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
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