ exaple: 0 0 of eigesae f 4

8 Sep : 0 U ˆ l e ie l 0 This sep ca be ipleeed as a sequece of sigle qubi ad wo- qubi gaes, where he uber of gaes is a polyoial fucio of. For Isig odel i which he Hailoia cosiss of couig pair-wise ieracios, his decoposiio is eleeary. For he case of ocouig ers, Troer-Suzuki expasio ca be eployed. eige-eergy Sep 3: e ie l 0 W ˆ e ie l 0 0 Tr e i l g l orhogoal copoes c A ubiased esiaor for g l ca be obaied by repeaig he algorih ay ies ad couig he uber of ies all qubis are foud i he logical 0 sae. This probabiliy is equal o g l. If a eseble of ideical icroscopic quau copuers execues he sae algorih, he eseble easurees for all qubis reveal g l by oly oe copuaio: However, g l is isufficie o recosruc E. We eed Re g l ad. I g l failure 43

9 Modified cofiguraio (successful case) oe qubi acilla ˆ W Sep : 0 a 0 q Sep : W a, ˆ 0 a q CUˆ l 0 Sep 3: R ˆ i x i 0 a Tr ei l 0 q W ˆ 0 i a 0 q ig l 0 iel a e q a 0 0 q 0 0 i ig l a q 0 orhogoal a q copoes ˆ R x : roaio abou x-axis by 90 i 0 i 44

10 Sep 3 : R ˆ y 0 W ˆ 0 a 0 q a Tr ei l g l 0 0 gl a q 0 q 0 a q orhogoal copoes ˆ R y : roaio abou y-axis by 90 0 ig l Pr X,0,0 PrX,,0 PrY,0,0 ig l g l PrY,,0 g l Re g l Pr Y,0,0 Pr Y,,0 I g l Pr X,,,, 0 Pr X,0,0,, success 45

11 .6.3 Geeral spi Hailoia Heiseberg odel: J ˆ ˆ h ˆ i, i exaple: elecro spi ework wih exchage couplig XXZ odel: i i i i i, i i i i z J ˆ ˆ ˆ ˆ ˆ ˆ ˆ x x x y y Jzz z hiz exaple: elecro/uclear hoo-spi ework wih dipolar couplig Hailoias are o diagoal i he copuaioal basis. E. Kill ad R. Laflae, Phys. Rev. Le. 8, 567 (998) i o eed o easure! The acilla qubi is iiialized i qubis are i a fully ixed sae: ˆ I ˆ a ˆ a z ˆ 0 a 0 I q sae, while he reaiig ˆ I q ˆ I q l l l I ˆ : decoposiio of uiy orhooral se of saes for q q 46

12 We ca choose he eige-saes of as l. The iiial desiy arix ca be cosidered as a icohere ixure of he eige-saes of. l We choose he eigesaes of he give Hailoia as. Eve hough we do o kow explicily l, he iiial desiy arix ˆ I q ˆ I q ˆ I q ca be cosidered as a icohere ixure of eigesaes for ay Hailoia. ˆ R x a ˆ z R ˆ a y ˆ z I g l Re g l eseble averaged resul The required accuracy for easurig ˆ a z scales expoeially ( z ) as he proble size. a ~ f () g() # of olecules spis The copuaioal ie scales polyoially wih he proble size, 0( ). C.P. Maser e al., Phys. Rev. A 67, 033 (003) 47

13 .6.4 Jorda-Wiger rasforaio I order o siulae a ay-ferio syse, we eed a appig bewee he algebra of ferioic syse ad he algebra of spi- syse. a ˆ, a ˆ k k, a ˆ, ˆ a k ˆ appig oo spi-operaors (isoorphis) a, a ˆ k 0 aˆ ˆ ˆ ˆ ˆ ˆ ˆ aˆ ˆ ˆ ˆ ˆ ˆ ˆ l z zz z l l z zz z exaple: -D spiless ferios.. aˆ aˆ aˆ aˆ U ˆ ˆ uelig uual Coulob ieracio ˆ x ˆ x ˆ y y 4U ˆ ˆ z z U 0 : aisoropic Heiseberg (XXZ) odel Eseble quau algorih usig ixed sae qubis 48

14 .6.5 Efficiecy of he eseble quau algorih How ay saples of g() do we eed o accuraely calculae Z? g ie widow : T 0 fiie eergy widow E e E : Bolza facor e ~ T 0 saplig ie ierval high eergy lii for esiaig E broadeig fucio be E si c e e e : eergy resoluio T Recagular ie widow T 0 error Z Ee E de E axiu E eergy Nyquis saplig heore has a error due o low eergy side lobes T 0 e T T 0 T 0 Gaussia ie widow 0 E 0 E evelope falls off b E si c E e e E e The error ca be iiized by sall ad large, bu he uber of ieraios icreases. How does N scale wih he uber of spis? N T 0 The oal ie scale polyoially wih he uber of spis ( ~ O ). 49

Basic Magnetism Thorsten Glaser, University of Münster. Basic Magnetism. 1. Paramagnetism and Diamagnetism (macroscopic)

Basic Magnetism Thorsten Glaser, University of Münster. Basic Magnetism. 1. Paramagnetism and Diamagnetism (macroscopic) asic Mageis Thorse Glaser Uiversiy of Müser asic Mageis. Paraageis ad Diaageis (acroscopic) exeral ageic field H r diaageic saple paraageic saple r : ageic iducio (ageic r r field r iesiy iside saple)