Adversary lower bounds for nonadaptive quantum algorithms

A dv ersary lo w er b ounds for nonadaptiv e quan tum algorithms P asal K oiran 1 , Jürgen Landes 2 , Nata ha P ortier 1 , and P engh ui Y ao 3 1 LIP ⋆⋆ , Eole Normale Sup érieure de Ly on, Univ ersité de Ly on 2 S ho ol of Mathematis, Univ ersit y of Man hester 3 State Key Lab oratory of Computer Siene, Chinese A adem y of Sienes Abstrat. W e presen t general metho ds for pro ving lo w er b ounds on the query omplexit y of nonadaptiv e quan tum algorithms. Our results are based on the adv ersary metho d of Am bainis. 1 In tro dution In this pap er w e presen t general metho ds for pro ving lo w er b ounds on the query omplexit y of nonadaptiv e quan tum algorithms. A nonadap- tiv e algorithm mak es all its queries sim ultaneously . By on trast, an un- restrited (adaptiv e) algorithm ma y  ho ose its next query based on the results of previous queries. In lassial omputing, lasses of problems for whi h adaptivit y do es not help ha v e b een iden tied [4 ,10 ℄ and it is kno wn that this question is onneted to a longstanding op en prob- lem [15 ℄ (see [10 ℄ for a more extensiv e disussion). In quan tum omputing, the study of nonadaptiv e algorithms seems esp eially relev an t sine some of the b est kno wn quan tum algorithms (namely , Simon's algorithms and some other hidden subgroup algorithms) are nonadaptiv e. This is nev er- theless a rather understudied sub jet in quan tum omputing. The pap er that is most losely related to the presen t w ork is [14 ℄ (and [8℄ is another related pap er). In [14 ℄ the authors use an algorithmi argu- men t (this is a kind of K olmogoro v argumen t) to giv e lo w er b ounds on the nonadaptiv e quan tum query omplexit y of ordered sear h, and of generalizations of this problem. The mo del of omputation that they onsider is less general than ours (more on this in setion 2). The t w o metho ds that ha v e pro v ed most suessful in the quest for quan- tum lo w er b ounds are the p olynomial metho d (see for instane [5,2 ,11 ,12 ℄) and the adv ersary metho d of Am bainis. It is not lear ho w the p olyno- mial metho d migh t tak e the nonadaptivit y of algorithms in to aoun t. Our results are therefore based on the adv ersary metho d, in its w eigh ted v ersion [3℄. W e pro vide t w o general lo w er b ounds whi h yield optimal results for a n um b er of problems: sear h in an ordered or unordered list, elemen t distintness, graph onnetivit y or bipartiteness. T o obtain our rst lo w er b ound w e treat the list of queries p erformed b y a nonadaptiv e algorithm as one single sup er query. W e an then apply the adv ersary ⋆⋆ UMR 5668 ENS Ly on, CNRS, UCBL asso iée à l'INRIA. W ork done when Landes and Y ao w ere visiting LIP with nanial supp ort from the Mathlogaps program. metho d to this 1-query algorithm. In terestingly , the lo w er b ound that w e obtain is v ery losely related to the lo w er b ounds on adaptive proba- bilisti query omplexit y due to Aaronson [1℄, and to Laplan te and Mag- niez [13 ℄. Our seond lo w er b ound requires a detour through the so-alled minimax (dual) metho d and is based on the fat that in a nonadaptiv e algorithm, the probabilit y of p erforming an y giv en query is indep enden t of the input. 2 Denition of the Mo del In the bla k b o x mo del, an algorithm aesses its input b y querying a funtion x (the blak b ox ) from a nite set Γ to a (usually nite) set Σ . A t the end of the omputation, the algorithm deides to aept or rejet x , or more generally pro dues an output in a (usually nite) set S ′ . The goal of the algorithm is therefore to ompute a (partial) funtion F : S → S ′ , where S = Σ Γ is the set of bla k b o xes. F or example, in the Unor der e d Se ar h problem Γ = [ N ] = { 1 , . . . , N } , Σ = { 0 , 1 } and F is the OR funtion: F ( x ) = _ 1 ≤ i ≤ N x ( i ) . Our seond example is Or der e d Se ar h . The sets Γ and Σ are as in the rst example, but F is no w a partial funtion: w e assume that the bla k b o x satises the promise that there exists an index i su h that x ( j ) = 1 for all j ≥ i , and x ( j ) = 0 for all j < i . Giv en su h an x , the algorithm tries to ompute F ( x ) = i . A quan tum algorithm A that mak es T queries an b e formally de- srib ed as a tuple ( U 0 , . . . , U T ) , where ea h U i is a unitary op erator. F or x ∈ S w e dene the unitary op erator O x (the all to the bla k b o x) b y O x | i i| ϕ i| ψ i = | i i| ϕ ⊕ x ( i ) i| ψ i . The algorithm A omputes the nal state U T O x U T − 1 . . . U 1 O x U 0 | 0 i and mak es a measuremen t of some of its qubits. The result of this measure is b y denition the outome of the omputation of A on input x . F or a giv en ε , the query omplexit y of a funtion F , denoted Q 2 ,ε , is the smallest query omplexit y of a quan tum algorithm omputing F with probabilit y of error at most ε . In the sequel, the quan tum algorithms as desrib ed ab o v e will also b e alled ad adaptive to distinguish them from nonadaptiv e quan tum algo- rithms. Su h an algorithm p erforms all its queries at the same time. A nonadaptiv e bla k-b o x quan tum algorithm A that mak es T queries an therefore b e dened b y a pair ( U, V ) of unitary op erators. F or x ∈ S w e dene the unitary op erator O T x b y O T x | i 1 , . . . , i T i| ϕ 1 , . . . , ϕ T i| ψ i = | i 1 , . . . , i T i| ϕ 1 ⊕ x ( i 1 ) , . . . , ϕ T ⊕ x ( i T ) i| ψ i . The algorithm A omputes the nal state V O T x U | 0 i and mak es a mea- suremen t of some of its qubits. As in the adaptiv e ase, the result of this measure is b y denition the outome of the omputation of A on input x . F or a giv en ε , the nonadaptiv e query omplexit y of a funtion F , denoted Q na 2 ,ε , is the smallest query omplexit y of a nonadaptiv e quan tum algorithm omputing F with probabilit y of error at most ε . Our mo del is more general than the mo del of [14 ℄. In that mo del, the | ϕ i register m ust remain set to 0 after appliation of U . After appliation of O T x , the on ten t of this register is therefore equal to | x ( i 1 ) , . . . , x ( i T ) i rather than | ϕ 1 ⊕ x ( i 1 ) , . . . , ϕ T ⊕ x ( i T ) i . It is easy to v erify that for ev ery nonadaptiv e quan tum algorithm A of query omplexit y T there is an adaptiv e quan tum algorithm A ′ that mak es the same n um b er of queries and omputes the same funtion, so that Q 2 ,ε ≤ Q na 2 ,ε . Indeed, onsider for ev ery k ∈ [ T ] the unitary op erator A k whi h maps the state | i 1 , . . . , i T i| ϕ 1 , . . . , ϕ T i to | i k i| ϕ k i| i 1 , . . . , i k − 1 , i k +1 , . . . i T i| ϕ 1 , . . . , ϕ k − 1 , ϕ k +1 , . . . , ϕ T i . If the nonadaptiv e algorithm A is dened b y the pair of unitary op erators ( U, V ) , then the adaptiv e algorithm A ′ dened b y the tuple of unitary op erators ( U 0 , . . . , U T ) = ( A 1 U, A 2 A − 1 1 , . . . , A T A T − 1 T − 1 , V A − 1 T ) omputes the same funtion. 3 A Diret Metho d 3.1 Lo w er Bound Theorem and Appliations The main result of this setion is Theorem 3. It yields an optimal Ω ( N ) lo w er b ound on the nonadaptiv e quan tum query omplexit y of Unordered Sear h and Elemen t Distintness. First w e reall the w eigh ted adv ersary metho d of Am bainis and some related denitions. The onstan t C ε = (1 − 2 p ε (1 − ε )) / 2 will b e used throughout the pap er. Denition 1. The funtion w : S 2 → R + is a v alid w eigh t funtion if every p air ( x, y ) ∈ S 2 is assigne d a non-ne gative weight w ( x, y ) = w ( y , x ) that satises w ( x, y ) = 0 whenever F ( x ) = F ( y ) . W e then dene for al l x ∈ S and i ∈ Γ : wt ( x ) = P y w ( x, y ) and v ( x, i ) = P y : x ( i ) 6 = y ( i ) w ( x, y ) . Denition 2. The p air ( w , w ′ ) is a v alid w eigh t s heme if:  Every p air ( x, y ) ∈ S 2 is assigne d a non-ne gative weight w ( x, y ) = w ( y , x ) that satises w ( x, y ) = 0 whenever F ( x ) = F ( y ) .  Every triple ( x, y , i ) ∈ S 2 × Γ is assigne d a non-ne gative weight w ′ ( x, y , i ) that satises w ′ ( x, y , i ) = 0 whenever x ( i ) = y ( i ) or F ( x ) = F ( y ) , and w ′ ( x, y , i ) w ′ ( y , x, i ) ≥ w 2 ( x, y ) for al l x, y , i with x ( i ) 6 = y ( i ) . W e then dene for al l x ∈ S and i ∈ Γ w t ( x ) = P y w ( x, y ) and v ( x, i ) = P y w ′ ( x, y , i ) . Of ourse these denitions are relativ e to the partial funtion F . R emark 1. Let w b e a v alid w eigh t funtion and dene w ′ su h that if x ( i ) 6 = y ( i ) then w ′ ( x, y , i ) = w ( x, y ) and w ′ ( x, y , i ) = 0 otherwise. Then ( w , w ′ ) is a v alid w eigh t s heme and the funtions wt and v dened for w in Denition 1 are exatly those dened for ( w , w ′ ) in Denition 2. Theorem 1 (w eigh ted adv ersary metho d of Am bainis [3 ℄) Given a pr ob ability of err or ε and a p artial funtion F , the quantum query  om- plexity Q 2 ,ε ( F ) of F as dene d in se tion 2 satises: Q 2 ,ε ( F ) ≥ C ε max ( w,w ′ ) valid min x,y ,i w ( x,y ) > 0 x ( i ) 6 = y ( i ) s wt ( x ) w t ( y ) v ( x, i ) v ( y , i ) . A probabilisti v ersion of this lo w er b ound theorem w as obtained b y Aaronson [1 ℄ and b y Laplan te and Magniez [13 ℄. Theorem 2 Fix the pr ob ability of err or to ε = 1 / 3 . The pr ob abilisti query  omplexity P 2 ( F ) of F satises the lower b ound P 2 ( F ) = Ω ( L P ( F )) , wher e L P ( F ) = max w min x,y ,i w ( x,y ) > 0 x ( i ) 6 = y ( i ) max „ wt ( x ) v ( x, i ) , wt ( y ) v ( y , i ) « . Her e w r anges over the set of valid weight funtions. W e no w state the main result of this setion. Theorem 3 (nonadaptiv e quan tum lo w er b ound, diret metho d) The nonadaptive query  omplexity Q na 2 ,ε ( F ) of F satises the lower b ound Q na 2 ,ε ( F ) ≥ C 2 ε L na Q ( F ) , wher e L na Q ( F ) = max w max s ∈ S ′ min x,i F ( x )= s wt ( x ) v ( x, i ) . Her e w r anges over the set of valid weight funtions. The follo wing theorem, whi h is an un w eigh ted adv ersary metho d for nonadaptiv e algorithm, is a onsequene of Theorem 3. Theorem 4 L et F : Σ Γ → { 0; 1 } , X ⊆ F − 1 (0) , Y ⊆ F − 1 (1) and let R ⊂ X × Y b e a r elation suh that:  for every x ∈ X ther e ar e at le ast m elements y ∈ Y suh that ( x, y ) ∈ R ,  for every y ∈ Y ther e ar e at le ast m ′ elements x ∈ X suh that ( x, y ) ∈ R ,  for every x ∈ X and every i ∈ Γ ther e ar e at most l elements y ∈ Y suh that ( x, y ) ∈ R and x ( i ) 6 = y ( i ) ,  for every y ∈ X and every i ∈ Γ ther e ar e at most l ′ elements x ∈ X suh that ( x, y ) ∈ R and x ( i ) 6 = y ( i ) . Then Q na 2 ,ε ( F ) ≥ C 2 ε max( m l , m ′ l ′ ) . Pr o of. As in [3 ℄ and [13 ℄ w e set w ( x, y ) = w ( y , x ) = 1 for all ( x, y ) ∈ R . Then wt ( x ) ≥ m for all x ∈ A , wt ( y ) ≥ m ′ for all y ∈ B , v ( x, i ) ≤ l and v ( y , i ) ≤ l ′ .  F or the Unordered Sear h problem dened in Setion 2 w e ha v e m = N and l = l ′ = m ′ = 1 . Theorem 4 therefore yields an optimal Ω ( N ) lo w er b ound. The same b ound an b e obtained for the Elemen t Distintness problem. Here the set X of negativ e instanes is made up of all one-to- one funtions x : [ N ] → [ N ] and Y on tains the funtions y : [ N ] → [ N ] that are not one-to-one. W e onsider the relation R su h that ( x, y ) ∈ R if and only if there is a unique i su h that x ( i ) 6 = y ( i ) . Then m = 2 , l = 1 , m ′ = N ( N − 1) and l ′ = N − 1 . As p oin ted out in [13 ℄, the Ω (max( m/l , m ′ /l ′ )) lo w er b ound from Theo- rem 4 is also a lo w er b ound on P 2 ( F ) . There is a further onnetion: Prop osition 1. F or any funtion F we have L P ( F ) ≥ L na Q ( F ) . That is, ignoring  onstant fators, the lower b ound on P 2 ( F ) given by The or em 2 is at le ast as high as the lower b ound on Q na 2 ,ε ( F ) given by The or em 3. Pr o of. Pi k a w eigh t funtion w Q whi h is optimal for the diret metho d of Theorem 3. That is, w Q a hiev es the lo w er b ound L na Q ( F ) dened in this theorem. Let s Q b e the orresp onding optimal  hoie for s ∈ S ′ . W e need to design a w eigh t funtion w P whi h will sho w that L P ( F ) ≥ L na Q ( F ) . One an simply dene w P b y: w P ( x, y ) = w Q ( x, y ) if F ( x ) = s Q or F ( y ) = s Q ; w P ( x, y ) = 0 otherwise. Indeed, for an y i and an y pair ( x, y ) su h that w P ( x, y ) > 0 w e ha v e F ( x ) = s Q or F ( y ) = s Q , so that max( w t ( x ) /v ( x, i ) , wt ( y ) /v ( y , i )) ≥ L na Q ( F ) .  The nonadaptiv e quan tum lo w er b ound from Theorem 3 is therefore rather losely onneted to adaptiv e probabilisti lo w er b ounds: it is sandwi hed b et w een the w eigh ted lo w er b ound of Theorem 2 and its un- w eigh ted max( m/l, m ′ /l ′ ) v ersion. Prop osition 1 also implies that The- orem 3 an at b est pro v e an Ω (log N ) lo w er b ound on the nonadaptiv e quan tum omplexit y of Ordered Sear h. Indeed, b y binary sear h the adaptiv e probabilisti omplexit y of this problem is O (log N ) . In se- tion 4 w e shall see that there is in fat a Ω ( N ) lo w er b ound on the nonadaptiv e quan tum omplexit y of this problem. R emark 2. The onnetion b et w een nonadaptiv e quan tum omplexit y and adaptiv e probabilisti omplexit y that w e ha v e p oin ted out in the paragraph ab o v e is only a onnetion b et w een the lower b ounds on these quan tities. Indeed, there are problems with a high probabilisti query omplexit y and a lo w nonadaptiv e quan tum query omplexit y (for in- stane, Simon's problem [16 ,10 ℄). Con v ersely , there are problems with a lo w probabilisti query omplexit y and a high nonadaptiv e quan tum query omplexit y (for instane, Ordered Sear h). 3.2 Pro of of Theorem 3 As men tioned in the in tro dution, w e will treat the tuple ( i 1 , . . . , i k ) of queries made b y a nonadaptiv e algorithm as a single sup er query made b y an ordinary quan tum algorithm (iniden tally , this metho d ould b e used to obtain lo w er b ounds on quan tum algorithm that mak e sev eral rounds of parallel queries as in [8℄). This motiv ates the follo wing deni- tion. Denition 3. L et Σ , Γ and S b e as in se tion 2. Given an inte ger k ≥ 2 , we dene:  k Σ = Σ k , k Γ = Γ k and k S = ` Σ k ´ Γ k .  T o the blak b ox x ∈ S we asso iate the sup er b ox k x ∈ k S suh that if I = ( i 1 , . . . , i k ) ∈ Γ k then k x ( I ) = ( x ( i 1 ) , . . . , x ( i k )) .  k F ( k x ) = F ( x ) .  If w is a weight funtion for F we dene a weight funtion W for k F by W ( k x, k y ) = w ( x, y ) . Assume for instane that Σ = { 0; 1 } , Γ = [3] , k = 2 , and that x is dened b y: x (1) = 0 , x (2) = 1 and x (3) = 0 . Then w e ha v e 2 x (1 , 1) = (0 , 0) , 2 x (1 , 2) = (0 , 1) , 2 x (1 , 3) = (0 , 0) . . . Lemma 1. If w is a valid weight funtion for F then W is a valid weight funtion for k F and the minimal numb er of queries of a quantum algorithm  omputing k F with err or pr ob ability ε satises: Q 2 ,ε ( k F ) ≥ C ε · min k x, k y ,I W ( k x, k y ) > 0 k x ( I ) 6 = k y ( I ) s W T ( k x ) W T ( k y ) V ( k x, I ) V ( k y , I ) . Pr o of. Ev ery pair ( x, y ) ∈ S 2 is assigned a non-negativ e w eigh t W ( k x, k y ) = W ( k y , k x ) = w ( x, y ) = w ( y , x ) that satises W ( k x, k y ) = 0 whenev er F ( x ) = F ( y ) . Th us w e an apply Theorem 1 and w e obtain the an- nouned lo w er b ound.  Lemma 2. L et x b e a blak-b ox and w a weight funtion. F or any inte ger k and any tuple I = ( i 1 , . . . , i k ) we have W T ( k x ) V ( k x, I ) ≥ 1 k min j ∈ [ k ] wt ( x ) v ( x, i j ) . Pr o of. Let m = min j ∈ [ k ] wt ( x ) v ( x ,i j ) . W e ha v e W T ( k x ) = w t ( x ) and: V ( k x, I ) = X k y : k x ( i ) 6 = k y ( i ) W ( k x, k y ) ≤ X y : x ( i 1 ) 6 = y ( i 1 ) w ( x, y ) + · · · + X y : x ( i k ) 6 = y ( i k ) w ( x, y ) = v ( x, i 1 ) + · · · + v ( x, i k ) ≤ k ma x j ∈ [ k ] v ( x, i j ) .  Lemma 3. If w is a valid weight funtion: Q na 2 ,ε ( F ) ≥ C 2 ε min x,y F ( x ) 6 = F ( y ) max „ min i wt ( x ) v ( x, i ) , min i wt ( y ) v ( y , i ) « . Pr o of. Let w b e an arbitrary v alid w eigh t funtion and k b e an in teger su h that k < C 2 ε min x,y F ( x ) 6 = F ( y ) max „ min i wt ( x ) v ( x, i ) , min i wt ( y ) v ( y , i ) « . W e sho w that an algorithm omputing k F with probabilit y of error ≤ ε m ust mak e stritly more one than query to the sup er b o x k x . This will pro v e that for ev ery su h k w e ha v e Q na 2 ,ε ( F ) > k and th us our result. F or ev ery x and I w e ha v e W T ( k x ) V ( k x, I ) ≥ 1 and th us b y lemma 2 for ev ery x , y and I = ( i 1 , . . . , i k ) : W T ( k x ) V ( k x, I ) W T ( k y ) V ( k x, I ) = min „ W T ( k x ) V ( k x, I ) , W T ( k y ) V ( k x, I ) « max „ W T ( k x ) V ( k x, I ) , W T ( k y ) V ( k x, I ) « ≥ max „ W T ( k x ) V ( k x, I ) , W T ( k y ) V ( k x, I ) « ≥ 1 k max „ min j ∈ [ k ] wt ( x ) v ( x, i j ) , min l ∈ [ k ] wt ( y ) v ( x, i l ) « . In order to apply Lemma 1 w e observ e that: min k x, k y ,I W ( k x, k y ) > 0 k x ( I ) 6 = k y ( I ) W T ( k x ) W T ( k y ) V ( k x, I ) V ( k y , I ) ≥ 1 k min x,y ,i 1 ,...,i k w ( x,y ) > 0 ∃ m x ( i m ) 6 = y ( i m ) max „ min j ∈ [ k ] wt ( x ) v ( x, i j ) , min l ∈ [ k ] wt ( y ) v ( x, i l ) « ≥ 1 k min x,y F ( x ) 6 = F ( y ) max „ min i wt ( x ) v ( x, i ) , min i wt ( y ) v ( x, i ) « By h yp othesis on k , this expression is greater than 1 /C 2 ε . Th us aording to Lemma 1 w e ha v e Q 2 ,ε ( k F ) > 1 , and Q na 2 ,ε ( F ) > k .  W e an no w omplete the pro of of Theorem 3. Supp ose without loss of generalit y that F ( S ) = [ m ] and dene for ev ery l ∈ [ m ] : a l = C 2 ε min x,i F ( x )= l wt ( x ) v ( x, i ) . Supp ose also without loss of generalit y that a 1 ≤ · · · ≤ a m . It follo ws immediately from the denition that a 2 = C 2 ε min x,y F ( x ) 6 = F ( y ) max „ min i wt ( x ) v ( x, i ) , min i wt ( y ) v ( x, i ) « , and a m = C 2 ε max l ∈ F ( S ) min x,i F ( x )= l wt ( x ) v ( x, i ) . By Lemma 3 w e ha v e Q na 2 ,ε ( F ) ≥ a 2 , but w e w ould lik e to sho w that Q na 2 ,ε ( F ) ≥ a m . W e pro eed b y redution from the ase when there are only t w o lasses (i.e., m = 2 ). Let G b e dened b y G (1) = · · · = G ( m − 1) = 1 and G ( m ) = m . Applying Lemma 3 to GoF , w e obtain that Q na 2 ,ε ( GoF ) ≥ a m . But b eause the funtion GoF is ob viously easier to ompute than F , w e ha v e Q na 2 ,ε ( F ) ≥ Q na 2 ,ε ( GoF ) and th us Q na 2 ,ε ( F ) ≥ a m as desired. 4 F rom the Dual to the Primal Our starting p oin t in this setion is the minimax metho d of Laplan te and Magniez [13 ,17 ℄ as stated in [9 ℄: Theorem 5 L et p : S × Σ → R + b e the set of | S | pr ob ability distributions suh that p x ( i ) is the aver age pr ob ability of querying i on input x , wher e the aver age is taken over the whole  omputation of an algorithm A . Then the query  omplexity of A is gr e ater or e qual to: C ε max x,y F ( x ) 6 = F ( y ) 1 P i x ( i ) 6 = y ( i ) p p x ( i ) p y ( i ) . Theorem 5 is the basis for the follo wing lo w er b ound theorem. It an b e sho wn that up to onstan t fators, the lo w er b ound giv en b y Theorem 6 is alw a ys as go o d as the lo w er b ound giv en b y Theorem 3. Theorem 6 (nonadaptiv e quan tum lo w er b ound, primal-dual metho d) L et F : S → S ′ b e a p artial funtion, wher e as usual S = Σ Γ is the set of blak-b ox funtions. L et DL ( F ) = min p max x,y F ( x ) 6 = F ( y ) 1 P i x ( i ) 6 = y ( i ) p ( i ) and P L ( F ) = max w P x,y w ( x, y ) max i P x,y x i 6 = y i w ( x, y ) wher e the min in the rst formula is taken over al l pr ob ability distribu- tions p over Γ , and the max in the se  ond formula is taken over al l valid weight funtions w . Then DL ( F ) = P L ( F ) and we have the fol lowing nonadaptive query  omplexity lower b ound: Q 2 ,ε ( F ) ≥ C ε DL ( F ) = C ε P L ( F ) . Pr o of. W e rst sho w that Q 2 ,ε ( F ) ≥ C ε DL ( F ) . Let A b e a nonadaptiv e quan tum algorithm for F . Sine A is nonadaptiv e, the probabilit y p x ( i ) of querying i on input x is indep enden t of x . W e denote it b y p ( i ) . Theorem 5 sho ws that the query omplexit y of A is greater or equal to C ε max x,y F ( x ) 6 = F ( y ) 1 P i x ( i ) 6 = y ( i ) p ( i ) . The lo w er b ound Q 2 ,ε ( F ) ≥ C ε DL ( F ) follo ws b y minimizing o v er p . It remains to sho w that DL ( F ) = P L ( F ) . Let L ( F ) = min p max x,y F ( x ) 6 = F ( y ) X i x ( i )= y ( i ) p ( i ) . W e observ e that L ( F ) is the optimal solution of the follo wing linear program: minimize µ sub jet to the onstrain ts ∀ x, y su h that f ( x ) 6 = f ( y ) : µ − X i x ( i ) 6 = y ( i ) p ( i ) ≥ 0 , and to the onstrain ts N X i =1 p ( i ) = 1 and ∀ i ∈ [ N ] : p ( i ) ≥ 0 . Clearly , its solution set is nonempt y . Th us L ( f ) is the optimal solution of the dual linear program: maximize ν sub jet to the onstrain ts ∀ i ∈ [ N ] : ν − X x,y x i = y i w ( x, y ) ≤ 0 ∀ x, y : w ( x , y ) ≥ 0 , and w ( x, y ) = 0 if F ( x ) = F ( y ) and to the onstrain t X x,y w ( x, y ) = 1 . Hene L ( F ) = max w min i P x i = y i w ( x, y ) P x,y w ( x, y ) and DL ( F ) = 1 1 − L ( F ) = P L ( F ) .  4.1 Appliation to Ordered Sear h and Connetivit y Prop osition 1 F or any err or b ound ε ∈ [0 , 1 2 ) we have Q na 2 ,ε ( Or der e d Se ar h ) ≥ C ε ( N − 1) . Pr o of. Consider the w eigh t funtion w ( x, y ) = ( 1 if | F ( y ) − F ( x ) | = 1 , 0 otherwise . Th us w ( x, y ) = 1 when the leftmost 1's in x and y are adjaen t. Hene P x,y w ( x, y ) = 2( N − 2) + 2 . Moreo v er, if w ( x, y ) 6 = 0 and x i 6 = y i then { F ( x ) , F ( y ) } = { i, i + 1 } . Therefore, max i P x,y x i 6 = y i w ( x, y ) = 2 and the result follo ws from Theorem 6.  Our seond appliation of Theorem 6 is to the graph onnetivit y prob- lem. W e onsider the adjaeny matrix mo del: x ( i, j ) = 1 if ij is an edge of the graph. W e onsider undireted, lo opless graph so that w e an as- sume j < i . F or a graph on n v erties, the bla k b o x x therefore has N = n ( n − 1) / 2 en tries. W e denote b y G x the graph represen ted b y x . Theorem 7 F or any err or b ound ε ∈ [0 , 1 2 ) , we have Q na 2 ,ε ( Conne tivity ) ≥ C ε n ( n − 1) / 8 . Pr o of. W e shall use essen tially the same w eigh t funtion as in ([6 ℄, The- orem 8.3). Let X b e the set of all adjaeny matries of a unique yle, and Y the set of all adjaeny matries with exatly t w o (disjoin t) y- les. F or x ∈ X and y ∈ Y , w e set w ( x, y ) = 1 if there exist 4 v erties a, b, c, d ∈ [ n ] su h that the only dierenes b et w een G x and G y are that: 1. ab, cd are edges in G x but not in G y . 2. ac, bd are edges in G y but not in G x . W e laim that max ij X x ∈ X,y ∈ Y x ( i,j ) 6 = y ( i,j ) w ( x, y ) = 8 n ( n − 1) X x ∈ X,y ∈ Y x ( i,j ) 6 = y ( i,j ) w ( x, y ) . (1) The onlusion of Theorem 7 will then follo w diretly from Theorem 6. By symmetry , the funtion that w e are maximizing on the left-hand side of (1) is in fat indep enden t of the edge ij . W e an therefore replae the max o v er ij b y an a v erage o v er ij : the left-hand side is equal to 1 N X x ∈ X,y ∈ Y w ( x, y ) |{ ij ; x ( i, j ) 6 = y ( i, j ) }| . No w, the ondition x ( i, j ) 6 = y ( i, j ) holds true if and only if ij is one of the 4 edges ab , cd , ac , bd dened at the b eginning of the pro of. This nishes the pro of of (1), and of Theorem 7.  A similar argumen t an b e used to sho w that testing whether a graph is bipartite also requires Ω ( n 2 ) queries. 5 Some Op en Problems F or the 1-to-1 v ersus 2-to-1 problem, one w ould exp et a higher quan- tum query omplexit y in the nonadaptiv e setting than in the adap- tiv e setting. This ma y b e diult to establish sine the adaptiv e lo w er b ound [2 ℄ is based on the p olynomial metho d. Hidden T ranslation [7℄ (a problem losely onneted to the dihedral hidden subgroup problem) is another problem of in terest. No lo w er b ound is kno wn in the adap- tiv e setting, so it w ould b e natural to lo ok rst for a nonadaptiv e lo w er b ound. Finally , one w ould lik e to iden tify some lasses of problems for whi h adaptivit y do es not help quan tum algorithms. A  kno wledgemen ts: This w ork has b eneted from disussions with Sophie Laplan te, T ro y Lee, F rédéri Magniez and Vinen t Nesme. Email addresses: [Pasal.Koiran, Nataha.Portier ℄ens - ly on.fr , juergen_landesyahoo.de , phyao1985gmail.om . Referenes 1. S. Aaronson. Lo w er b ounds for lo al sear h b y quan tum argumen ts. In Pr o . STOC 2004 , pages 465474. A CM, 2004. 2. S. Aaronson and Y. Shi. Quan tum Lo w er Bounds for the Colli- sion and the Elemen t Distintness Problems. Journal of the A CM , 51(4):595605, July 2004. 3. Andris Am bainis. P olynomial degree vs. quan tum query omplexit y . J. Comput. Syst. Si. , 72(2):220238, 2006. 4. Z. Bar-Y ossef, R. Kumar, and D. Siv akumar. Sampling Algorithms: lo w er b ounds and appliations. In Pr o . STOC 2001 , pages 266275. A CM, 2001. 5. R. Beals, H. Buhrman, R. Clev e, M. Mosa, and R. de W olf. Quan- tum lo w er b ounds b y p olynomials. Journal of the A CM , 48(4):778 797, 2001. 6. Christoph Dürr, Mark Heiligman, P eter Høy er, and Mehdi Mhalla. Quan tum query omplexit y of some graph problems. SIAM J. Com- put. , 35(6):13101328, 2006. 7. K. F riedl, G. Iv an y os, F. Magniez, M. San tha, and P . Sen. Hidden translation and orbit oset in quan tum omputing. In STOC '03: Pr o  e e dings of the thirty-fth annual A CM symp osium on The ory of  omputing , 2003. 8. Lo v K. Gro v er and Jaikumar Radhakrishnan. Quan tum sear h for m ultiple items using parallel queries. arXiv, 2004. 9. P eter Ho y er and Rob ert Spalek. Lo w er b ounds on quan tum query omplexit y . EA TCS Bul letin , 87:78103, o tob er 2005. 10. P . K oiran, V. Nesme, and N. P ortier. On the probabilisti query omplexit y of transitiv ely symmetri problems. h ttp://p erso.ens- ly on.fr/pasal.k oiran. 11. P . K oiran, V. Nesme, and N. P ortier. A quan tum lo w er b ound for the query omplexit y of Simon's problem. In Pr o . ICALP 2005 , v olume 3580 of L e tur e Notes in Computer Sien e , pages 12871298. Springer, 2005. 12. P . K oiran, V. Nesme, and N. P ortier. The quan tum query omplexit y of ab elian hidden subgroup problems. The or eti al Computer Sien e , 380:115126, 2007. 13. S. Laplan te and F. Magniez. Lo w er b ounds for randomized and quan- tum query omplexit y using K olmogoro v argumen ts. SIAM journal on Computing , to app ear. 14. Harumi hi Nishim ura and T omo yuki Y amak ami. 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