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Z𣢧jC 0e0e     A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab 322@8  ʚ;p8ʚ;g4dd0B 0ppp@ <4dddd\pC 0- <4!d!d\hG 0<4BdBd\hG 0D<___PPT10 pp6 Quantum Searching _August 1, 2005, Universit de Montral The Fifth Canadian Summer School on Quantum Information 6_B96   8 Quadratic speed-up  Cost was O(M)(h- D@   Grover s Problem OA A query PCRemember the input QB Reversible R@ Unitary Query SD Unitary Query TEComputational Cost \NAlternative query model '$Grover s Algorithm *Standard Analysis  :Grover s Algorithm Rotation"( (  Amplitude Amplification  A A = some algorithm p = success probability of ADB  B  Amplitude Amplification A A = some algorithm p = success probability of ADB  B ^PGeneral Setting  Algorithm A,. @   UFGeneral Setting  Algorithm A,. @   %General Setting  Algorithm A,. @   &General Setting  Algorithm A,. @   (2-dimensional subspace  WIAmplitude Amplification Algorithm A,. @   ,Amplitude Amplification  1#Maximizing the success prob. 7': De-randomization (p known)   5&If m is not known?4$Guessing when p unknown cUBizarre example E4 Kiil, the cat ;+Guessing when p unknown VGQ Rotation$     9)Q Rotation$     ]O Cost of Q F5 The Recipe @   Algorithm A,(@  <,Complete Grover =-General rotation  >.General Analysis @0General Analysis B1General Analysis C2General Analysis D3General Analysis `T The Recipe @   Algorithm A,(@  1  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@(fz?" dd@  " @ ` n?" dd@   @@``PL    @ ` ` pH @`H @` (    0H "`9  T Click to edit Master title style! !$   0 " `i  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     SNZ  & # "  H-ff?"6@`NNN?N &  Bf?"6@`NNN?N/ b  H?"6@`NNN?N* b B B?"6@`NNN?N+ B  s *޽h ? 33___PPT10u..֎t+D=' = @B + CSSQI'05  `0(    0 "`  T Click to edit Master title style! !  0 "f G  W#Click to edit Master subtitle style$ $1  < "W&  'Peter Hyer www.cpsc.ucalgary.ca/~hoyer2 H(  :B  s *޽h ? 33___PPT10u..ݎ<+D=' = @B + *(  x  c $0  r  S    H  0޽h ? 33___PPT10i.ގ@+D=' = @B +PN     @D (  r  S \Y9     S XZ P ` <$ 0    s *,"` +,$ 0 Z* Actual running time on a quantum computer might be incomparable to running time on a classical computer. Availability of quantum computers is limited. Additional error correction not included. Not applicable in conjunction with measurements. Verifier required.  ;  B(] ?"6@ NNN?N,$  0 W*  D CP #   pS ,$@ 0  N` jJ?"6@`NNN?NCPn >"  GB  BDjJ?"0@NNN?N.  S 0Arollbackwcp,$@ 0  0d }'g ,$ 0 h Now only O( M)(h-D@    0h PY  gCost: O(M)(h-D@  H  0޽h ? 33,D$D___PPT10D. c+!dDB' = @B D{B' = @BA?%,( < +O%,( < +D=' =%(Dx=' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' D=-s6Bwipe(down)*<3<*%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*%(xD' =0l9 BBBpB*<3<*%(D' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%( D' =0l9 BBBB*<3<*%(:))?D' =0l9 BBBB*<3<*%(jD' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%(D' =0l9 BBBB*<3<*%(*))?D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' D=-s6Bwipe(down)*<3<*%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*%(xD' =0l9 BBBpB*<3<*%(D' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%( D' =0l9 BBBB*<3<*%(:))?D' =0l9 BBBB*<3<*%(jD' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%(D' =0l9 BBBB*<3<*%(*))?D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' D=-s6Bwipe(down)*<3<*%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*%(xD' =0l9 BBBpB*<3<*%(D' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%( D' =0l9 BBBB*<3<*%(:))?D' =0l9 BBBB*<3<*%(jD' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%(D' =0l9 BBBB*<3<*%(*))?D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(++0+ ++0+  ++0+  ++0+ ++0+ +}"  !!#g !(  x  c $\59   |  `] ] #" @ `] 3  T=_ж_ж ?"6@ NNN?N`] i022  @`3  TD_ж_ж ?"6@ NNN?N] i022  @`3  T蒘_ж_ж ?"6@ NNN?NP] i122  @`3  T<_ж_ж ?"6@ NNN?NP] i022  @`3  T_ж_ж ?"6@ NNN?N ] i022  @`3   T_ж_ж ?"6@ NNN?N@  ] i022  @`3   T_ж_ж ?"6@ NNN?N @ ] i022  @`3   TD_ж_ж ?"6@ NNN?N ] i122  @`3   T$ʘ_ж_ж ?"6@ NNN?N0] i022  @`3   TӘ_ж_ж ?"6@ NNN?N0] i022  @`3  Tܘ_ж_ж ?"6@ NNN?N] i122  @`3  TL_ж_ж ?"6@ NNN?N ] i022  @`B  Z_ж_ж221 ?"0@NNN?N  ]B  Z_ж_ж221 ?"0@NNN?N``]B  <221 ?"0@NNN?N]B  <221 ?"0@NNN?N]B  <221 ?"0@NNN?N00]B  <221 ?"0@NNN?N]B  <221 ?"0@NNN?N  ]B  <221 ?"0@NNN?N@ @ ]B  <221 ?"0@NNN?N  ]B  <221 ?"0@NNN?N]B  <221 ?"0@NNN?NPP]B  <221 ?"0@NNN?N]B  <221 ?"0@NNN?N]B  Z_ж_ж221 ?"0@NNN?N `B  Z_ж_ж221 ?"0@NNN?N ]`]  B ?"6@ NNN?Nzs W1 22    B ?"6@ NNN?Nz WN 22  D B ?"6@ NNN?N2  hInput: Problem:  G  E 6 ?"6@ NNN?N24   2f:{1,2,,N}{0,1} Find integer i such that f(i)=1x3 GGGG'GB     g B ?"6@ NNN?N    dYou can ask questions of the form:  What is f(j)?  3 3G.,   H  0޽h ? 33___PPT10i.+D=' = @B +'   > 6 `  %L(  Lx L c $p9   F M/  L M/  L N ?"6@ NNN?NG =j  G L N ?"6@ NNN?N/  Zf(j)  G@ L N  ?"6@ NNN?NMXm | What is f(j)?   G N G L G L N$ ?"6@ NNN?NAa AQuery  G  L H 1?"6@ NNN?NB  LB HD1?"0@NNN?NGB  LB HD1?"0@NNN?NB  LB HD1?"0@NNN?NGPPH L 0޽h ? 33___PPT10i.`|+D=' = @B +   Tp(  Tx T c $\,9   F M/  T M/  T N. ?"6@ NNN?NG =j  GMMM T N 2 ?"6@ NNN?N/  Zf(j)  GMMM@ T N46 ?"6@ NNN?NMXm | What is f(j)?   GMMM N G T G T N9 ?"6@ NNN?NAa AQuery  GMMM  T H 1?"6@ NNN?NB  TB HD1?"0@NNN?NGB  TB HD1?"0@NNN?NB  TB HD1?"0@NNN?NGPP F k M  T Mk  T N> ?"6@ NNN?Nk Mm ORemember the input:  G T NB ?"6@ NNN?N ?j" G T NA ?"6@ NNN?N   6" G  T NI ?"6@ NNN?Nn \f(j)" G T NM ?"6@ NNN?N{6 AQuery  G T H 1?"6@ NNN?NpFB TB HD1?"0@NNN?N PpPB TB HD1?"0@NNN?NFB TB HD1?"0@NNN?NF T N4R ?"6@ NNN?N   =j  GH T 0޽h ? 33___PPT10i.`|+D=' = @B +X,  o+g+p 3OP0 *(  Px P c $ <  TX#    QX N$ ?"6@ NNN?N[;   J", C K RX Np ?"6@ NNN?N>B <  ;j=1 C SX N ?"6@ NNN?N{ g  9N C~T > <  UX# e    VX N ?"6@ NNN?N[;   J", C K WX NL ?"6@ NNN?N>B <  ;j=1 C XX N ?"6@ NNN?N{ g  9N CI@   F bXw K/  ^X H ?"6@ NNN?N   W1 C  _X HX ?"6@ NNN?N L F Z"N C B aX TD1?"0@NNN?N h hKN   F cX ; y 1  dX B ?"6@ NNN?N   W1 C  eX B ?"6@ NNN?N L F Z"N C B fX TD1?"0@NNN?N h h gX N  ?"6@ NNN?N   |j|f(j)Z C2  n iX NDM ?"6@ NNN?N ,,$ 0 ~F Apply function f in superposition $ $CH X 0޽h ? 33e ] ___PPT10= .`|+ӹD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(Db' =%(D ' =4@BBBB%()?)?D^' =.&7 BBBBB_M -5.55556E-7 -1.85185E-6 L -0.20469 -0.39907 *3>*B ppt_xB ppt_y=0BBAA<*&XD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*KX%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*hX%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*iX%(+8+0+iX +   6`(  `x ` c $9   yF K GB &`  '` N0 ?"6@ NNN?N   4  G (` N8 ?"6@ NNN?NP " B 6" G )` N ?"6@ NNN?N  6" G; *` N@ ?"6@ NNN?Ny {  w Uf4 GO +` H 1?"6@ NNN?NnM D B ,`B HD1?"0@NNN?N n B -`B HD1?"0@NNN?ND  B .`B HD1?"0@NNN?ND? ? 8 /` N4 ?"6@ NNN?NR  ?  t|j 4  B 0`B HD1?"0@NNN?N = o= 8 1` NH ?"6@ NNN?NK  t|b 4   2` N ?"6@ NNN?N G@ 6" G 3` N( ?"6@ NNN?N   6" G8 4` N ?"6@ NNN?N Q=  t|j 4  z 5` N4 ?"6@ NNN?N G  |bf(j) Z G& N 6`  f @~ @~  ?"6@`NNN?N u  z*1 query 1 unit cost4 GH ` 0޽h ? 33___PPT10i.`|+D=' = @B +G  9x9 >I9(  x  c $9   8 (  4 YS  T ?"6@ NNN?N 4 T  SfF GGO  N 1?"6@ NNN?N  B  B ND1?"0@NNN?N  B  B ND1?"0@NNN?N ^ >   T ?"6@ NNN?N3 4 t|j 4  r  T(  ?"6@ NNN?N: (2 (-1)f(j)|j X O  yF K GB   +  Np ?"6@ NNN?N   4  G  NH ?"6@ NNN?NP " B 6" G  N ?"6@ NNN?N  6" G;  N< ?"6@ NNN?Ny {  w Uf4 GO  H 1?"6@ NNN?NnM D B B HD1?"0@NNN?N n B B HD1?"0@NNN?ND  B B HD1?"0@NNN?ND? ? 8  N" ?"6@ NNN?NR  ?  t|j 4  B B HD1?"0@NNN?N = o= 8   N`' ?"6@ NNN?NK  t|b 4   ! N, ?"6@ NNN?N G@ 6" G " N/ ?"6@ NNN?N   6" G8 # N3 ?"6@ NNN?N Q=  t|j 4  z $ N4 ?"6@ NNN?N G  |bf(j) Z G& 0 % N> ?"6@ NNN?N ,$ 0 @"  f & N C ?"6@ NNN?N/ | O  4  G!l *L Y > +p ,$@ 0 ' NhG ?"6@ NNN?NN 6" G ( NHE ?"6@ NNN?NOo 6" G; ) NdN ?"6@ NNN?N t  w Uf4 GO * H 1?"6@ NNN?N IB +B HD1?"0@NNN?N a 8 , NT ?"6@ NNN?N*   t|j 4  B -B HD1?"0@NNN?N  8 . NY ?"6@ NNN?N8 t|1 4   / N ^ ?"6@ NNN?N 6" G 0 Nd\ ?"6@ NNN?NMPm 6" Gl 1 Nd ?"6@ NNN?Nk Y  (-1)f(j)|j X O  8 2 Nj ?"6@ NNN?N6 t|1 4  )T ) <   3#  & J 4 No ?"6@ NNN?N8 <  =H  G 5 N 1?"6@ NNN?N) 9  )T ) <   6# ] +pO 7 Ns ?"6@ NNN?N8 <  =H  G 8 N 1?"6@ NNN?N) 9  B 9B HD1?"0@NNN?NPB :B HD1?"0@NNN?Nk B ; HD1?"0@NNN?N H tR B < HD1?"0@NNN?NgH R  = H 1?"6@ NNN?NL P ? N@x ?"6@ NNN?Np78 ,$ 0 `Proof of :,  Cz j0 @ 6t,$D 0 A N| ?"6@ NNN?NF  0 2 CB BB HD1?"0@NNN?N } 2 C NL ?"6@ NNN?NjL  n |b ,  %T ) <   D#    E N ?"6@ NNN?N8 <  9H C F N 1?"6@ NNN?N) 9  B GB HD1?"0@NNN?N ~ ~ H N ?"6@ NNN?Nb  |0+(-1)b|1 n C(  x I N| ?"6@ NNN?NZ ,$ 0 L Compute the value of f in the phases  ' 'GH  0޽h ? 33  ___PPT10 .`|+=D ' = @B DK ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*I%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(DY' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(++0+% ++0+? ++0+I ++0+I +M  }7u78@47(  4x 4 c $Ԣ9     z` 4# #" = ` / 4 T_ж_ж ?"6@ NNN?Nz` e0  @`/ 4 T_ж_ж ?"6@ NNN?Nz e0  @`/ 4 T_ж_ж ?"6@ NNN?NPz e1  @`/ 4 T_ж_ж ?"6@ NNN?NzP e0  @`/ 4 T_ж_ж ?"6@ NNN?N z e0  @`/  4 T_ж_ж ?"6@ NNN?N@ z  e0  @`/  4 T _ж_ж ?"6@ NNN?N z@  e0  @`/  4 T_ж_ж ?"6@ NNN?Nz  e1  @`/  4 T_ж_ж ?"6@ NNN?N0z e0  @`/  4 T_ж_ж ?"6@ NNN?Nz0 e0  @`/ 4 TP_ж_ж ?"6@ NNN?Nz e1  @`/ 4 T_ж_ж ?"6@ NNN?N z e0  @`B 4 Z_ж_ж ?"0@NNN?N z`zB 4 Z_ж_ж ?"0@NNN?N `B 4 Z_ж_ж ?"0@NNN?N z B 4 Z_ж_ж ?"0@NNN?N`z`B 4 Bg  ?"0@NNN?NzB 4 Bg  ?"0@NNN?NzB 4 Bg  ?"0@NNN?N0z0B 4 Bg  ?"0@NNN?NzB 4 Bg  ?"0@NNN?N z B 4 Bg  ?"0@NNN?N@ z@ B 4 Bg  ?"0@NNN?N z B 4 Bg  ?"0@NNN?NzB 4 Bg  ?"0@NNN?NPzPB 4 Bg  ?"0@NNN?NzB 4 Bg  ?"0@NNN?Nz 4 B,  ?"6@ NNN?N0sP S1    4 B ?"6@ NNN?N0P SN   !4 Bl ?"6@ NNN?N z { . m "4 B  ?"6@ NNN?N} u,$ 0 Suppose t solutions (here t=3): H(  T #4 B ?"6@ NNN?N R*,$  0 p Grover:0 @2CG D $4 6d$ ?"6@ NNN?N 8 f *,$ 0 l 1/p0 @2CG > w 5 %4#  5,$@  0 &4 HX) jJ?"6@`NNN?Nw 5 >"  GB '4 BDjJ?"0@NNN?Ng  ] (4 B, ?"6@ NNN?N V,,$ 0 y = N/t queries0 @2CG > w 5 )4#  5,$@  0 *4 H|2 jJ?"6@`NNN?Nw 5 >"  GB +4 BDjJ?"0@NNN?Ng  W ,4 B6 ?"6@ NNN?N  ,$  0 s Repetition:0 @2 CG  C -4 6|; ?"6@ NNN?N 8 f ,$ 0 k1/p0 @2CG \ .4 B? ?"6@ NNN?N V ,$ 0 x= N/t queries0 @2CG ` /4 6xD ?"6@ NNN?N  ,$ 0 4The  classical algorithm:$ H(G  l  9)  9496 ,$@ 0 14 HpH ?"6@ NNN?Ne 9  ]N$ O GT  # 24#  ()  34 BM ?"6@ NNN?N7 } [1" G  44 BR ?"6@ NNN?NX# ^"N" G B 54 BD1?"0@NNN?N:ggu 64 6dV ?"6@ NNN?N6 V ,$ 0 Success probability = p = t/NH H(GGG  74 6^ ?"6@ NNN?Nm , ,$ 0  A|0 = |y0 = "i=1|i B H(%G GH(     @4 c 6A rollbackf )R,$@  0H 4 0޽h ? 33xp___PPT10P.+D5D' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"4%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*94%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/4%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*64%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*74%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.4%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-4%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,4%(Ds' =%(Dr' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%4%(D' D=-s6Bwipe(down)*<3<*%4%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*%4%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*%4%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*%4%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*%4%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*%4%(xD' =0l9 BBBpB*<3<*%4%(D' =0l9 BBBB*<3<*%4%())?D' =0l9 BBBB*<3<*%4%( D' =0l9 BBBB*<3<*%4%(:))?D' =0l9 BBBB*<3<*%4%(jD' =0l9 BBBB*<3<*%4%())?D' =0l9 BBBB*<3<*%4%(D' =0l9 BBBB*<3<*%4%(*))?D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)4%(D' D=-s6Bwipe(down)*<3<*)4%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*)4%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*)4%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*)4%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*)4%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*)4%(xD' =0l9 BBBpB*<3<*)4%(D' =0l9 BBBB*<3<*)4%())?D' =0l9 BBBB*<3<*)4%( D' =0l9 BBBB*<3<*)4%(:))?D' =0l9 BBBB*<3<*)4%(jD' =0l9 BBBB*<3<*)4%())?D' =0l9 BBBB*<3<*)4%(D' =0l9 BBBB*<3<*)4%(*))?D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@4%(D' D=-s6Bwipe(down)*<3<*@4%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*@4%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*@4%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*@4%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*@4%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*@4%(xD' =0l9 BBBpB*<3<*@4%(D' =0l9 BBBB*<3<*@4%())?D' =0l9 BBBB*<3<*@4%( D' =0l9 BBBB*<3<*@4%(:))?D' =0l9 BBBB*<3<*@4%(jD' =0l9 BBBB*<3<*@4%())?D' =0l9 BBBB*<3<*@4%(D' =0l9 BBBB*<3<*@4%(*))?D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#4%(D' D=-s6Bwipe(down)*<3<*#4%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*#4%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*#4%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*#4%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*#4%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*#4%(xD' =0l9 BBBpB*<3<*#4%(D' =0l9 BBBB*<3<*#4%())?D' =0l9 BBBB*<3<*#4%( D' =0l9 BBBB*<3<*#4%(:))?D' =0l9 BBBB*<3<*#4%(jD' =0l9 BBBB*<3<*#4%())?D' =0l9 BBBB*<3<*#4%(D' =0l9 BBBB*<3<*#4%(*))?D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(4%(D' D=-s6Bwipe(down)*<3<*(4%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*(4%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*(4%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*(4%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*(4%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*(4%(xD' =0l9 BBBpB*<3<*(4%(D' =0l9 BBBB*<3<*(4%())?D' =0l9 BBBB*<3<*(4%( D' =0l9 BBBB*<3<*(4%(:))?D' =0l9 BBBB*<3<*(4%(jD' =0l9 BBBB*<3<*(4%())?D' =0l9 BBBB*<3<*(4%(D' =0l9 BBBB*<3<*(4%(*))?D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$4%(D' D=-s6Bwipe(down)*<3<*$4%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*$4%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*$4%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*$4%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*$4%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*$4%(xD' =0l9 BBBpB*<3<*$4%(D' =0l9 BBBB*<3<*$4%())?D' =0l9 BBBB*<3<*$4%( D' =0l9 BBBB*<3<*$4%(:))?D' =0l9 BBBB*<3<*$4%(jD' =0l9 BBBB*<3<*$4%())?D' =0l9 BBBB*<3<*$4%(D' =0l9 BBBB*<3<*$4%(*))?D' =4@BBBB%( D' =1:Bhidden*o3>+B#style.visibility<*@4%(+0+0+"4 ++0+#4 ++0+$4 ++0+(4 ++0+,4 ++0+-4 ++0+.4 ++0+/4 ++0+64 ++0+74 +  ~x~PH$ }(  Hx H c $9     z` H# #" = ` / H T_ж_ж ?"6@ NNN?Nz` e0  @`/ H T`_ж_ж ?"6@ NNN?Nz e0  @`/ H T_ж_ж ?"6@ NNN?NPz e1  @`/ H T_ж_ж ?"6@ NNN?NzP e0  @`/ H T_ж_ж ?"6@ NNN?N z e0  @`/  H Tؾ_ж_ж ?"6@ NNN?N@ z  e0  @`/  H T_ж_ж ?"6@ NNN?N z@  e0  @`/  H Tp_ж_ж ?"6@ NNN?Nz  e1  @`/  H T_ж_ж ?"6@ NNN?N0z e0  @`/  H Tx_ж_ж ?"6@ NNN?Nz0 e0  @`/ H T_ж_ж ?"6@ NNN?Nz e1  @`/ H T@_ж_ж ?"6@ NNN?N z e0  @`B H Z_ж_ж ?"0@NNN?N z`zB H Z_ж_ж ?"0@NNN?N `B H Z_ж_ж ?"0@NNN?N z B H Z_ж_ж ?"0@NNN?N`z`B H Bg  ?"0@NNN?NzB H Bg  ?"0@NNN?NzB H Bg  ?"0@NNN?N0z0B H Bg  ?"0@NNN?NzB H Bg  ?"0@NNN?N z B H Bg  ?"0@NNN?N@ z@ B H Bg  ?"0@NNN?N z B H Bg  ?"0@NNN?NzB H Bg  ?"0@NNN?NPzPB H Bg  ?"0@NNN?NzB H Bg  ?"0@NNN?Nz H B ?"6@ NNN?N0sP S1    H B ?"6@ NNN?N0P SN  B )H <D?"0@NNN?NpVpB +H <D?"0@NNN?Nf ''f l t 9  AH ,$@ 0B /H HD)?"0@NNN?Nt t9 B 0H HD)?"0@NNN?N 8 B 1H HD)?"0@NNN?N 8 %l    BH` G,$@ 0 .H B ?"6@ NNN?NG   ?1/N  EB 3H BDjJ?"0@NNN?N | z t 9  CH (  ,$@  0B DH BD)?"0@NNN?Nt t9 B EH BD)?"0@NNN?N 8 B FH BD)?"0@NNN?N 8 3z    GH   ,$@  0 HH B ?"6@ NNN?NG   ?2/N  EB IH BDjJ?"0@NNN?N | z t 9  JH s ) ,$@ 0B KH BD)?"0@NNN?Nt t9 B LH BD)?"0@NNN?N 8 B MH BD)?"0@NNN?N 8 3z    NH   ,$@ 0 OH B<  ?"6@ NNN?NG   ?3/N  EB PH BDjJ?"0@NNN?N | l U HU,$@ 0B *H BDjJ?"0@NNN?NUB ,H HDjJ?"0@NNN?NU}U  XH B ?"6@ NNN?Ng} QSuccess probability" G6 YH B` ?"6@ NNN?N ,$ 0 RClassical repetition" G( ZH B ?"6@ NNN?N,$ 0 DGrover" Gfz t 9  [H  *,$@ 0B \H BDf)?"0@NNN?Nt t9 B ]H BDf)?"0@NNN?N 8 B ^H BDf)?"0@NNN?N 8 l  H9  kH + ,$@ 0lN t 9  _H  G9 B `H BDf)?"0@NNN?Nt t9 B aH BDf)?"0@NNN?N 8 B bH BDf)?"0@NNN?N 8 lN t 9  cH  H B dH BDf)?"0@NNN?Nt t9 B eH BDf)?"0@NNN?N 8 B fH BDf)?"0@NNN?N 8 lN t 9  gH  H B hH BDf)?"0@NNN?Nt t9 B iH BDf)?"0@NNN?N 8 B jH BDf)?"0@NNN?N 8 3z    pH 5  ,$@ 0 qH B ?"6@ NNN?NG   ?4/N  EB rH BDjJ?"0@NNN?N |  l p  H2+ ,$@ 0p@ E   HE  B uH HDf)?"0@NNN?NF  B vH HDf)?"0@NNN?N? E ? B wH HDf)?"0@NNN?NE  p@  E  H E B yH HDf)?"0@NNN?N E B zH HDf)?"0@NNN?N@ @ D B {H HDf)?"0@NNN?N D p@    H  B }H HDf)? "0@NNN?N  B ~H HDf)? "0@NNN?N@ @ B H HDf)? "0@NNN?N  p@ &   H&  B H HDf)?"0@NNN?N&  B H HDf)?"0@NNN?N? & ? B H HDf)?"0@NNN?N&  p@ p&  Hp& B H HDf)?"0@NNN?Nq& B H HDf)?"0@NNN?N@ p@ % B H HDf)?"0@NNN?Np%  l   Hg ,$@  0N    H    H B* ?"6@ NNN?NG   ?5/N  EB H BDjJ?"0@NNN?N | N    H !   H B. ?"6@ NNN?NG   ?6/N  EB H BDjJ?"0@NNN?N | N    H pW  H B@3 ?"6@ NNN?NG   ?7/N  EB H BDjJ?"0@NNN?N | N    H  H B`7 ?"6@ NNN?NG   ?8/N  EB H BDjJ?"0@NNN?N | N    H  H B; ?"6@ NNN?NG   ?9/N  EB H BDjJ?"0@NNN?N | F    H m ,6 N  9)  H c   H Bt@ ?"6@ NNN?Ne 9  ]N$ O GT  # H#  ()  H BPD ?"6@ NNN?N7 } [1" G  H BpH ?"6@ NNN?NX# ^"N" G B H BD1?"0@NNN?N:gg H 6N ?"6@ NNN?N    A|0 = |y0 = "i=1|i B H(%G GH(    % H BY ?"6@ NNN?N # ,$  0 A(1)" G% H B] ?"6@ NNN?N ' ,$  0 A(2)" G% H Bda ?"6@ NNN?N ' ,$ 0 A(3)" G% H B,$D! 0B H BDf1?"0@NNN?N~o  o B H BDf1?"0@NNN?Nt [t B H BDf1?"0@NNN?N[z z  l  Hw,$D 0B H HD1?"0@NNN?NB H HD1?"0@NNN?NB H HD1?"0@NNN?NzzB H HD1?"0@NNN?N11B H HD1?"0@NNN?NB H HD1?"0@NNN?N  B H HD1?"0@NNN?N@ @ B H HD1?"0@NNN?N  B H HD1?"0@NNN?NB H HD1?"0@NNN?NQQB H HD1?"0@NNN?NB H HD1?"0@NNN?NH H 0޽h ? 33< <___PPT10;.+RD 9' = @B D8' = @BA?%,( < +O%,( < +D' =%(Db' =%(D ' =4@BBBB%()?)?D^' =.&7 BBBBB_M -3.05556E-6 -3.33333E-6 L -0.11024 -0.17847 *3>*B ppt_xB ppt_y=0BBAA<*HD' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D~' =%(D&' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*YH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*AH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*BH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D|' =%(D$' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*CH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*GH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D|' =%(D$' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*JH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*NH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*ZH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*[H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D|' =%(D$' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*kH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*pH%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D|' =%(D$' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(++0+YH ++0+ZH ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H +  p(  r  S H  `V   H  0޽h ? 33___PPT10i. 8a+D=' = @B +  xp0(  x  c $9   x  c $ M)   B  <D?"0@NNN?NF :gF ,  0 2d ,$ 0 m repetitions success probability H" mp (provided mp d" 2/3, say)SBFFBB<   H  0޽h ? 33___PPT10. #+ DO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+  +4  zr (  x  c $9   x  c $ M)   B  <D?"0@NNN?NF :gF   0 2d  m repetitions success probability H" mp (provided mp d" 2/3, say)SBFFBB<   Z  B ?"6@ NNN?N j ,$ 0 v(Solutions must be verifiable)   2l %yE  <o\,$@ 0  H  ?"6@ NNN?N%E @"" GB   HDV?"0@NNN?NPyP  C *Arollbackm'Y,$@ 0H  0޽h ? 33++___PPT10+. #+D?+' = @B D*' = @BA?%,( < +O%,( < +D'' =%(%(Dp'' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' D=-s6Bwipe(down)*<3<* %(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<* %(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<* %(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<* %(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<* %(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<* %(xD' =0l9 BBBpB*<3<* %(D' =0l9 BBBB*<3<* %())?D' =0l9 BBBB*<3<* %( D' =0l9 BBBB*<3<* %(:))?D' =0l9 BBBB*<3<* %(jD' =0l9 BBBB*<3<* %())?D' =0l9 BBBB*<3<* %(D' =0l9 BBBB*<3<* %(*))?D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' D=-s6Bwipe(down)*<3<* %(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<* %(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<* %(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<* %(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<* %(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<* %(xD' =0l9 BBBpB*<3<* %(D' =0l9 BBBB*<3<* %())?D' =0l9 BBBB*<3<* %( D' =0l9 BBBB*<3<* %(:))?D' =0l9 BBBB*<3<* %(jD' =0l9 BBBB*<3<* %())?D' =0l9 BBBB*<3<* %(D' =0l9 BBBB*<3<* %(*))?D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +@  // 0/(  x  c $$/9   x  c $  / }    0x ]/ ,$ 0 q Verifier f,. @   3  0,1 ]0 ,$ 0 f:{1,2,,N}{0,1}V@ BCBCB   087 0 } \6A|0 = |y0 = "i=1 ai|i @ <       BD ?"6@ NNN?N3 YN  K   NI ?"6@ NNN?N$  4  G AY~   #" =Q Y ,$@ 03 ( T|T_ж_ж ?"6@ NNN?NAY~  i0  @`3 ) TK_ж_ж ?"6@ NNN?NA~  i0  @`3 * Tf_ж_ж ?"6@ NNN?NIA~  i1  @`3 + TLp_ж_ж ?"6@ NNN?NAI~  i0  @`3 , Tx_ж_ж ?"6@ NNN?N A~  i0  @`3 - T_ж_ж ?"6@ NNN?N9 A ~  i0  @`3 . TX_ж_ж ?"6@ NNN?N A9 ~  i0  @`3 / T_ж_ж ?"6@ NNN?NA ~  i1  @`3 0 T_ж_ж ?"6@ NNN?N)A~  i0  @`3 1 TH_ж_ж ?"6@ NNN?NyA)~  i0  @`3 2 T_ж_ж ?"6@ NNN?NAy~  i1  @`3 3 T_ж_ж ?"6@ NNN?NA~  i0  @`B 6  `_ж_ж1 ?"0@NNN?NA~ B 7  `_ж_ж1 ?"0@NNN?NYAY~ B 4  `_ж_ж1 ?"0@NNN?NAYAB 5  `_ж_ж1 ?"0@NNN?N~ Y~ B d N1 ?"0@NNN?NA~ B i N1 ?"0@NNN?NyAy~ B n N1 ?"0@NNN?N)A)~ B s N1 ?"0@NNN?NA~ B x N1 ?"0@NNN?N A ~ B } N1 ?"0@NNN?N9 A9 ~ B  N1 ?"0@NNN?N A ~ B  N1 ?"0@NNN?NA~ B  N1 ?"0@NNN?NIAI~ B  N1 ?"0@NNN?NA~ B  N1 ?"0@NNN?NA~ ; C B ?"6@ NNN?N ~l ,$ 0 W1  ; D B ?"6@ NNN?N  ,$ 0 WN  K F 0  8 ,$ 0 f:{1,2,,N}{0,1}n@ FGFGF l s 2   s2 ,$@ 0 E 6 ?"6@ NNN?Ns 2   A|0 = |y0 = "i=1|iB H(%G GH(    N  9)    H   B ?"6@ NNN?Ne 9  ]N$ O GT  # #  ()   B  ?"6@ NNN?N7 } [1" G   B ?"6@ NNN?NX# ^"N" G B  BD1?"0@NNN?N:gg;  N ?"6@ NNN?N}T-,$  0 KExample: Grover  GH  0޽h ? 33___PPT10.0+D' `Tw= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D$ ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+P+0+ ++0+ ++0+C ++0+D ++0+F ++0+ +-  a%Y% &1d $(  dx d c $p9   x d c $q / }   d 0 ]/  q Verifier f,. @     d 0 ]0  f:{1,2,,N}{0,1}V@ BCBCB   d 0 0 } \6A|0 = |y0 = "i=1 ai|i @ <       d BP& ?"6@ NNN?N3 YN  K l T e | 1d  ,$D 0.  d # l*@~ @~  ?"6@`NNN?NT[ q L * - d # l/@~ @~  ?"6@`NNN?N  e | -1 application of A 1 query to Uf 1 unit cost ! - - GO z K GB d % 73 ,$@ 0 d N: ?"6@ NNN?N   4  G d NL< ?"6@ NNN?NP " B 6" G d N4@ ?"6@ NNN?N  6" G; d ND ?"6@ NNN?Ny {  w Uf4 GO d H 1?"6@ NNN?NnM D B dB HD1?"0@NNN?N n B dB HD1?"0@NNN?ND  B dB HD1?"0@NNN?ND? ? 8 d NJ ?"6@ NNN?NR  ?  t|j 4  B dB HD1?"0@NNN?N = o= 8 d NB ?"6@ NNN?NK  t|b 4   d NM ?"6@ NNN?N G@ 6" G d NV ?"6@ NNN?N   6" G8 d NhZ ?"6@ NNN?N Q=  t|j 4  z d N^ ?"6@ NNN?N G  |bf(j) Z G&   d Te ?"6@ NNN?N$  4  G l 0  0d7 ,$D 0 !d Tli ?"6@ NNN?N  6" G "d Tl ?"6@ NNN?Nf9Y  6" G #d Txo ?"6@ NNN?N T A4 GO $d N 1?"6@ NNN?N3 B 'dB ND1?"0@NNN?NdLdB )dB ND1?"0@NNN?N bb> *d Tj ?"6@ NNN?NR t|0 4   +d Tx ?"6@ NNN?N&  6" G ,d T,w ?"6@ NNN?N7] W  6" Gh .d TD{ ?"6@ NNN?N!0   |y0 \  H d 0޽h ? 33^V___PPT106.0+@zD ' = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0d%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1d%(+   ,r(  ,x , c $9   x , c $ / }  1z   ,  ,$D 0 , 0 /  QLet@   , 0t   ~p|Good ~ "i:f(i)=1 ai|i |Bad ~ "i:f(i)=0 ai|iv9@ t      , 0 g ,$ 0 >Success prob. of A = p = sin2 d @ 6   , 0 ]/  q Verifier f,. @     , 0 ]0  f:{1,2,,N}{0,1}V@ BCBCB   , 6 0 } \6A|0 = |y0 = "i=1 ai|i @ <      , Hl ?"6@ NNN?N3 YN  K t  , 0X )  ,$ 0 PA|0 = |y0 = sin |Good + cos |Bad)@     , , N ?"6@ NNN?N5=,$ 0 < H , 0޽h ? 33  ___PPT10 .0+:VD' = @B Dx' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(++0+, ++0+ , ++0+, +$    0 (  0x 0 c $D9   x 0 c $ / }  F   0   0 0D /  QLet@   0 0   ~p|Good ~ "i:f(i)=1 ai|i |Bad ~ "i:f(i)=0 ai|iv9@ t     \ 0 0L g  >Success prob. of A = p = sin2 d @ 6   0 0$# ]/  q Verifier f,. @     0 0) ]0  f:{1,2,,N}{0,1}V@ BCBCB @  0 00 )   PA|0 = |y0 = sin |Good + cos |Bad)@      0 0' =A@BB BB0B%(D' =-o6Bdissolve*<3<* 0D' =1:Bhidden*o3>+B#style.visibility<* 0%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*0 D' =1:Bhidden*o3>+B#style.visibility<*0 %(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D>' =A@BB BB0B%(D' =-o6Bdissolve*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =%(Do' =%(D' =A@BBBB0B%()?)?D^' =.&7 BBBBB_M 3.88889E-6 4.81481E-6 L 3.88889E-6 -0.55649 *3>*B ppt_xB ppt_y=0BBAA<* 0++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 +a  --/18,(  8x 8 c $X9   L 8 0p  v] RA|0 = |y0 = sin |Good + cos |Bad *@     F TP@  8 ' 2 8 < ?"6@ NNN?N P Y2 8 B 1?"6@ NNN?N B 8 BD1?"0@NNN?NTF @ F B 8 BD1?"0@NNN?N;};*  8 H ?"6@ NNN?N{u t |Good0 C ,  8 H4 ?"6@ NNN?N y  v |Bad4 h C B  8 BD1?"0@NNN?N\ q  2  8 ENH0Q ? `T`TS`T`TS`T`TS`T"6@ NNN?N k D2  8 3 >B4CGENGx3H :JQ 1? `T`T4B,G`T`T4B,G4B,G"6@ NNN?N*/c  8 H4 ?"6@ NNN?N  ^" B 8@ <D?"0@NNN?N6 Z 6 B 8@ <D?"0@NNN?N6  6 B 8 <D?"0@NNN?N B 8@ HD?"0@NNN?Nzl {Z 8 HT ?"6@ NNN?N^ m    |y0 b C  z  d  8  d ,$@ 0B 8 <D?"0@NNN?N ~.$N TP@  8   2 8 < ?"6@ NNN?N P Y2 8 B 1?"6@ NNN?N B 8 BD1?"0@NNN?NTF @ F B 8 BD1?"0@NNN?N;};B 8 BD1?"0@NNN?N'u R2 8 C gRB5CLENGH2HK@JQ 1? `T`T5@$L`T`T5@$L5@$L"6@ NNN?N7l * 8 H ?"6@ NNN?Nv p l2" " B 8 <D?"0@NNN?NFH 8 Hl ?"6@ NNN?NPdp |YL G J  8 HЦ ?"6@ NNN?N E Q|YL G B !8 BD1?"0@NNN?Nd  "8 Bd ?"6@ NNN?NP hp,$ 0 <Operator Q rotates by angle 2" G B #8 BD1?"0@NNN?NCM q ,$@ 0B %8 BD1?"0@NNN?N{C ~ ,$@, 0B &8 BD1?"0@NNN?N t ,$@X  0B '8 BD1?"0@NNN?N a ,$@  0B (8 BD1?"0@NNN?N2 ,$@  0B )8 BD1?"0@NNN?Nr ,$@ 0x2 *8 3 >B4COENGx3HFJQ 1? `T`T4BO`T`T4BO4BO"6@ NNN?Np#D ,$@ 0V +8 H  ?"6@ NNN?NZ ,$ 0 l3" " B ,8 BD1?"0@NNN?Nr ,$@ 0x2 .8 3 >B4C]TENGx3HYJQ 1? `T`T4BW]T`T`T4BW]T4BW]T"6@ NNN?Nu$I ,$@ 0V /8 HP ?"6@ NNN?NV2v,$ 0 l5" " ~ 08 H  ?"6@ NNN?N"m B,$ 0  QA|0 P C  18 H ?"6@ NNN?N7 ,$ 0 QQA|0 P C H 8 0޽h ?_P 8 88*8.8 3333___PPT10o3.0+\DK1' = @B D1' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"8%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* 8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#8%(D' =4@BBBB%(,D' =1:Bhidden*o3>+B#style.visibility<*#8%(D' =4@BBBB%(,D' =1:Bvisible*o3>+B#style.visibility<*%8%(D' =4@BBBB%(XD' =1:Bhidden*o3>+B#style.visibility<*%8%(D' =4@BBBB%(XD' =1:Bvisible*o3>+B#style.visibility<*&8%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*&8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'8%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*'8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(8%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*(8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)8%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*8%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* 8%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*+8%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<**8%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*8%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*08%(Dt ' =%(D ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*)8%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<**8%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*+8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,8%(D' =A@BBBB0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*18%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*08%(++0+8 ++0+8 ++0+"8 ++0++8 ++0++8 ++0+/8 ++0+08 ++0+08 ++0+18 +x*  "" %(p ,"(  px p c $,k9   x p c $l / }  F   p   p 0m /  QLet@   p 0ds   ~p|Good ~ "i:f(i)=1 ai|i |Bad ~ "i:f(i)=0 ai|iv9@ t     \ p 0, g  >Success prob. of A = p = sin2 d @ 6   p 0 ]/  q Verifier f,. @     p 0 ]- f:{1,2,,N}{0,1}V@ BCBCB @  p 0, )   PA|0 = |y0 = sin |Good + cos |Bad)@       p 0 -} \6A|0 = |y0 = "i=1 ai|i @ <       p B, ?"6@ NNN?N3   YN  K B p BD?"0@NNN?N   l C<  pC< ,$D 0*T TP@  p# C% 2 p < ?"6@ NNN?N P Y2 p B 221?"6@`NNN?N B p BD221?"0@NNN?NTF @ F B p BD221?"0@NNN?N;};B p HD1?"0@NNN?NRW}X2 p S gRB5CLENGH2HK@JQ 1? `T`T5@$L`T`T5@$L5@$L"6@`NNN?N9 $ p B• ?"6@ NNN?N l2" " B p BD?"0@NNN?N`N p NÕ ?"6@ NNN?NI |YL G P p N̕ ?"6@ NNN?N(" Q|YL G B p HD1?"0@NNN?N&i@ p N8ҕ ?"6@ NNN?N4[^ | |Good8 G 6 p BLו ?"6@ NNN?Nes<G  ~ |Bad< h G Tz %m  p %m ,$D 02  p < ?"6@ NNN?N%D 2 !p B 221?"6@`NNN?Nm B "p <D?"0@NNN?N` #p N?"6@ NNN?NcC6 2 $p H?"6@ NNN?N7`C2 %p H?"6@ NNN?N`C2 &p H?"6@ NNN?N2 'p H?"6@ NNN?NVH p 0޽h ?p 33\T___PPT104.0+ntD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(+7  **p+bP`)(  Px P c $49   F TP@  P M' 2 P < ?"6@ NNN?N P Y2 P B 1?"6@ NNN?N B P BD1?"0@NNN?NTF @ F B P BD1?"0@NNN?N;};*  P H  ?"6@ NNN?N* t |Good0 C ,  P Hı ?"6@ NNN?NEi  v |Bad4 h C  2  P ENH0Q ? `T`TS`T`TS`T`TS`T"6@ NNN?N k B P@ <D?"0@NNN?N6 Z 6 B P@ <D?"0@NNN?N6  6 B P <D?"0@NNN?N Jl VOp  @POVp ,$D 0@ Wk :PWkB  P BD1?"0@NNN?NWN P H4 ?"6@ NNN?Nk \   P H< ?"6@ NNN?NJO I A|0 =|y0  G(   P 6IJ V    A|0 LH @   %P 0ɲ  p  @= sin |Good + cos |Bad !@ f  l DI BPDI,$D 0 /P 0ղ  I QQA|0 LH @ fff  0P 0ܲ  DI <= sin5 |Good + cos5 |Bad @ ffffffffff\  z@ W +B#style.visibility<*@P%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*AP%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*BP%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*aP%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*bP%(+8+0+bP* ++3  #&&(Nl@%(  lx l c $H;*9  * F TP@  l M' 2 l < ?"6@ NNN?N P Y2 l B 1?"6@ NNN?N B l BD1?"0@NNN?NTF @ F B l BD1?"0@NNN?N;};* l H>* ?"6@ NNN?N* t |Good0 C ,  l H(C* ?"6@ NNN?NEi  v |Bad4 h C B  l@ <D?"0@NNN?N6 Z 6 B  l <D?"0@NNN?N  *l B*00 ?"6@`NNN?NM [ |QmA |0 = sin((2m+1)) |Good + cos((2m+1)) |Bad? CK$CC C Cz   (L Wk -l# WkB .l BD1?"0@NNN?NWN /l H\* ?"6@ NNN?Nk \   0l Na* ?"6@ NNN?NOJ I A|0 =|y0  G(  l  '  >l'  ,$D 0T  F 5l#   6l Bdj* ?"6@ NNN?N F @ 4" eB 7l HD1?"0@NNN?Ndn 8l Bn* ?"6@ NNN?N& E   "p R  eE2  9l Tu* ?"6@ NNN?N '  :hTheorem: Set m= , then Prob[Bad] d" sin2 = p r5 @*GG< n  F ;l#   ,$@ 0& +B#style.visibility<*>l%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*El%(DJ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Kl%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ml%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*Ml%(+,  G"?"P#<P!(  x  c $*9  * F TP@   M' 2  < ?"6@ NNN?N P Y2  B 1?"6@ NNN?N B  BD1?"0@NNN?NTF @ F B  BD1?"0@NNN?N;};*  H* ?"6@ NNN?N* t |Good0 C ,   H|* ?"6@ NNN?NEi  v |Bad4 h C B  @ <D?"0@NNN?N6 Z 6 B   <D?"0@NNN?N    BH* ?"6@ NNN?NGg =   Gf   Zh*00 ?"6@`NNN?NM [ |QmA |0 = sin((2m+1)) |Good + cos((2m+1)) |Bad? CK$CC C Cz   (L Wk # WkB  BD1?"0@NNN?NWN  Hط* ?"6@ NNN?Nk \    HT* ?"6@ NNN?NOJ I A|0 =|y0  G(  `  HP* ?"6@ NNN?N'   lTheorem: If prob. p is known, we can  de-randomize . "7 @7GBL z # zB  HDf1?"0@NNN?NfB   HDf1?"0@NNN?NfF2 ! # BENGoHIxQ f1? cncnx`Tnx`T"6@ NNN?N1( " N* ?"6@ NNN?Nza d2 fll   - ,$@ 0BT  F # G &  NT* f?"6@`NNN?N F b 2" efB  HDf1?"0@NNN?N\  H* ?"6@ NNN?N  If (2m+1) is slightly more than , then choose slightly smaller angle  < such that (2m+1) IS equal to . % { @(JfffffGffGff @   BT  F )#  jp & * N* f?"6@`NNN?N F b 2" efB + HDf1?"0@NNN?NB , HDjJ?"0@NNN?N\_,$@ 08 9 T* ?"6@ NNN?N =3 ,$ 0 BProof:  GfN : Nh+ ?"6@ NNN?N3S ,$ 0 ^   B < HD?"0@NNN?NUwp,$D 0H  0޽h ?! 33P H ___PPT10( .0+cD ' = @B DG ' = @BA?%,( < +O%,( < +D~' =%(D&' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(+p+0+9+ ++0+:+ +  0(@ (  r  S +9  +   XA soufflez:   +F TP@   M' 2  < ?"6@ NNN?N P Y2  B 1?"6@ NNN?N B   BD1?"0@NNN?NTF @ F B   BD1?"0@NNN?N;};*   HD+ ?"6@ NNN?N* t |Good0 C ,   H+ ?"6@ NNN?NEi  v |Bad4 h C B   <D?"0@NNN?N (L Wk # WkB  BD1?"0@NNN?NWN  Ht+ ?"6@ NNN?Nk \  BL z # zB  HDf1?"0@NNN?NfB  HDf1?"0@NNN?NfF2  # BENGoHIxQ f1? cncnx`Tnx`T"6@ NNN?N1(  N$+ ?"6@ NNN?Nza d2 f  H)+ ?"6@ NNN?NOJ I A|0 =|y0  G(  H  0޽h ? 33___PPT10i.ՏPdQ+D=' = @B +2   |v(  |r | S H5+9  + F TP@  | M' 2 | < ?"6@ NNN?N P Y2 | B 1?"6@ NNN?N B | BD1?"0@NNN?NTF @ F B | BD1?"0@NNN?N;};(L Wk  |# \pB  | BD1?"0@NNN?NWN  | H8+ ?"6@ NNN?Nk \    | Hl:+ ?"6@ NNN?NSO M A|0 =|y0  G(  *  | H`E+ ?"6@ NNN?N* t |Good0 C , | HJ+ ?"6@ NNN?NEi  v |Bad4 h C |l cl  |cl ,$@ 0B | HD1?"0@NNN?NaB | HD1?"0@NNN?NWwB | HD1?"0@NNN?NaB | HD1?"0@NNN?Ns]fB |B HD1?"0@NNN?NcfB |B HD1?"0@NNN?Nal B | HD1?"0@NNN?Nf \ | TO+ ?"6@ NNN?N  ,$ 0 f*A random vector yields success prob. = 1/2 + +G| | TT+ ?"6@ NNN?N/ 5,$ 0 NSolution: obtain a near-random vector by classically guessing m.O OCH | 0޽h ? 33___PPT10.Տ@yN+mGD$' = @B D' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(+p+0+|+ ++0+|+ +   7/P (  r  S |a+9  +   Nb+ ?"6@ NNN?Ne$ YViewer discretion is advised...  CH  0޽h ? 33___PPT10i.0 {+D=' = @B +^  RR@XYR(  x  c $hj+9  + B  HD1?"0@NNN?N,  N1?"6@`NNN?NB @ HD1?"0@NNN?N,  N1?"6@`NNN?N  N1?"6@`NNN?N&  N1?"6@`NNN?N@   N1?"6@`NNN?N_   N1?"6@`NNN?Ny.   N1?"6@`NNN?NX    N1?"6@`NNN?N r    N1?"6@`NNN?N    N1?"6@`NNN?N    N1?"6@`NNN?N%    N1?"6@`NNN?N?L Y< # ^GA  N1?"6@ NNN?NY<B  HDo?"0@NNN?NJ  VA catM <$D 0   c ,A catc,$@2 0  C *Acatdrunk jV,$@ 0  C *Acatdrunk V,$DD 0 j* # #"F: d G  Ht+ ?"6@ NNN?N* > @`  H|+ ?"6@ NNN?N* > @`  H~+ ?"6@ NNN?N * > @`  Hd+ ?"6@ NNN?N *  > @`  H+ ?"6@ NNN?Nb *  > @`  H+ ?"6@ NNN?NC *b  > @`  H,+ ?"6@ NNN?N$ *C  > @`   HĤ+ ?"6@ NNN?N*$  > @` ! H\+ ?"6@ NNN?N* > @` " H+ ?"6@ NNN?N* > @` # H+ ?"6@ NNN?N* > @` $ H$+ ?"6@ NNN?N* > @` % H+ ?"6@ NNN?Nj* > @`B & Ho ?"0@NNN?Nj*jB ' B1 ?"0@NNN?N*B ( B1 ?"0@NNN?N*B ) B1 ?"0@NNN?N*B * B1 ?"0@NNN?N*B + B1 ?"0@NNN?N*B , B1 ?"0@NNN?N$ *$ B - B1 ?"0@NNN?NC *C B . B1 ?"0@NNN?Nb *b B / B1 ?"0@NNN?N * B 0 B1 ?"0@NNN?N * B 1 B1 ?"0@NNN?N*B 2 B1 ?"0@NNN?N*B 3 Ho ?"0@NNN?N*B 4 Ho ?"0@NNN?NjB 5 Ho ?"0@NNN?N*j* 6 N1?"6@`NNN?NT 2 7 T+ ?"6@ NNN?N&F,$ 0 < 2 8 T+ ?"6@ NNN?N1,$ 0 < 2 9 T0+ ?"6@ NNN?N ,$ 0 < 2 : T+ ?"6@ NNN?N  ,$  0 < 2 ; T+ ?"6@ NNN?N  ,$  0 < 2 < T+ ?"6@ NNN?N  ,$ 0 < 2 = T+ ?"6@ NNN?N ,$ 0 < 2 > T+ ?"6@ NNN?NVv,$  0 < 2 ? T+ ?"6@ NNN?N6V,$  0 < 2 @ Tp+ ?"6@ NNN?N!A,$  0 < 0 A N+ ?"6@ NNN?N ,$ 0 @" 3 B ThA ?"6@ NNN?N<j,$ 0 =2  3 C T@A ?"6@ NNN?N" j ,$ 0 =3  3 D T A ?"6@ NNN?N j ,$  0 =4  3 E TTA ?"6@ NNN?N j ,$  0 =5  3 F T8A ?"6@ NNN?N j ,$  0 =6  3 G TA ?"6@ NNN?Nj~ ,$ 0 =7  3 H TA ?"6@ NNN?NjU,$  0 =8  3 I T$A ?"6@ NNN?Naj5,$  0 =9  4 J T A ?"6@ NNN?NL ,$  0 >10  2 K N#A ?"6@ NNN?N,$ 0 B11$ 2 L TA ?"6@ NNN?N!A,$4 0 < 2 M TA ?"6@ NNN?N  ,$5 0 < 2 N T|-A ?"6@ NNN?NQq,$7 0 < 2 O TT1A ?"6@ NNN?N1Q,$= 0 < 2 P T$6A ?"6@ NNN?N<,$? 0 < 0 Q N|4A ?"6@ NNN?N,$A 0 @" 0 R N\+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D"' =%(D!' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =A@BBBB0B%( D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =A@BBBB0B%( D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =A@BBBB0B%( D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =A@BBBB0B%( D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =A@BBBB0B%( D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =A@BBBB0B%( D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*Y%(D&' =%(DU&' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*B%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*C%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*D%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*E%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*F%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*G%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*H%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*I%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*J%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*K%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*7%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*9%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*:%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*;%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*<%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*=%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*>%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*?%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*@%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*A%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(+x +0+7A ++0+7A ++0+8A ++0+8A ++0+9A ++0+9A ++0+:A ++0+:A ++0+;A ++0+;A ++0+<A ++0+<A ++0+=A ++0+=A ++0+>A ++0+>A ++0+?A ++0+?A ++0+@A ++0+@A ++0+AA ++0+AA ++0+BA ++0+BA ++0+CA ++0+CA ++0+DA ++0+DA ++0+DA ++0+EA ++0+EA ++0+FA ++0+FA ++0+GA ++0+GA ++0+HA ++0+HA ++0+HA ++0+IA ++0+IA ++0+IA ++0+JA ++0+JA ++0+JA ++0+KA ++0+KA ++0+KA ++0+LA ++0+MA ++0+NA ++0+OA ++0+PA ++0+QA ++0+RA ++0+SA ++0+YA ++0+YA +   % 3(  x  c $PA9  A F TP@   M' 2  < ?"6@ NNN?N P Y2  B 1?"6@ NNN?N B  BD1?"0@NNN?NTF @ F B  BD1?"0@NNN?N;};(L Wk # \pB   BD1?"0@NNN?NWN   HA ?"6@ NNN?Nk \     HA ?"6@ NNN?NSO M A|0 =|y0  G(  *   HA ?"6@ NNN?N* t |Good0 C ,   HA ?"6@ NNN?NEi  v |Bad4 h C l  j4  j  ,$D 0   TE ?"6@ NNN?N 4  B Set M=1, =1.99 Repeat forever: Set M=M Let mR[1,M] Do QmA|0 Measure, say |i If f(i)=1, then break Output iv @G)GG GOGG%x "   8X  TE ?"6@ NNN?N   K 1) 2) 3)" @G  TE ?"6@ NNN?N j F Algorithm:  GVF cl   cl B  HD1?"0@NNN?NaB   HD1?"0@NNN?NWwB ! HD1?"0@NNN?NaB " HD1?"0@NNN?Ns]fB #B HD1?"0@NNN?NcfB $B HD1?"0@NNN?Nal B % HD1?"0@NNN?Nf H  0޽h ? 33___PPT10n.Տ@yN+YDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+9  ++#+ +Gh*(  hx h c $A9  A * h H`A ?"6@ NNN?N{u t |Good0 C , h H E ?"6@ NNN?N y  v |Bad4 h C  2 h ENH0Q ? `T`TS`T`TS`T`TS`T"6@ NNN?N k B h@ <D?"0@NNN?N6 Z 6 B h@ <D?"0@NNN?N6  6 j F   h  B  h <D?"0@NNN?N~*T TP@   h# & 2  h < ?"6@ NNN?N P Y2  h B 1?"6@ NNN?N B  h BD1?"0@NNN?NTF @ F B h BD1?"0@NNN?N;};B h BD1?"0@NNN?NC u R2 h C gRB5CLENGH2HK@JQ 1? `T`T5@$L`T`T5@$L5@$L"6@ NNN?N(7l * h HD(E ?"6@ NNN?N,vjp l2" " B h <D?"0@NNN?N H h H<-E ?"6@ NNN?N P p |YL G J h H2E ?"6@ NNN?N] } Q|YL G B h BD1?"0@NNN?Nzd  z  8p -h A y,$D 0p .h B48E ?"6@ NNN?N <\ S0 |0 h CKC  /h B<@E ?"6@ NNN?N&87X 9= C4 0h BCE ?"6@ NNN?Nc@[` | |0 : C  1h B0EE ?"6@ NNN?N>^ 9  Cp 2h B8GE ?"6@ NNN?N L~l S0 |i h CKC  3h BTE ?"6@ NNN?N&Hh <= -  C4 4h BOE ?"6@ NNN?NcPp | |i : C > 5h B|]E ?"6@ NNN?NNn  (i`"0) B CC t z  ; s  6h D | ,$D 0z 7h Bh BE ?"6@ NNN?NQ q   (f(i)=0) B CC.   9 Dh BdE ?"6@ NNN?N f,$ 0 UQ = -4 GO Eh B ?"6@ NNN?N ,$@ 0P Fh BE ?"6@ NNN?N g,$ 0 lSf4 GOt Gh B`E ?"6@ NNN?Nf,$ 0 AS0A-1n GOGOOH h 0޽h ?/ hh 33 y ___PPT10Y .0+ƸD ' = @B D@ ' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Dh%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Eh%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Fh%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Gh%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-h%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6h%(++0+DhE ++0+FhE ++0+GhE +  @@=MP?(  x  c $A9  E *  H,E ?"6@ NNN?N{u t |Good0 C ,  HE ?"6@ NNN?N y  v |Bad4 h C  2  ENH0Q ? `T`TS`T`TS`T`TS`T"6@ NNN?N k B @ <D?"0@NNN?N6 Z 6 B @ <D?"0@NNN?N6  6 B   BD?"0@NNN?N~"L TP@   # & 2   < ?"6@ NNN?N P Y2   B 1?"6@ NNN?N B   BD1?"0@NNN?NTF @ F B  BD1?"0@NNN?N;};B  BD?"0@NNN?N  F  8p  A yp  B|E ?"6@ NNN?N <\ S0 |0 h CKC   BE ?"6@ NNN?N&8{X :=  C4  BE ?"6@ NNN?Nc@[` | |0 : C   B0E ?"6@ NNN?N>^ 9  Cp  BE ?"6@ NNN?N L~l S0 |i h CKC   BE ?"6@ NNN?N&Hh <= -  C4  BE ?"6@ NNN?NcPp | |i : C >  BPE ?"6@ NNN?NNn  (i`"0) B CC @ F  ; s    D | z ! BE ?"6@ NNN?N ? Z_  Sf |i h CKC&  " BE ?"6@ NNN?N&; {[  :=  C4 # BAS0A-1 reflection about |y0  GOGOG  , H+F ?"6@ NNN?N a',$  0 .\ Q rotation by 2  GG* B 0 BD1?"0@NNN?N\ q  1 H3F ?"6@ NNN?N  ^" l # u  3# u ,$D 0B  HD1?"0@NNN?NC u Z 2 N|9F ?"6@ NNN?N # C |YX GC B 5 NDjJ?"0@NNN?N{ ,$@ 0B 7@ HDjJ?"0@NNN?N| V ,$@  0B 8 HDjJ?"0@NNN?N{ rm B 9@ BD1?"0@NNN?Nv ,$@ 0~2 : C BKCTBENGu-HJTBQ jJ? `T'4K`T'4KTB'4KTB"6@ NNN?N ,$@ 02 ; < ?"6@ NNN?N  B >@ HDf1?"0@NNN?N x| ,$D 0B ? NDfjJ?"0@NNN?Nv  ,$D 0B @ NDfjJ?"0@NNN?Nv ,$D 0B A HDf1?"0@NNN?N|  ,$@ 0~2 B C SUBKCTBENGu-HJTBQ fjJ? `T'4K`T'4KTB'4KTB"6@ NNN?N  ,$@ 0V C NAF ?"6@ NNN?N" B,$ 0 f2  V D NFF ?"6@ NNN?N 7 ,$ 0 f2  fX E HJF ?"6@ NNN?No 9   |y0 ` G x2 F 3 BENGfPeHIFLQ jJ? L0/L0/FL`TL0/FL`T"6@ NNN?NYZ ,$D 0   HLPF ?"6@ NNN?N 0f SQ =4 GO G H ?"6@ NNN?N " H HUF ?"6@ NNN?N g lSf4 GOF I HHZF ?"6@ NNN?Nf AS0A-1n GOGOO~2 L C CcENGúHbI`TJ Q jJ? Gu cGu c`T  c`T "6@ NNN?N   ,$@  0 M B`F ?"6@ NNN?Nf Q-4 GOH  0޽h ?o`:;BF L 33==___PPT10=.0+RD:' = @B D:' = @BA?%,( < +O%,( < +DT' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D4' =%(D' =%(D' =A@BBBB0B%(D' =1:B#333399 [5]*z3>!Bstyle.color= `B<*HD' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(DB' =%(D' =%(D|' =A@BBBB0B%(D' =1:B#cc0000*z3>!Bstyle.color= `B<*HD' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*F%(D' =A@BBBB0B%(D' =1:B#333399 [5]*z3>!Bstyle.color= `B<*IDY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(DB' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*L%(D|' =A@BBBB0B%(D' =1:B#cc0000*z3>!Bstyle.color= `B<*ID' =A@BBBB0B%(D' =1:B#333399 [5]*z3>!Bstyle.color= `B<*MDY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(Du ' =%(D ' =%(D|' =A@BBBB0B%(D' =1:B#cc0000*z3>!Bstyle.color= `B<*MD' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*3%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*9%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*5%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*7%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*:%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*C%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(++0+*F ++0++F ++0+,F ++0+CF ++0+CF ++0+DF ++0+HF ++0+HF ++0+IF ++0+IF ++0+MF ++0+MF + *  N$F$ %(#(  r  S F9  F Y  BF ?"6@ NNN?NYj    Q = -AS0A-1Sf GOGOGOWF T e |   V(   f(F@~ @~  ?"6@`NNN?NT[ q L * -f   f8F@~ @~  ?"6@`NNN?N  e | -1 application of A 1 query to Uf 1 unit cost ! - -f GfOff fyF K GB   73 <   NF ?"6@ NNN?N   4  G   NF ?"6@ NNN?NP " B 6" G   NF ?"6@ NNN?N  6" G;   NXF ?"6@ NNN?Ny {  w Uf4 GO  H 1?"6@ NNN?NnM D B B HD1?"0@NNN?N n B B HD1?"0@NNN?ND  B B HD1?"0@NNN?ND? ? 8  N F ?"6@ NNN?NR  ?  t|j 4  B B HD1?"0@NNN?N = o= 8  N4F ?"6@ NNN?NK  t|b 4    NܹF ?"6@ NNN?N G@ 6" G  NF ?"6@ NNN?N   6" G8  NPF ?"6@ NNN?N Q=  t|j 4  z  NF ?"6@ NNN?N G  |bf(j) Z G& N F 0   7   NF ?"6@ NNN?N  6" G  N|F ?"6@ NNN?Nf9Y  6" G  NF ?"6@ NNN?N T A4 GO  H 1?"6@ NNN?N3 B B HD1?"0@NNN?NdLdB B HD1?"0@NNN?N bb8   NF ?"6@ NNN?NR t|0 4   ! N\F ?"6@ NNN?N&  6" G " NF ?"6@ NNN?N7] W  6" Gb # NF ?"6@ NNN?N!0   |y0 \  Wl G  'O GY,$@ 0 $ NF ?"6@ NNN?NG e 2  J1 application of Q C % NF ?"6@ NNN?NA   O2 applications of Uf + 2 applications of A + 1 application of S0 5 units cost t: -CK,CKC<6 ( N(F ?"6@ NNN?NL w T V,$ 0 F   -H  0޽h ? 33g____PPT10?.+*D' = @B D' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+8+0+(F +3  # #P%"(  x  c $,J9  J ~  s *J   i J   0HJ ]/  q Verifier f,. @      0 J ]0  f:{1,2,,N}{0,1}V@ BCBCB    0J 0 } \6A|0 = |y0 = "i=1 ai|i @ <        BJ ?"6@ NNN?N3 YN  K    B$J ?"6@ NNN?N y ,$D  0  Q = -AS0A-1Sf GOGOGO z  8p  s  ,$@ 0  Bp+J ?"6@ NNN?N <\ S0 |0  GOG   B4J ?"6@ NNN?N&8{X >=   G@  B4.J ?"6@ NNN?Nc@[`  |0 F G   B=J ?"6@ NNN?N>^ =   G  B:J ?"6@ NNN?N L~l S0 |i  GOG   BGJ ?"6@ NNN?N&H{h >=   G@  BxLJ ?"6@ NNN?NcPp  |i F G N  BMJ ?"6@ NNN?NNn  (i`"0) R GG  z  ; s     ,$@ 0  BWJ ?"6@ NNN?N ? Z_  Sf |i  GOG&   B_J ?"6@ NNN?N&; {[  >=   G@  B4cJ ?"6@ NNN?NcC c   |i F G   BdJ ?"6@ NNN?NA a  g (f(i)=1)   G  BmJ ?"6@ NNN?N O Zo  Sf |i  GOG&   BtJ ?"6@ NNN?N&K k  ?=   G@  BDpJ ?"6@ NNN?NcS s   |i F G e  B}J ?"6@ NNN?NQ q   (f(i)=0) R GG.   G   BJ ?"6@ NNN?N[ ,$ 0 cDefine reflections:4 GOU ! B J ?"6@ NNN?N9 @Y ,$ 0 q!Amplitude Amplification operator:4" !GO- " NJ ?"6@ NNN?N,$ 0 =-  G- # NJ ?"6@ NNN?N,$ 0 =-  G $ ^A  souffle 6C" JH  0޽h ? 33sk___PPT10K.0+\冴D' = @B D' = @BA?%,( < +O%,( < +D ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(++0+ J ++0+ J ++0+!J ++0+"J ++0+#J +K  KDCDPBBC(  x  c $LJ9  J   z` # #" = ` J/  TJ_ж_ж ?"6@ NNN?Nz` e0  @`/  TtJ_ж_ж ?"6@ NNN?Nz e0  @`/  TJ_ж_ж ?"6@ NNN?NPz e0  @`/  T|J_ж_ж ?"6@ NNN?NzP e0  @`/  TJ_ж_ж ?"6@ NNN?N z e0  @`/   TJ_ж_ж ?"6@ NNN?N@ z  e0  @`/   TxJ_ж_ж ?"6@ NNN?N z@  e0  @`/   T^ 9  Cp * BAK ?"6@ NNN?N L~l S0 |i h CKC  + BdIK ?"6@ NNN?N&Hh <= -  C4 , BMK ?"6@ NNN?NcPp | |i : C > - BRK ?"6@ NNN?NNn  (i`"0) B CC N T  ; s  .# G  z / BHXK ?"6@ NNN?N ? Z_  Sf |i h CKC&  0 Bx`K ?"6@ NNN?N&; {[  :=  C4 1 BdK ?"6@ NNN?NcC c  | |i : C  2 BiK ?"6@ NNN?NA a  c (f(i)=1)   Cz 3 BmK ?"6@ NNN?N O Zo  Sf |i h CKC&  4 BpvK ?"6@ NNN?N&K k  <= -  C4 5 BzK ?"6@ NNN?NcS s  | |i : C U 6 BK ?"6@ NNN?NQ q   (f(i)=0) B CC.   N 7z  7 7z  8 BK ?"6@ NNN?N  YN  K $ 9 6K ?"6@ NNN?N7z  xr A|0 = "i=1|i: H(%CC4(   ?N  :   ; BܕK ?"6@ NNN?Nw W1 C  < BK ?"6@ NNN?NU Z"N C B = HD1?"0@NNN?NiwiZz   > 2 X ,$@ 0 ? NK ?"6@ NNN?N %   Apply A on |0, gives |y Repeat O("N) times: Apply Q on |y Measure |y Return outcome |i Vh  G G(GGGl    )    8X  @ NTK ?"6@ NNN?N   H 1) 2) 3) 4)  G& A NK ?"6@ NNN?N a  b&Grover s algorithm:  G B N K ?"6@ NNN?N n ,$ 0 6Theorem: Prob[f(i)=1] e" 2/3  G  H  0޽h ? 33___PPT10.+ [NDg' = @B D"' = @BA?%,( < +O%,( < +DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(+8+0+BK +F  ))`"\P)(  x  c $K9  K B @ <D?"0@NNN?N6 Z 6 B @ <D?"0@NNN?N6  6 B  <D?"0@NNN?N~Y  BK ?"6@ NNN?Nl  Q = -AS0A-1Sf GOGOGOn < HK ?"6@ NNN?N  S0 |0 h CKC  = HK ?"6@ NNN?NI Z 9= C2 > HK ?"6@ NNN?N  | |0 : C  ? HK ?"6@ NNN?NW 9  Cn @ HK ?"6@ NNN?N  * S0 |i h CKC  A HK ?"6@ NNN?NI & :=  C2 B H\K ?"6@ NNN?N . | |i : C < C HL ?"6@ NNN?N Q,  (i`"0) B CC x E HL ?"6@ NNN?N, L Sf |i h CKC&  F HL ?"6@ NNN?N(O `H 9= C2 G HL ?"6@ NNN?N0P | |i : C  H HL ?"6@ NNN?N.IN c (f(i)=1)   Cx I HxL ?"6@ NNN?N< \ Sf |i h CKC&  J H!L ?"6@ NNN?N8O X :=  C2 K H$&L ?"6@ NNN?N@` | |i : C S L H(+L ?"6@ NNN?N>I^  (f(i)=0) B CC.    M H$1L ?"6@ NNN?Nf 9- C N Hp5L ?"6@ NNN?N%lE 9- CV P B3L ?"6@ NNN?N|,$ 0 rs4 GO X Q B=L ?"6@ NNN?N/O,$ 0 tf 4 GO z R < BL2,$ 0 s,f C* t GO GO K ,! S B0IL ?"6@ NNN?Nf,$ 0 =-  G! T B|ML ?"6@ NNN?N'lG,$ 0 =-  GC U NhQL ?"6@ NNN?N3x S ,$  0 SSuitable choices of phases: C W NLL ?"6@ NNN?N  ,$  0 s = f = -1l GO GO G, X N[L ?"6@ NNN?N  9 ,$ 0 "s = f = i = "-1l GO GO G,  Z N$dL ?"6@ NNN?N  ,$ 0 s = f = ei/3 GO GO GnoB [ NmL ?"6@ NNN?N ,,$  0 .6s = f such that Re(s)>0 GO GO GO GBH  0޽h ? 33___PPT10.0+IcDl' = @B D'' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*M%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*N%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =%(DI' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*S%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*T%(D' =%(DM' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(++0+MK ++0+NL ++0+PL ++0+QL ++0+RL ++0+SL ++0+SL ++0+TL ++0+TL ++0+UL ++0+WL ++0+XL ++0+ZL ++0+[L +#  p3-(  r  S hL9  L   0L 6V TA|0 = sin |Good + cos |Bad +@  r      0L 0I `Q(s,f)A|0 =  sin |Good +  cos |Bad ^1@ FN FN       & 0tL"`9@ ,$ 0  -sfj GO GO GB ' 0L"`* ,$ 0  -1+f-sf~GO GO GO Xn ( 0L"`| {,$ 0 1-s-fXGO GO B< ) 0L"` ,$ 0 -sDGO G, * 61f4% ,$@ 0 Ss  +# fq% ,$@ 0  , BlL?   sin2DGOG  - BL S?   cos2DGOGf . 610s  0s  /# f% ,$@ 0 0 BL:  j.fC 1 BL< @  d(Cf 2 610s  3 <L9 Y,$  0 5=H  0޽h ? 33___PPT10.+1 iDW' = @B D' = @BA?%,( < +O%,( < +DI ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(++0+&L ++0+'L ++0+(L ++0+)L ++0+3L +!  !,+(  x  c $ N9  N   0hN 6V TA|0 = sin |Good + cos |Bad +@  r      0xN 0I `Q(s,f)A|0 =  sin |Good +  cos |Bad ^1@ FN FN      N  0,*N"`9@   -sfj GO GO GB  0 2N"`*   -1+f-sf~GO GO GO X:  0(;N"`| { 1-s-fXGO GO B  04BN"`  -sDGO G,^   61f4% L Ss   # fq%     B`HN?   sin2DGOG    B0NN S?   cos2DGOGf   610s L 0s  # f%   B0TN:  j.fC  B YN< @  d(Cf  610s   <T]N9 Y 5=|  NaN ?"6@ NNN?N   s = f = -1X CK CK C,  0hN"`)  I  V-1C  0|lN"`} +  i -3*CK    0pN"`(  |H  e3*CK  ! 0\uN"`  U1C " <pyN1"` 5 2 CL Ss  ##  r $ BL}N?   sin28CKC % BxN S?   cos28CKCf & 610s L 0s  '#   ( B@N:  f*bC ) BN< @  `$Cf * 610s  + <dN  1=H  0޽h ? 33___PPT10.+D:' = @B D' = @BA?%,( < +O%,( < +D,' =%(D' =%(D|' =A@BBBB0B%(D' =1:B#cc0000*z3>!Bstyle.color= `B<* +8+0+ N +6  og!!(  x  c $`N9  N   0N 6V TA|0 = sin |Good + cos |Bad +@  r      0N 0I `Q(s,f)A|0 =  sin |Good +  cos |Bad ^1@ FN FN      N  0N"`9@   -sfj GO GO GB  0N"`*   -1+f-sf~GO GO GO X:  0N"`| { 1-s-fXGO GO B  0N"`  -sDGO G,^   61f4% L Ss   # fq%     BN?   sin2DGOG    BN S?   cos2DGOGf   610s L 0s  # f%   BN:  j.fC  BlN< @  d(Cf  610s   <N9 Y 5=  N4N ?"6@ NNN?N / ,$ 0 s = f = iX CK CK C,  0O"`)  I ,$ 0 U1C   08O"`} + ,$ 0 h i*CK    04 O"`(  |H ,$ 0 h1-2i*CK   08O"` ,$  0 V-iC  <O1"` 5,$D  0 2 C Ss  #  r,$@ 0  BdO?   sin28CKC  B@N S?   cos28CKCf  610s  0s  #  ,$@ 0  B O:  f*bC  B%O< @  `$Cf   610s  ! <*O ,$ 0 1=H  0޽h ? 33___PPT10.+c.D' = @B D@' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DV ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(DX' =%(D|' =A@BBBB0B%(D' =1:B#cc0000*z3>!Bstyle.color= `B<*D|' =A@BBBB0B%(D' =1:B#cc0000*z3>!Bstyle.color= `B<*++0+O ++0+O ++0+O ++0+O ++0+O ++0+O ++0+O ++0+O ++0+!O +5    !! (  x  c $CO9  O   0X 6V TA|0 = sin |Good + cos |Bad +@  r      0PO 0I `Q(s,f)A|0 =  sin |Good +  cos |Bad ^1@ FN FN      N  0eO"`9@   -sfj GO GO GB  0mO"`*   -1+f-sf~GO GO GO X:  0vO"`| { 1-s-fXGO GO B  0}O"`  -sDGO G,^   61f4% L Ss   # fq%     B O?   sin2DGOG    BO S?   cos2DGOGf   610s L 0s  # f%   BO:  j.fC  B̔O< @  d(Cf  610s   <O9 Y 5=  NO ?"6@ NNN?N`  ,$ 0 s = f = ei/3 CK CK CjkCB@  0 O"`)  I ,$ 0 -e2i/3@Cjk,   0O"`} + ,$ 0 h 0*CK B  0ԱO"`(  |H ,$ 0 1-2ei/3@ Cjk,>  0(O"` ,$  0  -ei/3@Cjk,  <O1"` 5,$D  0 2 C Ss  #  r,$@ 0  B$O?   sin28CKC  B\O S?   cos28CKCf  610s  0s  #  ,$@ 0  B(O:  f*bC  B0O< @  `$Cf   610s  ! <0O ,$ 0 1=H  0޽h ? 33 ___PPT10.+c.D' = @B D' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DV ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D,' =%(D' =%(D|' =A@BBBB0B%(D' =1:B#cc0000*z3>!Bstyle.color= `B<*++0+O ++0+O ++0+O ++0+O ++0+O ++0+O ++0+O ++0+!O +2  0!$X(  x  c $4O9  O   0|O 6V TA|0 = sin |Good + cos |Bad +@  r      0lP 0I `Q(s,f)A|0 =  sin |Good +  cos |Bad ^1@ FN FN      N  0P"`9@   -sfj GO GO GB  0P"`*   -1+f-sf~GO GO GO X:  0 (P"`| { 1-s-fXGO GO B  0/P"`  -sDGO G,^   61f4% L Ss   # fq%     BD5P?   sin2DGOG    B;P S?   cos2DGOGf   610s L 0s  # f%   BAP:  j.fC  BEP< @  d(Cf  610s   <8JP9 Y 5=  NNP ?"6@ NNN?N  ,$ 0 6s = f such that Re(s)>0v CK CK CK CB  0`WP"`} + ,$ 0 q <12GO   <[P1"` 5,$@ 0 2 C Ss  #  r,$@ 0  B_P?   sin28CKC  BeP S?   cos28CKCf  610s  0s  #  ,$@ 0  BlP:  f*bC  BgP< @  `$Cf   610s  ! <uP ,$ 0 1= " 00zP"`)  I ,$  0 U?C # 0}P"`(  |H ,$  0 e?*CK  $ 0 P"` ,$ 0 U?CH  0޽h ? 33___PPT10.+?mD' = @B D' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DV ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(++0+P ++0+P ++0+P ++0+!P ++0+"P ++0+#P ++0+$P +"  !!@ N!(  x  c $P9  P   s *P   i P "  0P ]/  q Verifier f,. @     0P ]0  f:{1,2,,N}{0,1}V@ BCBCB   0(P 0 } \6A|0 = |y0 = "i=1 ai|i @ <       BdP ?"6@ NNN?N3 YN  K Y  BlP ?"6@ NNN?N y   Q = -AS0A-1Sf GOGOGOM F  8p   s     BP ?"6@ NNN?N <\ S0 |0  GOG    BP ?"6@ NNN?N&8{X >=   G@   BP ?"6@ NNN?Nc@[`  |0 F G    BP ?"6@ NNN?N>^ =   G  BLP ?"6@ NNN?N L~l S0 |i  GOG   BdP ?"6@ NNN?N&H{h >=   G@  BP ?"6@ NNN?NcPp  |i F G N  B P ?"6@ NNN?NNn  (i`"0) R GG  F  ; s       BPP ?"6@ NNN?N ? Z_  Sf |i  GOG&   BTR ?"6@ NNN?N&; {[  >=   G@  B,R ?"6@ NNN?NcC c   |i F G   B R ?"6@ NNN?NA a  g (f(i)=1)   G  B R ?"6@ NNN?N O Zo  Sf |i  GOG&   BtR ?"6@ NNN?N&K k  ?=   G@  BR ?"6@ NNN?NcS s   |i F G e  B!R ?"6@ NNN?NQ q   (f(i)=0) R GG.     B(R ?"6@ NNN?N[  cDefine reflections:4 GO!  B-R ?"6@ NNN?N9 @Y  q!Amplitude Amplification operator:4" !GO  N2R ?"6@ NNN?N =-  G  N 6R ?"6@ NNN?N =-  G  ^A  souffle 6C" RH  0޽h ? 33___PPT10i.0+D=' = @B +r 4* 0m5:<2z; 0%@s*u,1g4 Q@79/;@zF:w @w-BP8O؅׻pSCOll(]#a3~\0p>p`clTeW cOh+'0 hp    $08Quantum Searchinge4Fifth Canadian Summer School on Quantum Informationiftift79tMicrosoft PowerPoint Sc@;@A֎@[$Gg  SZ  --$--'@"Verdana-. 2 ipPeter Hr!."System-@"Verdana-.  2 i.-@"Verdana-.  2 i0yer.-@"Verdana-. 02 www.cpsc.ucalgary.ca/~hoyer   .-@"Verdana-. f!2 Quantum Searching,#"##6&!"###.-@"Verdana-. 2 dMAugust 1, 2005,    .-@"Verdana-. 2 d8 Universit    .-@"Verdana-.  2 d.-@"Verdana-. 2 dde Montr  .-@"Verdana-.  2 dJ.-@"Verdana-.  2 dZal .-@"Verdana-. Z2 M7The Fifth Canadian Summer School on Quantum Information            .-2՜.+,08    On-screen Shown-s )O .ArialVerdanaSymbol Wingdings CSSQI'05Quantum SearchingQuadratic speed-upGrover’s Problem A queryRemember the input ReversibleUnitary QueryUnitary QueryComputational CostAlternative query modelGrover’s AlgorithmStandard Analysis Slide 13Amplitude AmplificationAmplitude AmplificationGeneral SettingGeneral SettingGeneral SettingGeneral Setting2-dimensional subspaceAmplitude AmplificationAmplitude AmplificationMaximizing the success prob."“De-randomization” (p known) If m is not known?Guessing when p unknownBizarre exampleKiil, the catGuessing when p unknownQ  RotationQ  Rotation Cost of Q The RecipeComplete GroverGeneral rotationGeneral AnalysisGeneral AnalysisGeneral AnalysisGeneral AnalysisGeneral Analysis The Recipe  Fonts UsedDesign Template Slide Titles)_[ L    !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%2Root EntrydO)PicturesCurrent UserSummaryInformation(PowerPoint Document( DocumentSummaryInformation8