Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics

A Numerical Quantum and Classical Adversary 2011

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A  S e m id e fi n it e  P ro g ra m m in g  A  S e m id e fi n it e  P ro g ra m m in g  F o rm u la ti o n  o f Q u a n tu m  a n d  F o rm u la ti o n  o f Q u a n tu m  a n d  C la s s ic a l O ra c le  P ro b le m s C la s s ic a l O ra c le  P ro b le m s M ik e  M u lla n  a n d  E m a n u e l M ik e  M u lla n  a n d  E m a n u e l K n ill K n ill U n iv e rs it y  o f C o lo ra d o  a t B o u ld e r a n d  t h e  U n iv e rs it y  o f C o lo ra d o  a t B o u ld e r a n d  t h e  N a ti o n a l In s ti tu te  o f S ta n d a rd s  a n d  T e c h n o lo g y N a ti o n a l In s ti tu te  o f S ta n d a rd s  a n d  T e c h n o lo g y O u tl in e O u tl in e 1 . 1 . S e m id e fi n it e  P ro g ra m m in g  F o rm u la ti o n  o f Q u a n tu m  S e m id e fi n it e  P ro g ra m m in g  F o rm u la ti o n  o f Q u a n tu m  C o m p u ta ti o n  C o m p u ta ti o n  [ 2 ,3 ] [2 ,3 ] •• C a n  c a lc u la te  C a n  c a lc u la te  e x a c t e x a c t e rr o r p ro b a b ili ti e s  f o r a  q u a n tu m  e rr o r p ro b a b ili ti e s  f o r a  q u a n tu m  c o m p u te r o p ti m a lly  s o lv in g  a rb it ra ry  q u a n tu m  o ra c le  c o m p u te r o p ti m a lly  s o lv in g  a rb it ra ry  q u a n tu m  o ra c le  p ro b le m s  p ro b le m s  2 . 2 . R e m o te  S ta te  P re p a ra ti o n , a n d  a  G e n e ra liz e d  R e m o te  S ta te  P re p a ra ti o n , a n d  a  G e n e ra liz e d  O b je c ti v e O b je c ti v e 3 . 3 . R e s tr ic te d  C o m p u ta ti o n R e s tr ic te d  C o m p u ta ti o n •• D e c o h e re n c e  a n d  a n a ly z in g  a  c o m p u ta ti o n  s u b je c t to  D e c o h e re n c e  a n d  a n a ly z in g  a  c o m p u ta ti o n  s u b je c t to  n o is e n o is e •• O p ti m a l a lg o ri th m s  u s in g  r e d u c e d  q u a n tu m  r e s o u rc e s  O p ti m a l a lg o ri th m s  u s in g  r e d u c e d  q u a n tu m  r e s o u rc e s  (b ri e fl y ) (b ri e fl y ) 4 . 4 . Q u a n tu m  C lo c k s Q u a n tu m  C lo c k s Q u a n tu m  A lg o ri th m s , G e n e ri c a ll y Q u a n tu m  A lg o ri th m s , G e n e ri c a ll y Q u a n tu m  Q u e ry  A lg o ri th m : A rb it ra ry  u n it a ry  o p e ra to rs  in te rs p e rs e d  w it h  o ra c le  o p e ra to rs  a c ti n g  o n  s o m e  i n it ia l s ta te , fo llo w e d  b y  a  m e a s u re m e n t S e m id e fi n it e  P ro g ra m m in g : L in e a r p ro g ra m m in g  w it h  s e m id e fi n it e  c o n s tr a in ts | ψ (t )〉 = U t Ω U t− 1 Ω .. .Ω U 0 | ψ (0 )〉 G o a l:  M in im iz e  t h e  n u m b e r o f q u e ri e s  o f a  q u a n tu m  q u e ry  a lg o ri th m v ia  a  g e n e ra liz a ti o n  o f A m b a in is ’ a d v e rs a ry  m e th o d  [1 ] (a n d  r e c e n t p ro g re s s  g iv e n  i n  [ 8 ]) T ry  t o  e x p re s s  a  q u a n tu m  q u e ry  a lg o ri th m a s  a  s e m id e fi n it e  p ro g ra m  ( S D P ) : G ro v e r G ro v e r ’’ s  A lg o ri th m s  A lg o ri th m 00 00 11 00 U =  I n v e rs io n  a b o u t th e  M e a n 1 . 2 . 3 . 4 . 5 .|ψ (0 )〉 = |0〉 U 0 |ψ (0 )〉 = 1 2 (|0 〉+ |1〉 + |2〉 + |3〉 ) U 1 Ω U 0 |ψ (0 )〉 = |1〉 Ω 3 Ω 0 Ω 1 Ω 2 Ω U 0 |ψ (0 ) 〉 = 1 2 ( |0 〉− |1 〉 + |2 〉 + |3 〉 ) Ω U 0 |ψ (0 ) 〉 = 1 2 ( ( −1 )Ω 0 |0 〉 + ( − 1 )Ω 1 |1 〉 + ( − 1) Ω 2 |2 〉 + ( − 1) Ω 3 |3 〉 ) a i → − a i + 2 ∗〈 a 〉 G o a l:  S e a rc h  – F in d  t h e  m a rk e d  e le m e n t Ω |ψ (t )〉 = ∑ ia iΩ |i〉 → ∑ ia i( −1 )Ω i |i〉 O th e r O ra c le s  A re  P o s s ib le O th e r O ra c le s  A re  P o s s ib le 00 00 00 11 00 00 11 00 W e  c a n  a ls o  h a v e : 11 00 00 00 00 11 00 00 A s s u m e  e a c h  o ra c le  c a n  b e  g iv e n  t o  o u r c o m p u te r w it h  s o m e  p ro b a b ili ty N e w  G o a l:  F in d  t h e  m a rk e d  e le m e n t → Id e n ti fy  t h e  a p p lie d  o ra c le Ω (0 ) Ω (1 ) Ω (2 ) Ω (3 ) ρ O = ∑ x,y√ p x √ p y |x〉 〈y | Q : Q u e ri e r s y s te m ρ Q = ∑ i,ja i, j |i〉 〈j| O : O ra c le  s y s te m P u re  s ta te  o v e r p o s s ib le  o ra c le s T h e  C o m p u te r T h e  C o m p u te r A : A n c ill a ρ O = ∑ x,y√ p x √ p y |x〉 〈y | Q : Q u e ri e r s y s te m ρ Q = ∑ i,ja i, j |i〉 〈j| O : O ra c le  s y s te m Ω O Q = ∑ x| x 〉〈x |Ω Q (x ) P u re  s ta te  o v e r p o s s ib le  o ra c le s T h e  C o m p u te r T h e  C o m p u te r A : A n c ill a ρ O = ∑ x,y√ p x √ p y |x〉 〈y | Q : Q u e ri e r s y s te m ρ Q = ∑ i,ja i, j |i〉 〈j| O : O ra c le  s y s te m Ω O Q = ∑ x| x 〉〈x |Ω Q (x ) P u re  s ta te  o v e r p o s s ib le  o ra c le s T h e  C o m p u te r T h e  C o m p u te r A : A n c ill a Ω O Q (|x 〉O ⊗ |ψ 〉Q ) → (|x 〉O ⊗ Ω Q (x )|ψ 〉Q ) T h e  I n it ia l J o in t D e n s it y  M a tr ix T h e  I n it ia l J o in t D e n s it y  M a tr ix ρ O Q (0 ) = ∑ x,y∑ i, j √ p x √ p y |x〉 〈y |⊗ a i, j |i〉 〈j | ρ O Q (0 ) =      √ p 0 √ p 0 ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) √ p 0 √ p 1 ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) √ p 1 √ p 0 ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) . . .       ρ Q (0 ) T h e  I n it ia l J o in t D e n s it y  M a tr ix T h e  I n it ia l J o in t D e n s it y  M a tr ix ρ O Q =      √ p 0 √ p 0 Ω (0 )† ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) Ω( 0) √ p 0 √ p 1 Ω (0 )† ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) Ω( 1) √ p 1 √ p 0 Ω (1 )† ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) Ω( 0) † . . .       ρ O Q (0 ) = ∑ x,y∑ i, j √ p x √ p y |x〉 〈y |⊗ a i, j |i〉 〈j | ρ O Q (0 ) =      √ p 0 √ p 0 ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) √ p 0 √ p 1 ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) √ p 1 √ p 0 ( a 0 ,0 a 0 ,1 a 1 ,0 . . . ) . . .       Ω † ρ O Q (0 )Ω → ∑ x,y∑ i, j √ p x √ p y | x〉 〈 y |⊗ Ω (x )† a i, j | i〉 〈 j | Ω (y ) ρ Q (0 ) Q u a n tu m  A lg o ri th m  Q u a n tu m  A lg o ri th m  →→ S D P  ( 1 ) S D P  ( 1 ) ρ O Q N e e d  b o th  a n  o b je c ti v e  a n d  a  s e t o f lin e a r c o n s tr a in ts  o v e r (S tr a ig h tf o rw a rd ) C o n s tr a in ts ρ O Q (t ) ≥ 0, ρ O (t ) ≥ 0 ρ O (t ) = tr Q (ρ O Q (t )) ρ O (0 ) = ρ 0 t ∈ {0 .. .t f } Q u a n tu m  A lg o ri th m  Q u a n tu m  A lg o ri th m  →→ S D P  ( 2 ) S D P  ( 2 ) |ψ (t )〉 = U tΩ U t− 1 Ω .. .Ω U 0 |ψ (0 )〉 ρ O (t ) = tr Q A (U † Q A t Ω † O Q ρ O Q A (t − 1) Ω O Q U Q A t ) Q u a n tu m  A lg o ri th m  Q u a n tu m  A lg o ri th m  →→ S D P  ( 2 ) S D P  ( 2 ) |ψ (t )〉 = U tΩ U t− 1 Ω .. .Ω U 0 |ψ (0 )〉 ρ O (t ) = tr Q A (U Q A t U † Q A t Ω † O Q ρ O Q A (t − 1 )Ω O Q ) ρ O (t ) = tr Q A (U † Q A t Ω † O Q ρ O Q A (t − 1) Ω O Q U Q A t ) Q u a n tu m  A lg o ri th m  Q u a n tu m  A lg o ri th m  →→ S D P  ( 2 ) S D P  ( 2 ) |ψ (t )〉 = U tΩ U t− 1 Ω .. .Ω U 0 |ψ (0 )〉 ρ O (t ) = tr Q (Ω † O Q ρ O Q (t − 1) Ω O Q ) ρ O (t ) = tr Q A (U Q A t U † Q A t Ω † O Q ρ O Q A (t − 1 )Ω O Q ) ρ O (t ) = tr Q A (U † Q A t Ω † O Q ρ O Q A (t − 1) Ω O Q U Q A t ) Q u a n tu m  A lg o ri th m  Q u a n tu m  A lg o ri th m  →→ S D P  ( 2 ) S D P  ( 2 ) L e a d s  t o  t h e  S D P  S E , c o rr e s p o n d in g  to  t h e  a p p lic a ti o n  o f o ra c le  a n d  u n it a ry  o p e ra to rs  a s  a b o v e |ψ (t )〉 = U tΩ U t− 1 Ω .. .Ω U 0 |ψ (0 )〉 S E =                 ρO Q (t ) ≥ 0 , ρ O (t ) ≥ 0 ρ O (t ) = tr Q (ρ O Q (t )) ρ O (t ) = tr Q (Ω † O Q ρ O Q (t − 1 )Ω O Q ) ρ O (0 ) = ρ 0 t ∈ {0 .. .t f } ρ O (t ) = tr Q (Ω † O Q ρ O Q (t − 1) Ω O Q ) ρ O (t ) = tr Q A (U Q A t U † Q A t Ω † O Q ρ O Q A (t − 1 )Ω O Q ) ρ O (t ) = tr Q A (U † Q A t Ω † O Q ρ O Q A (t − 1) Ω O Q U Q A t ) M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n T h e  f o llo w in g  p ro to c o l d e s c ri b e s  t h e  m e a s u re m e n t p ro c e s s  a t th e  e n d  o f th e  c o m p u ta ti o n  a t ti m e  t f. [6 ] 1 . Q  m a k e s  a  m e a s u re m e n t w it h  a n y  c o m p le te  P O V M {P Q A i } i M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n T h e  f o llo w in g  p ro to c o l d e s c ri b e s  t h e  m e a s u re m e n t p ro c e s s  a t th e  e n d  o f th e  c o m p u ta ti o n  a t ti m e  t f. [6 ] 1 . Q  m a k e s  a  m e a s u re m e n t w it h  a n y  c o m p le te  P O V M 2 . C o n d it io n a l o n  m e a s u re m e n t o u tc o m e  i , th e  o ra c le  s y s te m  s ta te  i s  ( u p  t o  n o rm a liz a ti o n ) {P Q A i } i σ i = tr Q A (P Q A i ρ O Q A (t f )) M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n T h e  f o llo w in g  p ro to c o l d e s c ri b e s  t h e  m e a s u re m e n t p ro c e s s  a t th e  e n d  o f th e  c o m p u ta ti o n  a t ti m e  t f. [6 ] 1 . Q  m a k e s  a  m e a s u re m e n t w it h  a n y  c o m p le te  P O V M 2 . C o n d it io n a l o n  m e a s u re m e n t o u tc o m e  i , th e  o ra c le  s y s te m  s ta te  i s  ( u p  t o  n o rm a liz a ti o n ) 3 .  W e  m e a s u re  a  c o s t fu n c ti o n  A i o n  t h is  r e m o te ly  p re p a re d  s ta te , w h ic h  i n d ic a te s  h o w  s u c c e s s fu l o u r a lg o ri th m  w a s {P Q A i } i σ i = tr Q A (P Q A i ρ O Q A (t f )) M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n M e a s u re m e n t a n d  R e m o te  S ta te  P re p a ra ti o n T h e  f o llo w in g  p ro to c o l d e s c ri b e s  t h e  m e a s u re m e n t p ro c e s s  a t th e  e n d  o f th e  c o m p u ta ti o n  a t ti m e  t f. [6 ] 1 . Q  m a k e s  a  m e a s u re m e n t w it h  a n y  c o m p le te  P O V M 2 . C o n d it io n a l o n  m e a s u re m e n t o u tc o m e  i , th e  o ra c le  s y s te m  s ta te  i s  ( u p  t o  n o rm a liz a ti o n ) 3 .  W e  m e a s u re  a  c o s t fu n c ti o n  A i o n  t h is  r e m o te ly  p re p a re d  s ta te , w h ic h  i n d ic a te s  h o w  s u c c e s s fu l o u r a lg o ri th m  w a s Q ’s  P O V M  i s  u n c o n s tr a in e d , b u t th e  s e t o f s ta te s  t h a t c a n  b e  p re p a re d  o n  O  i s  r e s tr ic te d  b y : {P Q A i } i σ i = tr Q A (P Q A i ρ O Q A (t f )) ∑ iσ O i = ρ O (t f ) T h e  C o m p le te  S D P T h e  C o m p le te  S D P S = S M ∪ S E S E =                 ρO Q (t ) ≥ 0 , ρ O (t ) ≥ 0 ρ O (t ) = tr Q (ρ O Q (t )) ρ O (t ) = tr Q (Ω † O Q ρ O Q (t − 1 )Ω O Q ) ρ O (0 ) = ρ 0 t ∈ {0 .. .t f } S M =             M in im iz e ∑ it r (σ iA i) s u bj e ct to : ∀ i σ i ≥ 0 ∑ iσ O i = ρ O (t f ) C o n v e n ti o n a l C o s t F u n c ti o n C o n v e n ti o n a l C o s t F u n c ti o n  E a c h  m e a s u re m e n t o u tc o m e  E a c h  m e a s u re m e n t o u tc o m e  ii c o rr e s p o n d s  t o  t h e  c o rr e s p o n d s  t o  t h e  q u e ri e r q u e ri e r ’’ ss g u e s s  o f w h ic h  o ra c le  w a s  a p p lie d g u e s s  o f w h ic h  o ra c le  w a s  a p p lie d  S e a rc h : S e a rc h : i =  2 i =  2 –– T h e  m a rk e d  e le m e n t is  i n  p o s it io n  T h e  m a rk e d  e le m e n t is  i n  p o s it io n  22  A v e ra g e  P ro b a b ili ty  o f E rr o r:  P ro b a b ili ty  t h a t y o u  h a d  A v e ra g e  P ro b a b ili ty  o f E rr o r:  P ro b a b ili ty  t h a t y o u  h a d  o ra c le  o ra c le  xx , m e a s u re d  o u tc o m e  , m e a s u re d  o u tc o m e  ii , a n d  t h a t y o u  s h o u ld n , a n d  t h a t y o u  s h o u ld n ’’ t  t m e a s u re  o u tc o m e  m e a s u re  o u tc o m e  ii o n  o ra c le  o n  o ra c le  xx  S e a rc h : S e a rc h : x  =  1 0 0 0 , i =  2  x  =  1 0 0 0 , i =  2   T h is  e rr o r p ro b a b ili ty  w ill  b e  T h is  e rr o r p ro b a b ili ty  w ill  b e  e x a c t e x a c t .  Q  i s  f re e  t o  c h o o s e  .  Q  i s  f re e  t o  c h o o s e  th e  P O V M  w h ic h  o p ti m iz e s  t h e  c o s t fu n c ti o n , a n d  t h e  th e  P O V M  w h ic h  o p ti m iz e s  t h e  c o s t fu n c ti o n , a n d  t h e  S D P  w ill  f in d  t h is  o p ti m u m  s u b je c t to  i ts  c o n s tr a in ts .  S D P  w ill  f in d  t h is  o p ti m u m  s u b je c t to  i ts  c o n s tr a in ts .  (σ i) x ,x = P (m e a s u r e d i ∩ o r a cl e = x ) ǫ = ∑ i∑ x :x  = i( σ i) x ,x 〈x |A i|x 〉= 1 − δ (i ,x ) R e s tr ic te d  C o m p u ta ti o n R e s tr ic te d  C o m p u ta ti o n  A ll e x p e ri m e n ta lly  a v a ila b le  q u a n tu m  A ll e x p e ri m e n ta lly  a v a ila b le  q u a n tu m  c o m p u te rs  a re  s u b je c t to  n o is e  c o m p u te rs  a re  s u b je c t to  n o is e   D e c o h e re n c e  c a n  r e d u c e  o r e lim in a te  a n y  D e c o h e re n c e  c a n  r e d u c e  o r e lim in a te  a n y  q u a n tu m  a d v a n ta g e . q u a n tu m  a d v a n ta g e .  A ll c u rr e n t,  g e n e ra l q u a n tu m  l o w e r b o u n d  A ll c u rr e n t,  g e n e ra l q u a n tu m  l o w e r b o u n d  m e th o d s  a s s u m e  p e rf e c t q u a n tu m  m e th o d s  a s s u m e  p e rf e c t q u a n tu m  c o m p u ta ti o n . c o m p u ta ti o n .  W e  w o u ld  l ik e  a  w a y  t o  c h a ra c te ri z e  h o w  W e  w o u ld  l ik e  a  w a y  t o  c h a ra c te ri z e  h o w  ““ q u a n tu m q u a n tu m ”” o u r d e v ic e  i s o u r d e v ic e  i s D e c o h e re n c e D e c o h e re n c e A ft e r e a c h  q u e ry , th e  e n v ir o n m e n t w ill  i n te ra c t w it h  th e  c o m p u te r a n d  w ill  d e c o h e re th e  q u e ri e r [1 1 ] P o s t U n it a ry , P re -Q u e ry |ψ (0 )〉O Q E = ∑ x,ia x x ,i |x 〉O |i〉 Q |ǫ 0 〉E 1 D e c o h e re n c e D e c o h e re n c e A ft e r e a c h  q u e ry , th e  e n v ir o n m e n t w ill  i n te ra c t w it h  th e  c o m p u te r a n d  w ill  d e c o h e re th e  q u e ri e r [1 1 ] P o s t U n it a ry , P re -Q u e ry P o s t Q u e ry T h e  s iz e  o f th e  e n v ir o n m e n t g ro w s  w it h  e v e ry  q u e ry , s o  a t ti m e  t : |ψ (0 )〉O Q E = ∑ x,ia x x ,i |x 〉O |i〉 Q |ǫ 0 〉E 1 |ψ (1 )〉O Q E = ∑ x,ia x i, a |x〉 O (− 1) Ω (x ) i |i〉 Q |ǫ i 〉E 1 |ǫ 0 〉E 2 E (t ) ≡ E = E 1 ⊗ E 2 .. . ⊗ E t A d d in g  t h e  E n v ir o n m e n t A d d in g  t h e  E n v ir o n m e n t Q : Q u e ri e r s y s te m A : A n c ill a O : O ra c le  S y s te m E 1 E t 2 E : T h e  e n v ir o n m e n t is  p a rt  o f th e  o ra c le  s y s te m  t h a t th e  q u e ri e r h a s  n o  a c c e s s  t o , b u t a c ts  o n  t h e  q u e ri e r d u ri n g  e a c h  q u e ry . T h e  E x te n d e d  S D P T h e  E x te n d e d  S D P M in im iz e tr (∑ i σ iA i) su b je ct to : ∀ i σ O E i ≥ 0 ρ O Q E (t ) ≥ 0 , ρ O E (t ) ≥ 0 ∑ iσ O E i = ρ O E (T ) ρ O E (t ) = tr Q (ρ O Q E (t )) ρ O E (t ) = tr Q (D † Q E t Ω † O Q ρ O Q E (t − 1 ) ⊗ |ǫ 0 〉E t E t 〈ǫ 0 |Ω O Q D Q E t ) ρ O E (0 ) = ρ 0 t ∈ {0 .. .t f } Q u a n tu m /C la s s ic a l In te rp o la ti o n Q u a n tu m /C la s s ic a l In te rp o la ti o n O n e  o r m o re  q u b it s  m a y  d e c o h e re in d e p e n d e n tl y  w it h  p ro b a b ili ty  p R e d : O n e  q u b it d e c o h e re n c e G re e n : T w o  q u b it d e c o h e re n c e |x〉 O |i〉 Q |ǫ 0 〉E → √ 1− p |x〉 O (− 1 )Ω (x ) i |i〉 Q |ǫ 0 〉E + √ p |x〉 O (− 1) Ω (x ) i |i〉 Q |ǫ( i) 〉E C o m m e n ts C o m m e n ts  T h e  s iz e  o f th e  S D P  g ro w s  r a p id ly , p a rt ic u la rl y  w h e n  T h e  s iz e  o f th e  S D P  g ro w s  r a p id ly , p a rt ic u la rl y  w h e n  m o d e lin g  m o d e lin g  d e c o h e re n c e d e c o h e re n c e .  (  <  1 0  q u b it s  ) .  (  <  1 0  q u b it s  )  S in c e  t h e  S D P  o u tp u ts  t h e  o p ti m a l d e n s it y  m a tr ix  a t S in c e  t h e  S D P  o u tp u ts  t h e  o p ti m a l d e n s it y  m a tr ix  a t e v e ry  t im e  s te p , w e  c a n  r e c o n s tr u c t th e  i n te r e v e ry  t im e  s te p , w e  c a n  r e c o n s tr u c t th e  i n te r -- q u e ry  q u e ry  u n it a ri e s u n it a ri e s , a n d  h e n c e  t h e  o p ti m a l a lg o ri th m , a n d  h e n c e  t h e  o p ti m a l a lg o ri th m  W e  h a v e  r e c e n tl y  e x p lo re d  t h e s e  i s s u e s  f ro m  a n  W e  h a v e  r e c e n tl y  e x p lo re d  t h e s e  i s s u e s  f ro m  a n  a lt e rn a te  a n g le  b y  a s k in g : a lt e rn a te  a n g le  b y  a s k in g : Is  t h e re  a n  e q u a lly  o p ti m a l Is  t h e re  a n  e q u a lly  o p ti m a l a lg o ri th m  t h a t u s e s  f e w e r q u a n tu m  r e s o u rc e s ? a lg o ri th m  t h a t u s e s  f e w e r q u a n tu m  r e s o u rc e s ?  U s e  a n  a lt e rn a te  S D P  w h ic h  t ri e s  t o  s p lit  t h e  o p ti m a l a lg o ri th m  U s e  a n  a lt e rn a te  S D P  w h ic h  t ri e s  t o  s p lit  t h e  o p ti m a l a lg o ri th m  in to  t w o  o r m o re  p a rt s  t h a t c a n  b e  c la s s ic a lly  c o m b in e d in to  t w o  o r m o re  p a rt s  t h a t c a n  b e  c la s s ic a lly  c o m b in e d  U s e  t h e  V o n  N e u m a n n  e n tr o p y  t o  q u a n ti fy  t h e  q u a n tu m  U s e  t h e  V o n  N e u m a n n  e n tr o p y  t o  q u a n ti fy  t h e  q u a n tu m  re s o u rc e s  n e e d e d  i n  e a c h  b ra n c h re s o u rc e s  n e e d e d  i n  e a c h  b ra n c h  S e e  s lid e s  a t e n d  o f ta lk  f o r d e ta ils S e e  s lid e s  a t e n d  o f ta lk  f o r d e ta ils C lo c k s C lo c k s  O u r S D P  f o rm u la ti o n  i s  h ig h ly  g e n e ra l,  a n d  c a n  O u r S D P  f o rm u la ti o n  i s  h ig h ly  g e n e ra l,  a n d  c a n  d e s c ri b e  q u a n tu m  d e s c ri b e  q u a n tu m  p ro c e s s e s p ro c e s s e s a s  w e ll a s  q u a n tu m  a s  w e ll a s  q u a n tu m  q u e ry  a lg o ri th m s q u e ry  a lg o ri th m s  S p e c if ic a lly , if  a n  e le m e n t d ra w n  f ro m  a  c o n ti n u o u s  S p e c if ic a lly , if  a n  e le m e n t d ra w n  f ro m  a  c o n ti n u o u s  s e t o f u n it a ry  o p e ra to rs , in d e x e d  b y  s o m e  s e t o f u n it a ry  o p e ra to rs , in d e x e d  b y  s o m e  p a ra m e te r,  a c ts  o n  a  s ta te , o u r fo rm u la ti o n  c a n  b e  p a ra m e te r,  a c ts  o n  a  s ta te , o u r fo rm u la ti o n  c a n  b e  u s e d  t o  o p ti m a lly  e s ti m a te  t h e  v a lu e  o f th is  u s e d  t o  o p ti m a lly  e s ti m a te  t h e  v a lu e  o f th is  p a ra m e te r p a ra m e te r  W e  w ill  u s e  t h is  i d e a  t o  o p ti m iz e  t h e  a c c u ra c y  o f W e  w ill  u s e  t h is  i d e a  t o  o p ti m iz e  t h e  a c c u ra c y  o f q u a n tu m  c lo c k s q u a n tu m  c lo c k s  G o a l G o a l :  E x p re s s  t h e  o p e ra ti o n  o f th e  a  q u a n tu m  : E x p re s s  t h e  o p e ra ti o n  o f th e  a  q u a n tu m  c lo c k  a s  a  b la c k  b o x  o ra c le  p ro b le m .  c lo c k  a s  a  b la c k  b o x  o ra c le  p ro b le m .  A to m ic  C lo c k  B a s ic s A to m ic  C lo c k  B a s ic s  C o u n t th e  t ic k s  o f s o m e  C o u n t th e  t ic k s  o f s o m e  p e ri o d ic  s y s te m  a n d  i n te rp re t p e ri o d ic  s y s te m  a n d  i n te rp re t a s  t im e . a s  t im e .  A to m s  h a v e  a  d is c re te  s e t o f A to m s  h a v e  a  d is c re te  s e t o f e n e rg y  l e v e ls  a n d  c a n  e n e rg y  l e v e ls  a n d  c a n  tr a n s it io n  b e tw e e n  t h e m  b y  tr a n s it io n  b e tw e e n  t h e m  b y  a b s o rb in g  o r e m it ti n g  p h o to n s .  a b s o rb in g  o r e m it ti n g  p h o to n s .  ( E  =  ( E  =  h f h f ))  A to m ic  c lo c k s  c o u n t th e  A to m ic  c lo c k s  c o u n t th e  o s c ill a ti o n s  o f a  l a s e r w h o s e  o s c ill a ti o n s  o f a  l a s e r w h o s e  fr e q u e n c y  i s  s ta b ili z e d  t o  s o m e  fr e q u e n c y  i s  s ta b ili z e d  t o  s o m e  a to m ic  t ra n s it io n a to m ic  t ra n s it io n  1  s e c o n d  1  s e c o n d  ªª 9 ,1 9 2 ,6 3 1 ,7 7 0  9 ,1 9 2 ,6 3 1 ,7 7 0  c y c le s  o f th e  r a d ia ti o n  c y c le s  o f th e  r a d ia ti o n  c o rr e s p o n d in g  t o  a  h y p e rf in e  c o rr e s p o n d in g  t o  a  h y p e rf in e  tr a n s it io n  i n  C e s iu m tr a n s it io n  i n  C e s iu m N IS T  F 1 S p e c tr o s c o p y S p e c tr o s c o p y • T h is  a c ts  a s  o u r b la c k  b o x /o ra c le  o p e ra ti o n . G o a l: P h a s e  E s ti m a ti o n  • C o n s id e r a  s e t o f N io n s  a n d  t h e  s e t o f s ta te s  l a b e le d  b y     w h ic h  h a v e  k io n s  i n  t h e  e x c it e d  s ta te  a n d  N  - k io n s  i n  t h e  g ro u n d  s ta te .  ( D ic k e s ta te s )  • O v e r ti m e , o u r la s e r/ c la s s ic a l o s c ill a to r w ill  d ri ft  f ro m  t h e  a to m ic  r e s o n a n c e .  W e  n e e d  t o  m e a s u re  a n d  c o rr e c t th is  d e tu n in g . • A ft e r a n  i n te rr o g a ti o n  p e ri o d  o f le n g th  T  w it h  a  l a s e r tu n e d  n e a r re s o n a n c e  [ 7 ] | k〉 |k 〉 → e− ik (f − f 0 )T |k 〉 F re q u e n c y  P ri o rs  a n d  t h e  C lo c k  O ra c le F re q u e n c y  P ri o rs  a n d  t h e  C lo c k  O ra c le G iv e n  f  - f 0 w e  k n o w  t h a t: W e  d o n ’t  k n o w  f  - f 0 , b u t w e  c a n  c o n s tr u c t a  p ri o r p ro b a b ili ty  d is tr ib u ti o n  o v e r f - f 0 . L ik e w is e , b e fo re , w e  d id n ’t  k n o w  w h ic h  o ra c le  o u r a lg o ri th m  w o u ld  b e  g iv e n , s o  w e  a s s ig n e d  e a c h  a  p ro b a b ili ty . |k〉 → e− ik (f − f 0 )T |k〉 T . R o s e n b a n d , N IS T C lo c k  S D P C lo c k  S D P A ft e r o n e  q u e ry Ω = ∑ x| x 〉〈x |e− i∆ f x σ z T / 2 ρ O = ∑ x,y√ p x √ p y |x〉 〈y | Ω † ρ O Q Ω → ∑ x,y∑ k ,l a x ,y ,k ,l |x〉 〈y |⊗ ei ∆ f x k T |k〉 〈l| e− i∆ f y lT ρ Q = ∑ k,la k |k 〉 〈l | G o a ls : 1 . D e te rm in e  o p ti m a l in it ia l s ta te  o f Q 2 . D e te rm in e  o p ti m a l m e a s u re m e n t C lo c k  C o s t F u n c ti o n C lo c k  C o s t F u n c ti o n  M e a s u re  w it h  a  d is c re te , c o m p le te  P O V M     M e a s u re  w it h  a  d is c re te , c o m p le te  P O V M     w h o s e  e le m e n ts  e a c h  c o rr e s p o n d  t o  a  f re q u e n c y  g u e s s , w h o s e  e le m e n ts  e a c h  c o rr e s p o n d  t o  a  f re q u e n c y  g u e s s , ∆∆ ff ii  R e m o te  S ta te  P re p a ra ti o n : R e m o te  S ta te  P re p a ra ti o n :  P e n a liz e  g u e s s e s  f a r fr o m  t h e  t ru e  f re q u e n c y P e n a liz e  g u e s s e s  f a r fr o m  t h e  t ru e  f re q u e n c y  F ro m  t h e  f in a l d e n s it y  m a tr ix  a n d  t h e  r e m o te ly  p re p a re d  F ro m  t h e  f in a l d e n s it y  m a tr ix  a n d  t h e  r e m o te ly  p re p a re d  m ix e d  s ta te s , w e  c a n  r e c o n s tr u c t th is  o p ti m a l P O V M   m ix e d  s ta te s , w e  c a n  r e c o n s tr u c t th is  o p ti m a l P O V M   σ i = tr Q (P Q i ρ O Q ) (σ i) x ,x = P (m ea su r ed ∆ f i ∩ f r eq u en cy = ∆ f x ) {P Q i } i. 〈 ∆ f x | A i| ∆ f x 〉 = (∆ f i − ∆ f x )2 D is c u s s io n D is c u s s io n  P ri o r w o rk  h a s  a s s u m e d  a  u n if o rm  p ro b a b ili ty  P ri o r w o rk  h a s  a s s u m e d  a  u n if o rm  p ro b a b ili ty  d is tr ib u ti o n  o v e r c lo c k  o ra c le s . d is tr ib u ti o n  o v e r c lo c k  o ra c le s . [4 ] [4 ]  T h is  t e c h n iq u e  i s  s u b je c t to  T h is  t e c h n iq u e  i s  s u b je c t to  d is c re ti z a ti o n d is c re ti z a ti o n e rr o r e rr o r  W o rk in g  o n  b o u n d s  W o rk in g  o n  b o u n d s  –– e rr o r a p p e a rs  s m a ll e rr o r a p p e a rs  s m a ll  A l A l++ C lo c k  a t N IS T  i s  a  n a tu ra l a p p lic a ti o n  o f th is  C lo c k  a t N IS T  i s  a  n a tu ra l a p p lic a ti o n  o f th is  te c h n iq u e .  U s e s  o n ly  1 te c h n iq u e .  U s e s  o n ly  1 -- 2  i o n s  a n d  i s  t h e  m o s t 2  i o n s  a n d  i s  t h e  m o s t a c c u ra te  c lo c k  i n  t h e  w o rl d  (  8 .6  x  1 0 a c c u ra te  c lo c k  i n  t h e  w o rl d  (  8 .6  x  1 0 -- 1 8 1 8 ) ) [5 ,9 ] [5 ,9 ] R e s u lt s R e s u lt s • O p ti m a l in it ia l s ta te s  h a v e  e q u a l a m p lit u d e s  f o r k io n s  i n  t h e  e x c it e d  s ta te  a n d  N  – k in  t h e  g ro u n d  s ta te .  e .g . fo r N  =  3 : • O n e  q u e ry  o n  2  i o n s  i s  e q u iv a le n t to  2  q u e ri e s  o n  o n e  i o n .  N o t n e c e s s a ri ly  t ru e  w it h  m o re  i o n s  a n d  m o re  q u e ri e s . |ψ 0 〉= a 0 (|0 〉+ |3〉 ) + a 1 (|1 〉+ |2〉 ) A  n a rr o w e r p o s te ri o r c o rr e s p o n d s  t o  a n  i n c re a s e  in  k n o w le d g e .  G re e n : P ri o rs R e d : P o s te ri o rs P (∆ f |∆ f i = 0) C o n c lu s io n s C o n c lu s io n s  T h is  S D P  f o rm u la ti o n  i s  a  p o w e rf u l a n d  h ig h ly  T h is  S D P  f o rm u la ti o n  i s  a  p o w e rf u l a n d  h ig h ly  g e n e ra l w a y  o f d e s c ri b in g  q u a n tu m  a lg o ri th m s  a n d  g e n e ra l w a y  o f d e s c ri b in g  q u a n tu m  a lg o ri th m s  a n d  q u a n tu m  p ro c e s s e s q u a n tu m  p ro c e s s e s  T h e  s iz e  o f th e  S D P  g ro w s  q u ic k ly , b u t th is  T h e  s iz e  o f th e  S D P  g ro w s  q u ic k ly , b u t th is  te c h n iq u e  r e m a in s  a p p lic a b le  t o  m a n y  i n te re s ti n g  te c h n iq u e  r e m a in s  a p p lic a b le  t o  m a n y  i n te re s ti n g  p ro b le m s . p ro b le m s .  H e re  w e  a n a ly z e d  q u a n tu m  c o m p u te rs  s u b je c t to  H e re  w e  a n a ly z e d  q u a n tu m  c o m p u te rs  s u b je c t to  d e c o h e re n c e d e c o h e re n c e , a n d  d e v e lo p e d  a  t e c h n iq u e  t o  , a n d  d e v e lo p e d  a  t e c h n iq u e  t o  e x a m in e  t h e  e x a m in e  t h e  ““ q u a n tu m n e s s q u a n tu m n e s s ”” o f a  d e v ic e . o f a  d e v ic e .  T h e  a p p lic a ti o n  o f o u r S D P  t o  q u a n tu m  c lo c k s  T h e  a p p lic a ti o n  o f o u r S D P  t o  q u a n tu m  c lo c k s  in d ic a te s  t h a t it  l ik e ly  h a s  m a n y  o th e r a p p lic a ti o n s in d ic a te s  t h a t it  l ik e ly  h a s  m a n y  o th e r a p p lic a ti o n s R e fe re n c e s R e fe re n c e s 1 . 1 . A . A . A m b a in is A m b a in is , , Q u a n tu m  l o w e r b o u n d s  b y  q u a n tu m  a rg u m e n ts Q u a n tu m  l o w e r b o u n d s  b y  q u a n tu m  a rg u m e n ts 2 . 2 . H . B a rn u m , M . S a k s , a n d  M . H . B a rn u m , M . S a k s , a n d  M . S z e g e d y S z e g e d y , , Q u a n tu m  q u e ry  c o m p le x it y  Q u a n tu m  q u e ry  c o m p le x it y  a n d  s e m id e fi n it e  p ro g ra m m in g a n d  s e m id e fi n it e  p ro g ra m m in g 3 . 3 . H . B a rn u m , H . B a rn u m , S e m id e fi n it e  p ro g ra m m in g  c h a ra c te ri z a ti o n  a n d  s p e c tr a l S e m id e fi n it e  p ro g ra m m in g  c h a ra c te ri z a ti o n  a n d  s p e c tr a l a d v e rs a ry  m e th o d  f o r q u a n tu m  c o m p le x it y  w it h  a d v e rs a ry  m e th o d  f o r q u a n tu m  c o m p le x it y  w it h  n o n c o m m u ti n g n o n c o m m u ti n g u n it a ry  q u e ri e s u n it a ry  q u e ri e s 4 . 4 . V . V . B u z e k B u z e k , R . , R . D e rk a D e rk a , a n d  S . , a n d  S . M a s s a r M a s s a r ,  , O p ti m a l q u a n tu m  c lo c k s O p ti m a l q u a n tu m  c lo c k s 5 . 5 . C .W . C h o u , e t.  a l. , C .W . C h o u , e t.  a l. , F re q u e n c y  c o m p a ri s o n  o f tw o  h ig h F re q u e n c y  c o m p a ri s o n  o f tw o  h ig h -- a c c u ra c y  A l a c c u ra c y  A l++ o p ti c a l c lo c k s o p ti c a l c lo c k s 6 . 6 . L . L . H u g h s to n H u g h s to n , R . , R . J o z s a J o z s a , a n d  W . , a n d  W . W o o te rs W o o te rs , , A  c o m p le te  c la s s if ic a ti o n  o f A  c o m p le te  c la s s if ic a ti o n  o f q u a n tu m  e n s e m b le s  h a v in g  a  g iv e n  d e n s it y  m a tr ix q u a n tu m  e n s e m b le s  h a v in g  a  g iv e n  d e n s it y  m a tr ix 7 . 7 . N .F . R a m s e y , N .F . R a m s e y , M o le c u la r B e a m s M o le c u la r B e a m s 8 . 8 . B . B . R e ic h a rd t R e ic h a rd t ,  R e fl e c ti o n s  f o r q u a n tu m  q u e ry  c o m p le x it y : , R e fl e c ti o n s  f o r q u a n tu m  q u e ry  c o m p le x it y : T h e  g e n e ra l T h e  g e n e ra l a d v e rs a ry  b o u n d  i s  t ig h t fo r e v e ry  a d v e rs a ry  b o u n d  i s  t ig h t fo r e v e ry  b o o le a n b o o le a n fu n c ti o n fu n c ti o n 9 . 9 . P .O . S c h m id t,  e t.  a l. , P .O . S c h m id t,  e t.  a l. , S p e c tr o s c o p y  u s in g  q u a n tu m  l o g ic S p e c tr o s c o p y  u s in g  q u a n tu m  l o g ic 1 0 . 1 0 . B . S c h u m a c h e r,  B . S c h u m a c h e r,  Q u a n tu m  c o d in g Q u a n tu m  c o d in g 1 1 . 1 1 . W . W . Z u re k Z u re k , , D e c o h e re n c e , D e c o h e re n c e , e in s e le c ti o n e in s e le c ti o n a n d  t h e  q u a n tu m  o ri g in s  o f th e  a n d  t h e  q u a n tu m  o ri g in s  o f th e  c la s s ic a l c la s s ic a l A lg o ri th m  R e c o n s tr u c ti o n A lg o ri th m  R e c o n s tr u c ti o n • S D P  o u tp u ts  a  d e n s it y  m a tr ix  c o rr e s p o n d in g  t o  t h e  s ta te  o f th e  c o m p u te r a t e v e ry  t im e s te p • P u ri fy  t h e  f o llo w in g  p a ir s  o f s ta te s  i n to  A , w h e re  | A | =  | O || Q |E | • S in c e  w e  k n o w  t h a t w e  c a n  s o lv e  f o r th e  i n te r- q u e ry  u n it a ri e s .  S in c e  t h e  S D P  y ie ld s  t h e  s o lu ti o n  w it h  t h e  o p ti m a l p ro b a b ili ty  o f s u c c e s s , th is  re c o n s tr u c ts  a n  o p ti m a l a lg o ri th m . (q u a n tu m  o r c la s s ic a l) D † Q E t Ω † Q O ρ O Q E (t )Ω Q O D Q E t → ρ ′O Q E A (t ) ρ O Q E (t + 1 ) → ρ O Q E A (t + 1 ) ρ O Q E A (t + 1) = U (t )† Q A ρ ′ O Q E A (t )U (t )Q A M e a s u ri n g  Q u a n tu m  R e s o u rc e s M e a s u ri n g  Q u a n tu m  R e s o u rc e s  O u r S D P  f o rm u la ti o n  w ill  y ie ld  t h e  a lg o ri th m  w it h  O u r S D P  f o rm u la ti o n  w ill  y ie ld  t h e  a lg o ri th m  w it h  th e  o p ti m a l p ro b a b ili ty  o f s u c c e s s . th e  o p ti m a l p ro b a b ili ty  o f s u c c e s s .  B u t c a n  w e  f in d  a n  B u t c a n  w e  f in d  a n  ““ a lt e rn a te a lt e rn a te ”” a lg o ri th m  w it h  t h e  a lg o ri th m  w it h  t h e  s a m e  p ro b a b ili ty  o f s u c c e s s  t h a t u s e s  f e w e r s a m e  p ro b a b ili ty  o f s u c c e s s  t h a t u s e s  f e w e r q u a n tu m  r e s o u rc e s ? q u a n tu m  r e s o u rc e s ?  Q u a n ti fy  q u a n tu m  r e s o u rc e s  v ia  V o n  N e u m a n n  Q u a n ti fy  q u a n tu m  r e s o u rc e s  v ia  V o n  N e u m a n n  e n tr o p y e n tr o p y  B y  S c h u m a c h e r B y  S c h u m a c h e r ’’ s  q u a n tu m  c o d in g , w e  k n o w  t h is  s  q u a n tu m  c o d in g , w e  k n o w  t h is  is  t h e  m in im u m  n u m b e r o f q u b it s  n e e d e d  t o  is  t h e  m in im u m  n u m b e r o f q u b it s  n e e d e d  t o  re p re s e n t a  q u a n tu m  s ta te  re p re s e n t a  q u a n tu m  s ta te  [ 1 0 ] [1 0 ] S (ρ ) ≡ tr (ρ lg (ρ )) S e p a ra te  C o m p u ta ti o n a l P a th s S e p a ra te  C o m p u ta ti o n a l P a th s  L e t th e  o p ti m a l a lg o ri th m  r u n  n o rm a lly  u n ti l ti m e  L e t th e  o p ti m a l a lg o ri th m  r u n  n o rm a lly  u n ti l ti m e  tt ss , th e n  m e a s u re , th e n  m e a s u re  Q u a n ti fy  t h e  q u a n tu m  r e s o u rc e s  n e e d e d  t o  r e a c h  Q u a n ti fy  t h e  q u a n tu m  r e s o u rc e s  n e e d e d  t o  r e a c h  ti m e s te p ti m e s te p tt ss +  1 +  1  T ry  t o  e x p re s s        a s  t h e  i n c o h e re n t s u m  o f    T ry  t o  e x p re s s        a s  t h e  i n c o h e re n t s u m  o f     T h e s e  i n d e p e n d e n t c o m p u ta ti o n a l p a th s  c a n  b e  T h e s e  i n d e p e n d e n t c o m p u ta ti o n a l p a th s  c a n  b e  c la s s ic a lly  c o m b in e d . c la s s ic a lly  c o m b in e d .  C a lc u la te  e n tr o p y  a s : C a lc u la te  e n tr o p y  a s :  S in c e  t h e  f u ll d e n s it y  m a tr ix  i s  p u re : S in c e  t h e  f u ll d e n s it y  m a tr ix  i s  p u re : ρ O (t s ) ρ O 1 (t s ), ρ O 2 (t s ) .. .ρ O n (t s ) S T = tr (ρ O 1 )S ( ρ O 1 tr (ρ O 1 ) ) + tr (ρ O 2 )S ( ρ O 2 tr (ρ O 2 ) ) + .. . S (ρ O ) = S (ρ Q A ) S p li tt in g  S D P S p li tt in g  S D P  A d d  t h e  f o llo w in g  c o n s tr a in ts  t o  a  n e w  S D P A d d  t h e  f o llo w in g  c o n s tr a in ts  t o  a  n e w  S D P  A d d  a n  o b je c ti v e  w h ic h  m a x im iz e s  t h e  d is ta n c e  b e tw e e n  A d d  a n  o b je c ti v e  w h ic h  m a x im iz e s  t h e  d is ta n c e  b e tw e e n  th e  t w o  s u b p a rt s  o f th e  o ra c le  d e n s it y  m a tr ix . th e  t w o  s u b p a rt s  o f th e  o ra c le  d e n s it y  m a tr ix .  T h is  w ill  b e  a  n o n lin e a r o b je c ti v e  s o  w e  u s e  a n  i te ra ti v e  T h is  w ill  b e  a  n o n lin e a r o b je c ti v e  s o  w e  u s e  a n  i te ra ti v e  s tr a te g y s tr a te g y w it h  t h e  u s u a l s e ts  o f c o n s tr a in ts  o n  t h e s e  n e w  d e n s it y  w it h  t h e  u s u a l s e ts  o f c o n s tr a in ts  o n  t h e s e  n e w  d e n s it y  m a tr ic e s .  H e re , th e  o ra c le  d e n s it y  m a tr ix  i s  t h e  o p ti m a l m a tr ic e s .  H e re , th e  o ra c le  d e n s it y  m a tr ix  i s  t h e  o p ti m a l s o lu ti o n  t o  o u r o ri g in a l S D P . s o lu ti o n  t o  o u r o ri g in a l S D P . tr Q (ρ O Q 1 (t )) = ρ O 1 (t ) tr Q (ρ O Q 2 (t )) = ρ O 2 (t ) ρ O o p t( t) = ρ O 1 (t ) + ρ O 2 (t ) t ≥ t s S p li tt in g  R e s u lt s S p li tt in g  R e s u lt s  R e c u rs iv e ly  p e rf o rm  t h is  s p lit ti n g , c a lc u la ti n g  t h e  R e c u rs iv e ly  p e rf o rm  t h is  s p lit ti n g , c a lc u la ti n g  t h e  e n tr o p y  a t e a c h  i te ra ti o n . e n tr o p y  a t e a c h  i te ra ti o n .  P a ri ty  l o o k s  c la s s ic a l in  t h e  P a ri ty  l o o k s  c la s s ic a l in  t h e  H a d a m a rd H a d a m a rd b a s is .  b a s is .   E n tr o p y  i s  a  b a s is  i n d e p e n d e n t q u a n ti ty . E n tr o p y  i s  a  b a s is  i n d e p e n d e n t q u a n ti ty . 8  E le m e n t S e a rc h /O R : 0 .  2 .0 5 6 2 .  2 .0 0 9 6 4 .  2 .0 0 9 3 8 .  2 .0 0 9 3 4  E le m e n t P A R IT Y 0 .  2 2 .  1 4 .  0 8 .  0

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