Average case quantum lower bounds for computing the boolean mean

Transcription

1 Average case quatum lower bouds for computig the boolea mea A. Papageorgiou Departmet of Computer Sciece Columbia Uiversity New York, NY 1007 Jue 003 Abstract We study the average case approximatio of the Boolea mea by quatum algorithms. We prove geeral query lower bouds for classes of probability measures o the set of iputs. We pay special attetio to two probabilities, where we show specific query ad error lower bouds ad the algorithms that achieve them. We also study the worst expected error ad the average expected error of quatum algorithms ad show the respective query lower bouds. Our results exted the optimality of the algorithm of Brassard et al. 1 Itroductio Quatum computers ca solve certai problems sigificatly faster tha classical computers. Oe of these problems is the approximatio of the mea of a Boolea fuctio or, equivaletly, the approximatio of the mea of Boolea variables. Suppose that the iput is preseted as a black-box or a oracle, which the algorithm queries []. Classical algorithms require Θ((1 ε)) evaluatios (or queries) i the worst case, for error at most ε. Classical radomized algorithms solve this problem faster by requirig Θ(mi{ε, }) evaluatios. Quatum algorithms solve the problem i the worst case with high probability ad are superior because they require oly Θ(mi{ε 1, }) queries. More specifically, Brassard et al.[4] exhibited a algorithm achievig accuracy ε with a umber of queries proportioal to mi{ε 1, }. This algorithm is based o Grover s quatum search algorithm; see [9] for a descriptio of Grover s algorithm ad for details about quatum computig. The lower bouds of Nayak ad Wu [10] establish the asymptotic optimality of the algorithm of Brassard et al. i the worst case. Istead of the worst case error, we ca cosider the average error of quatum algorithms with respect to a probability measure o the class of the iputs. The average case is importat 1

2 for two reasos. The first oe is that it may reduce the query complexity. The secod is that if we kow that the querey complexity is ot reduced the the worst case results ad the optimality of kow algorithms is exteded. It is also importat to derive classes of measures for which similar complexity results hold. I this paper we deal with these issues. I particular, for the approximatio of the mea of Boolea variables with uiform distributio o the set of iputs, the average error of ay quatum algorithm, with T queries of order o(), is Ω(mi{ 1/, T 1 }). The query complexity is zero as log as ε is ω( 1/ ) 1. Whe ε = Θ( 1/ ) the query complexity remais zero as log as the asymptotic costat is large, but whe this costat is small the query complexity is Ω( 1/ ). The query complexity becomes asymptotically equal to that of the worst case whe ε is o( 1/ ). O the other had, if all possible values of the mea are uiformly distributed the the average error of ay algorithm is Ω(T 1 ). I this case, the query complexity is asymptotically equal to that of the worst case for all values of ε. We geeralize our results by showig coditios o classes of measures uder which the query complexity is asymptotically equal to that of the worst case as log as ε is appropriately small. Our results exted the optimality of the quatum algorithm of Brassard et al. whe high accuracy is importat. Quatum algorithms are probabilistic i ature. For a give iput, they ca produce various outcomes, each with a certai probability. Typically, we wat them to achieve a give accuracy with probability greater tha 1 ad, therefore, we study their probabilistic error. O the other had, we ca study the expected error of a quatum algorithm by cosiderig its average error with respect to all outcomes resultig from a give iput. Therefore, for a class of iputs we study the worst expected error. This is also a ituitive error criterio ad is similar to the way we measure the error i Mote Carlo itegratio. We show that ay algorithm with worst expected error at most ε must make Ω(ε 1, ) queries. Therefore, the algorithm of Brassard et al. with repetitios as described i [6] is asymptotically optimal. We also show that the query lower bouds that hold for the average case remai valid whe we cosider the average expected error of quatum algorithms. I this case we cosider a probability measure o the set of iputs, ad for each iput we cosider the expected error of the algorithm with respect to all possible outcomes. Fially, it is easy to see that a algorithm approximatig the Boolea mea ca be used to approximately cout the umber of oes amog Boolea variables. Therefore, all our results directly extet to approximate coutig ad we exhibit the correspodig query ad error lower bouds. Problem Defiitio Let B = {0, 1} deote all tuples of Boolea variables. We assume that ay X = (x 1,..., x ) B is give by a oracle or a black box, which o iput i outputs x i. Oracle 1 f() is ω(g()) g() is o(f()).

3 access of this type is called a query. We wat to compute the mea of X, i.e., a X = X, with X = x i. I this paper we cosider the quatum query model of Beals et al.[], where the cost of a algorithm is the umber of its queries. A quatum algorithm applies a sequece of uitary trasformatios, which iclude queries, to a iitial state, ad at the ed the fial state is measured. See [, 5, 9] for the details of the model of computatio, which we summarize below to the extet ecessary for this paper. A quatum algorithm has the form i=1 U T Q X U T 1 Q X U 1 Q X U 0 ψ 0 =: ψ, where U 0,..., U T are uitary trasformatios that do ot deped o the iput X, the operator Q X is also a uitary trasformatio ad correspods to a query to the oracle, the iteger T is the umber of times Q X is applied, that is the umber of queries, ψ 0 is the iitial state o which the sequece of trasformatios is applied, ad ψ is the fial state of the algorithm which is measured. The states ψ 0 ad ψ are uit vactors of H m = C } {{ C }, m for some appropriately chose m N. The measuremet produces oe of M outcomes. Outcome j {0,..., M 1} occurs with probability p X (j), which depeds o j ad the iput X. I priciple, quatum algorithms may have may measuremets applied betwee sequeces of uitary trasformatios of the form above. However, ay algorithm with may measuremets ad a total of T queries ca be simulated by a algorithm with oly oe measuremet that has T queries [5]. Hece, without loss of geerality we cosider the cost of algorithms with a sigle measuremet. Give a outcome j, we approximate a X by a umber â X (j). Note that â X (j) depeds o the iput X ad the outcome of the measuremet. Give a probability p > 1, the error of a quatum algorithm with T queries o iput X is defied by e(x, T, p) = if γ : p X (j) p. j: a X â X (j) γ The worst probabilistic error of a quatum algorithm with T queries i the class B is defied by e wp (B, T, p) = max X B e(x, T, p). As we have metioed, Brassard et al.[4] show a quatum summatio algorithm (QS) for computig the Boolea mea ad study its properties usig this error criterio. The query lower boud Ω(mi(ε 1, )) of Nayak ad Wu [10] also holds i the worst case. I this paper we cosider the average probabilistic error of a quatum algorithm i the class B which we defie by e ap (B, T, p) = X B e(x, T, p)µ(x), 3

4 where µ is a probability measure o the set of iputs B. I the ext sectio we exhibit coditios for classes of probability measures ad prove the correspodig query lower bouds. We will pay special attetio to the followig two measures µ 1 (X) =, X B 1 µ (X) = ( + 1) ( ), for X B with X = k. k The first measure correspods to the case where all iputs are equally likely, while the secod measure correspods to the case where all possible values of the mea are equally likely. We ow defie the worst expected error of a quatum algorithm with T queries i the class B as { M 1 } 1/q e (q) (X, T ) = a X â X (j) q p X (j) j=0 e we (q, B, T ) = max X B e (q) (X, T ), with 1 q <, where the summatio is over all possible outcomes. Note that i this case we cosider all the outcomes ad ot just outcomes that occur with probability p > 1. We also cosider the average with respect to the iputs of the expected error, with respect to the outcomes, of a quatum algorithm with T queries i the class B. We call this the average expected error ad we defie it by e ae (q, B, T ) = X B e (q) (X, T )µ(x), with 1 q <, where µ is a probability measure o the set of iputs B. Fially, Nayak ad Wu [10] show query lower bouds for a umber of differet problems. Oe of them is the computatio of a -approximate cout, i.e., a umber ˆt X such that t x ˆt X <, where t X = X = a X for X B. This problem is directly related to the approximatio of the Boolea mea. We study it o the average by appropriately defiig the error of a quatum algorithm. I particular, we set e 1 (X, T, p) = e(x, T, p) e ap 1 (B, T, p) = e ap (B, T, p) e we 1 (q, B, T ) = e we (q, B, T ), with 1 q <, e ae 1 (q, B, T ) = e ae (q, B, T ), with 1 q <. Our results cocerig the mea directly exted to approximate coutig by settig = ε. 3 Average probabilistic error Kwas ad Woźiakowski [7] show that the QS algorithm has zero worst probabilistic error whe the umber of its queries is greater tha 3 8 π, for p. Trivially, QS also has zero π 4

5 average probabilistic error i that case. Therefore, we study the error of quatum algorithms whe the umber of their queries is of order o(). It is coveiet to deal with arbitrary measures first ad the use the results i the study of µ 1 ad µ. So we begi by defiig classes of probability measures ad derivig the correspodig query lower bouds. Roughly speakig, all the measures µ i a class satisfy the same lower boud for µ(x) as log X belogs to a certai subset I of {0,..., }. This lower boud depeds o X, o I ad, particularly, its cardiality I. For example, for the set I = {,..., + }, which has cardiality I = +1, we ca use c[ ( X ) ] 1, (where c > 0 is a costat ad X I,) as the lower boud defiig the class of measures. Observe, that due to Lemma 6.1 i Appedix µ 1 asymptotically satisfies this lower boud o I. Similarly, for the set I = {,..., 3}, which has cardiality I = + 1, we ca use 4 4 c[ ( X ) ] 1 (where c > 0, is a costat ad X I) as the lower boud i the defiitio of the class of measures. Note that µ asymptotically satisfies this lower boud o I. Clearly these two choices of I distiguish two classes of probability measures. The cardiality of the set I as a fuctio of is importat i our aalysis. We will assume that I as as i the previous two examples. We also cosider the umber of queries T ad the desired accuracy ε as fuctios of ad carry out a asymptotic aalysis. I this ad i the followig sectios the implied asymptotic costats are absolute costats. Theorem 3.1. Cosider the approximatio of the Boolea mea. Let I {0,..., } be a set of cosecutive idices, such that its cardiality I is ω(1) as a fuctio of. Assume that k( k) is Θ( ) for every k I. Let µ be a probability measure o B such that µ(x) = Ω( I 1 ) 1 ), for every X B with X I. ( X The for ay ε > 0 of order o( I 1 ), the coditio e ap (B, T, p) ε implies that T must be Ω(mi(ε 1, )). Proof: We will prove the Theorem for ε 1/. The case ε < 1/ will the follow immediately. Cosider a quatum algorithm with T queries that has error e ap (B, T, P ) ε. Usig the lower boud o µ(x) i the assumptio of this theorem, we have ε e(x, T, p)µ(x) c 1 ( I e(x, T, p), X B k) k I X: X =k where c > 0 is a costat. We multiply both sides of the iequality by ad defie = c 1 ε ad (k) = X =k e(x, T, p)/( k) ad use the Markov iequality to obtai 1 I = 1 I k I (k) 1 I +, k: (k)< (k) + 5 k: (k) (k)

6 where + is the umber of idices for which (k). Clearly + 1 I ad, therefore, := I + 1 I. Thus for at least half of the idices i I we have (k) <. We defie J to be the set of all these idices. Note that is o( I ) because ε is o( I 1 ). Without loss of geerality we assume that is a iteger, sice otherwise we ca replace it by its ceilig, which does ot chage its order of magitude. Now cosider k J so that (k) < ad let δ(x, k) = e(x, T, p) for X = k. The for m, which we will further specify later, we have > ( 1 ) k = 1 ( k) X =k 1 ( k)m ñ +, δ(x, k) X =k: δ(x,k)<m δ(x, k) + X =k: δ(x,k) m δ(x, k) where ñ + is the umber of strigs X with X = k, for which δ(x, k) m. Clearly ñ + ( ) k m 1 ad, therefore, ñ := ( ) k ñ+ (1 m 1 ) ( k). Thus for at least ñ may strigs X we have δ(x, T, p) := δ(x, k) < m. Sice I is ω(1) we claim that for sufficietly large there exist k 1, k J that are at least 4m apart whose distace does ot exceed O( ), i.e., k 1 k 4m ad k 1 k is O( ). Ideed, if we assume otherwise, we have that k 1 k < 4m or k 1 k is ω( ). Cosider the idices i I i ascedig order. Let i 1 I be the first idex that belogs to the set J (recall that at least half of the idices i I belog to J). Based o our assumptio, the ext idex (greater tha i 1 ) that belogs to J is either at a distace less tha 4m away from i 1 or at a distace ω( ) away from i 1. We group together all the idices that are at a distace less tha 4m away from i 1. Clearly there are o more tha 4m idices i the group. Now we cosider the first idex i I that belogs to J ad is at a distace ω( ) away from i 1. We repeat the same procedure usig i i the place of i 1, ad we form a secod group of idices that belog to J ad are at a distace less tha 4m away from i. As we iterate this procedure we form groups of idices that belog to J, where each group is at a distace ω( ) away from the group before it. We stop whe we exhaust the idices i J. It is clear that betwee every two groups we have ω( ) elemets of I that do ot belog to J. It is also clear that we have to repeat the above procedure at least J /(4m ) I /(8m ) times i order to exhaust the idices i J. Cosiderig the idices betwee the cosecutive groups that do ot belog to J we coclude that the cardiality of I must be at least ω( )[ I /(8m )] = ω( I ), which is a cotradictio. Now we use the algorithm that approximates the mea to derive aother algorithm that approximates the partial Boolea fuctio { 1 if X = k1 f k1,k (X) = 0 if X = k, where, without loss of geerality, we ca assume that k 1 > k. 6

7 The descriptio of the ew algorithm A is as follows: O iput X, where X = k 1 or k, we ru the algorithm that approximates the mea ad if the value of the result â X (j) satisfies k 1 â X (j) < m the the ew algorithm outputs 1. It outputs 0 otherwise. Let s look at the success probability Pr{A(X) = f k1,k (X)} of the ew algorithm for the differet iputs for which f k1,k is defied. If X = k 1 we have Pr{A(X) = f k1,k (X)} = Pr{A(X) = 1} X =k 1 X =k 1 Pr{A(X) = 1} = X =k 1, δ(x,t,p)<m X =k 1, δ(x,t,p)<m ( ) (1 m 1 ) p, k 1 Pr{ k 1 â X (j) < m } because δ(x, T, p) < m is equivalet to e(x, T, p) < m 1, which implies that k 1 / â X (j) < m 1 holds with with probability at least p. The fact that ñ (1 m 1 ) ( ) k 1 yields the fial iequality. Therefore, the probability that algorithm A fails o ay iput X for which X = k 1 satisfies ( ) ( Pr{A(X) f k1,k (X)} = Pr{A(X) = 0} 1 p + p ). (1) k 1 m X =k 1 X =k 1 Let c = 1 p + p/m. We choose m i a way that that c < 1. From [] we kow that the acceptace probability q(x) = Pr{A(X) = 1} of a quatum algorithm A is a real multiliear polyomial of degree at most T, where T is the umber of its queries. Recall that the symmetrizatio of q is the polyomial q sym (X) = π q(x π(1),..., x π() ), X = (x 1,..., x ) B, ()! where the sum is over all permutatios of the itegers 1,...,. Misky ad Papert [8] show that there is a represetatio of q sym as a uivariate polyomial i X of degree at most that of q sym. For simplicity, with a slight abuse of otatio we deote this uivariate polyomial usig the same symbol, i.e., q sym ( X ). I particular, for X = k 1 we have q(x) = 1 Pr{A(X) = 0}, which implies Pr{A(X) = 0} = 1 q(x) = f k1,k (X) q(x). Thus ( ) Pr{A(X) = 0} = q(x) k 1 X =k 1 X =k 1 ( ) ( ) = q sym ( Y ) k 1 k ( ) 1 = [f k1,k (Y ) q sym ( Y )], Y = k 1, Y B. k 1 7

8 The secod equality holds because whe X has k 1 oes i particular locatios ad k 1 zeros i the remaiig locatios the the k 1!( k 1 )! permutatios of X (whe oly the k 1 oes or the k 1 zeros are permuted) yield tuples that are idetical to X. Therefore, every term i X =k 1 q(x) appears k 1!( k 1 )! times i the π q(x π(1),..., x π() ) of all permutatios. Thus cosiderig () we have π q(x π(1),..., x π() ) = k 1!( k 1 )! X =k 1 q(x). Usig (1) ad the last equality cocerig the probability of failure of A we obtai c f k1,k (X) q sym ( X ), X = k 1. (3) We work similarly whe X = k. We have Pr{A(X) = f k1,k (X)} = Pr{A(X) = 0} X =k X =k = X =k, δ(x,t,p)<m X =k, δ(x,t,p)<m X =k, δ(x,t,p)<m ( 1 1 ) ( ) p, m k Pr{A(X) = 0} Pr{ k 1 â X (j) m } Pr{ k â X (j) < m } because δ(x, T, p) < m is equivalet to e(x, T, p) < m 1, which implies that k / â X (j) < m 1 holds with probability at least p. The fact that ñ (1 m 1 ) ( ) k yields the fial iequality. Therefore, the probability that algorithm A fails o ay iput X for which X = k satisfies ( ) ( Pr{A(X) f k1,k (X)} = Pr{A(X) = 1} 1 p + p ). (4) k m X =k X =k I terms of q(x) ad its symmetrizatio the last iequality becomes ( ) c ( ) q(x) = q sym ( Y ), Y = k, Y B, k k X =k where the iequality is obtaied from (4) with c = 1 p + p/m, ad the equality holds for the same reasos as those cocerig the permutatios of oly oes or zeros i X which we explaied before. This implies c q sym ( Y ) = f k1,k (Y ) q sym ( Y ), Y = k. (5) We combie (3) ad (5) to obtai f k1,k (X) q sym ( X ) c < 1, 8

9 for all the X for which this partial Boolea fuctio is defied. Recall that symmetrizatio does ot icrease the degree of a polyomial, which implies that T is greater tha or equal to the degree of q sym. Usig the results of Nayak ad Wu [10] cocerig lower bouds for the degree of polyomials approximatig the partial Boolea fuctio f k1,k, ad our assumptio that k( k) = Θ( ), for all k I, we obtai that the degree of q sym is ( ) κ( κ) Ω k 1 k +, k 1 k where κ {k 1, k } which maximizes κ. Therefore, the umber of queries of the origial algorithm is Ω( 1 ), which, i tur, is Ω(ε 1 ). Theorem 3.1 exteds the optimality properties of QS to the average probabilistic case whe high accuracy is importat. It shows that QS is asymptotically optimal i computig the Boolea mea as log as µ satisfies certai properties. The rage of possible values of ε has to be appropriately small ad this depeds o the class of measures through the cardiality of the set I. The larger this set is the larger the rage of ε for which Theorem 3.1 holds ad QS is asymptotically optimal. O the other had, as we are about to see, whe there is demad for relatively low accuracy there ca be other algorithms faster tha QS. Let us ow cosider µ 1 where all elemets X B are equally likely havig probability. Kwas ad Woźiakowski [7] show that, with probability p = 1, the algorithm that outputs 1 o ay iput without ay queries at all, i.e., T = 0, has error e ap (B, 0, 1) = (π) 1/ (1 + o(1)). (6) However, reducig the error further requires Ω( 1/ ) queries, as we see below. Lemma 3.1. Cosider the measure µ 1. There exists a costat c > 0 such that the coditio e ap (B, T, p) c 1/, p > 1, implies that T is Ω(1/ ). Proof: The proof is very similar to that of Theorem 3.1. We poit out the differeces ad we refer to the proof of Theorem 3.1 for the idetical parts. Recall that i the proof of Theorem 3.1 equatio (1) lead us to select m such that 1 p + p/m < 1. Cosider ay such m here. We set c = e 6(m+1) (π) 1/. We cosider the sets I 1 = { +m,..., +(m+1) } ad I = { (m + 1),..., m }. Note that for the idices k I 1 I we have k( k) = Θ( ). Assume that c 1/ e(x, T, p)µ 1 (X) e(x, T, p), j = 1,, X B X I j sice µ 1 (X) =, for every X B. From Lemma 6.1, i the Appedix, we have that ( X ) > c 1/, whe X I 1 I, X B. We multiply by both sides of the iequality above, ad defie = c 1/ ad (k) = X =k e(x, T, p)/( k) to obtai > 1/ k I j (k), j = 1,. 9

10 Thus, there exist k j I j, such that (k j ) <, j = 1,. Let δ(x, k j ) = e(x, T, p), X = k j, j = 1,. The we have > ( 1 ) δ(x, k j ) k j X =k j = ( 1 ) k j X =k j : δ(x,k j )<m 1 ( k j )m ñ j,+, j = 1,, δ(x, k j ) + X =k j : δ(x,k j ) m δ(x, k j ) where ñ j,+ is the umber of strigs X with X = k j, for which δ(x, k j ) m. Just like i the proof of Theorem 3.1 we coclude that the umber of strigs X for which δ(x, k j ) < m satisfies ñ j, (1 m 1 ) ( ) k j, j = 1,. Now we use the algorithm that approximates the mea to derive aother algorithm that approximates the partial Boolea fuctio f k1,k (X) = { 1 if X = k1 0 if X = k. From this poit o the proof is idetical to the proof of Theorem 3.1. ad we omit the details. The coclusio is that the algorithm that approximates f k1,k ad, therefore, the origial algorithm must make Ω(/ ) or, equivaletly, Ω( 1/ ) queries. Thus we eed to study the error of the algorithm whe the umber of queries is Ω( 1/ ). The followig theorem deals with this case ad also summarizes our results with respect to µ 1. Theorem 3.. Cosider that approximatio of the Boolea mea ad the average probabilistic error of a quatum algorithm with respect to µ 1. The followig two statemets hold. 1. Let T be o(). The the error of ay quatum algorithm with T queries satisfies e ap (B, T, p) = Ω ( mi{ 1/, T 1 } ).. Let ε > 0 be o( 1/ ). The the umber of queries T (ε) for error at most ε satisfies T (ε) = Ω ( mi{ε 1, } ). Proof: The secod statemet directly follows from Theorem 3.1. Ideed, i the proof of Lemma 3.1 we saw a lower boud for µ 1 (X) whe X belogs to sets of 1/ may idices close to / (sets like I 1 ad I ). Thus the coditios of Theorem 3.1 hold for µ 1 ad the query lower boud is immediate. Now we prove the first statemet. From Lemma 3.1 we kow that error less tha c 1/ requires Ω( 1/ ) queries. Hece, whe the umber of queries is o( 1/ ) the the error is bouded from below by a quatity proportioal to 1/. 10

11 Let us ow cosider T to be Ω( 1/ ). The T 1 is O( 1/ ) ad mi{ 1/, T 1 } = Θ(T 1 ). We prove the first statemet by cotradictio. Assume that is sufficietly large. Suppose that the error lower boud is ot Ω ( mi{ 1/, T 1 } ) but that e ap (B, T, p) (T g(t )) 1, where g is a fuctio such that g(t ) = ω(1). Set ε = (T g(t )) 1 ad observe that ε is o( 1/ ). The use the secod statemet of this theorem coclude that T must be Ω(T g(t )), which is a cotradictio. Kwas ad Woźiakowski [7] show that for µ 1, the average probabilistic error of QS is O(mi{ 1/, T 1 }) whe the umber of its queries is divisible by four. Usig Theorem 3. we coclude: QS is a asymptotically optimal error algorithm. QS makes a asymptotically optimal umber of queries for accuracy ε, whe ε is ω( 1/ ). QS requires at least four queries for error O( 1/ ) whe ε is ω( 1/ ), while the optimal umber of queries is zero, ad is achieved by a costat algorithm. We ow cosider µ which correspods to the case that all values of the mea are equally likely. As we shall see, computig the mea i the average probabilistic case with µ is just as hard as computig the mea i the worst probabilistic case. Theorem 3.3. Cosider the approximatio of the Boolea mea ad the average probabilistic error of a quatum algorithm with respect to µ. The followig two statemets hold. 1. Let T be o(). The the error of ay quatum algorithm with T queries satisfies e ap (B, T, p) = Ω ( T 1).. Let ε > 0 be o(1). The the umber of queries T (ε) for error at most ε satisfies T (ε) = Ω ( mi{ε 1, } ). Proof: Trivially µ satisfies the coditios of Theorem 3.1 for a set I of Θ() may cosecutive idices, e.g., I = {,..., 3 }. Therefore, the secod statemet is immediate. 4 4 We show the first statemet by cotradictio. If T is O(1) the the error is bouded from below by a costat. Ideed, if we assume that e ap (B, T, p) 1/g() for some fuctio g satisfyig g() = ω(1), the Theorem 3.1 yields that T = Ω(g()), which is a cotradictio. I cotrast to µ 1, the measure µ does ot make the problem easier. Whe T is ω(1), suppose that e ap (B, T, p) is o(t 1 ). Let be sufficietly large. The there exists a fuctio g with g(t ) = ω(1) such that e ap (B, T, p) (T g(t )) 1. Set ε = (T g(t )) 1 ad observe that ε = o(1), as the secod statemet of the theorem requires. This leads us to coclude that T must be Ω(T g(t )) ad, therefore, we get a cotradictio. For µ, Theorem 3.3 ad the results of [4] ad [7] (for the worst probabilistic error of QS) imply that QS is a asymptotically optimal error ad query algorithm. Hece, i terms of 11

12 error ad umber of ecessary queries, computig the Boolea mea o the average with µ is as difficult as i the worst probabilistic case. We ed this sectio by extedig our results to -approximate cout. We preset three corollaries. We omit their proofs sice they are immediate from the correspodig theorems above. Corollary 3.1. Cosider -approximate cout. Let I {0,..., } be a set of idices, such that its cardiality I, as a fuctio of, is ω(1), ad k( k) is Θ( ) for every k I. Assume that µ is a probability measure o B such that µ(x) = Ω( I 1 ) 1 ), for every X I, X B. ( X The for ay > 0 of order o( I ), e ap 1 (B, T, p) implies that T = Ω(mi(/, )). Corollary 3.. Cosider -approximate cout ad the average probabilistic error of a quatum algorithm with respect to µ 1. The followig two statemets hold. 1. Let T be o(). The the error of ay quatum algorithm with T queries satisfies e ap 1 (B, T, p) = Ω ( mi{ 1/, /T } ).. Let > 0 be o( 1/ ). The the umber of queries T ( ) for error at most satisfies T ( ) = Ω (mi{/, }). Corollary 3.3. Cosider -approximate cout ad the average probabilistic error of a quatum algorithm with respect to µ. The followig two statemets hold. 1. Let T be o(). The the error of ay quatum algorithm with T queries satisfies e ap 1 (B, T, p) = Ω (/T ).. Let > 0 be o(). The the umber of queries T ( ) for error at most satisfies T ( ) = Ω (mi{/, }). 4 Worst expected error I this sectio we cosider quatum algorithms with a worst expected error criterio. We show query lower bouds for ay quatum algorithm computig the Boolea mea ad for ay quatum algorithm computig a -approximate cout. 1

13 Theorem 4.1. Cosider ay algorithm that computes the Boolea mea with worst expected error satisfyig e we (q, B, T ) ε, for a fixed q [1, ). The the umber of queries of this algorithm satisfies T (ε) = Ω(mi{ε 1, }). Proof: Cosider e we (q, B, T ) ε ad raise both sides to the power q ad multiply them by q. The set = ε ad (X, j) = a X â X (j) to obtai q M 1 j=0 (X, j) q p X (j), X B. For ay δ > 0 usig the Markov iequality we have q δ q p X (j), X B. (X,j) δ Choose a umber a > ad set δ = a ad p = 1 a q, p ( 1, 1). Defie p loss(x) = (X,j) δ p X(j) for X B. The 1 p = a q = q δ p loss(x). q This implies that p wi (X) = 1 p loss (X) p > 1, X B. The there exist outcomes j for which δ > (X, j) with probability p wi (X) = p X (j) p > 1/, X B. a X â X (j) <δ Hece, the probabilistic error of δ-approximate cout is e 1 (X, T, p) < δ, for all X B. Now take ay X such that X ( X ) = Θ( ) ad use the results of [10] to see that the umber of ecessary queries is ( ) X ( X ) Ω δ +. δ Therefore, the umber of queries satisfies Ω ( mi{ε 1, } ). It has bee recetly show i [6] that if oe repeats ( q + 1) times the QS algorithm of Brassard et al. (with T queries) the the media of the outputs has worst expected error of order O(T 1 ). Usig the theorem above we coclude that this is a asymptotically optimal algorithm. The followig query lower boud for -approximate cout is a direct cosequece of Theorem 4.1. Corollary 4.1. Cosider ay algorithm that computes a -approximate cout with worst expected error satisfyig e we 1 (q, B, T ), for fixed q [1, ). The the umber of queries of this algorithm satisfies T ( ) = Ω(mi{/, }). 13

14 5 Average expected error I this sectio we cosider the average expected error of quatum algorithms. For brevity we call this the average expected settig. Recall that we are cosiderig the average with respect to a probability measure o the set of iputs B ad for each of the iputs we cosider the expected error of the quatum algorithm with respect to all possible oucomes. We show query lower bouds for ad quatum algorithm computig the Boolea mea ad for ay quatum algorithm computig a -approximate cout. We deal oly with the measures of Theorem 3.1 sice µ 1 ad µ are special cases that ca be dealt with i the same way. I fact, Theorem 3.1 holds for the average expected error as well. The proof is based o that of Theorem 3.1. Theorem 5.1. Cosider the approximatio of the Boolea mea. Let I {0,..., } be a set of cosecutive idices, such that its cardiality I, as a fuctio of, is ω(1). Assume that k( k) is Θ( ) for every k I. Let µ be a probability measure o B such that µ(x) = Ω( I 1 ) 1 ), for every X B with X I. ( X Cosider a fixed q [1, ). The for ay ε > 0 of order o( I 1 ), the coditio e ae (q, B, T ) ε implies that T must be Ω(mi(ε 1, )). Proof: The proof is almost idetical to that of Theorem 3.1 ad we will oly poit out the differeces. I particular, for 1 q < cosider a quatum algorithm with average expected error at most ε, i.e., ε e ae (B, T ) = X B e (q) (X, T )µ(x). We follow the first part of the proof of Theorem 3.1 replacig e(x, T, p) by e (q) (X, T ) ad redefiig the rest of the quatities accordigly. After the two applicatios of the Markov iequality we kow that the umber of strigs X for which δ(x, k) < m, X = k, is ñ (1 m 1 ) ( k), ad m. Recall that δ(x, k) = e (q) (X, T ). Usig the Markov iequality as i Theorem 4.1 to derive the probabilistic error from the expected error, we coclude that the probabilistic error of approximate cout satisfies e 1 (X, T, p) < ma, with probability p 1 a q > 1/ for a chose a >. Now we retur to the proof of Theorem 3.1. We have that there exist k 1, k J that are at least 4ma apart whose distace does ot exceed O( ), i.e., k 1 k 4ma ad k 1 k = O( ), ad e 1 (X, T, p) ma, X = k 1, or k. We use the origial algorithm that approximates the mea to derive a ew algorithm that approximates the partial Boolea fuctio { 1 if X = k1 f k1,k (X) = 0 if X = k. where, without loss of geerality, we ca assume that k 1 > k. 14

15 The descriptio of the ew algorithm A is as follows: O iput X, where X = k 1 or k, we ru the algorithm that approximates the mea ad if the value of the result â X (j) satisfies k 1 â X (j) < ma the the ew algorithm outputs 1. It outputs 0 otherwise. There is oe more differece betwee this proof ad the proof of Theorem 3.1. It cocers the derivatio of the success/failure probability of the ew algorithm ad we explai this differece below. Let s look at the success probability Pr{A(X) = f k1,k (X)} of the ew algorithm for the differet iputs for which f k1,k is defied. If X = k 1 we have Pr{A(X) = f k1,k (X)} = Pr{A(X) = 1} X =k 1 X =k 1 Pr{A(X) = 1} = X =k 1, δ(x,k)<m X =k 1, δ(x,k)<m ( ) (1 m 1 ) (1 a q ), k 1 Pr{ k 1 â X (j) < ma } because we saw that whe the expected error satisfies δ(x, k) < m the this implies that the probabilistic error satisfies e 1 (X, T, p) < ma with probability p 1 a q > 1/. The fact that ñ (1 m 1 ) ( ) k 1 yields the fial iequality. Therefore, the probability that algorithm A fails o ay iput X for which X = k 1 satisfies ( ) Pr{A(X) f k1,k (X)} = Pr{A(X) = 0} (a q + 1 a q k 1 m X =k 1 X =k 1 Let c = a q + (1 a q )/m, where a >. Just like i the proof of Theorem 3.1, we choose m i a way that c < 1. This leads us to the equivalet of (1) of Theorem 3.1. I the same way we derive the equatio cocerig the probability of failure of the ew algorithm o iput X = k which correspods to equatio (4) of Theorem 3.1. The remaiig steps are idetical to those of Theorem 3.1 ad complete the proof. Theorem 5.1 shows that QS algorithm with repetitios [6] is asymptotically optimal i the average expected case whe the required accuracy is high. I fact, the query lower bouds of sectio 3 that deped either o Theorem 3.1 directly or have bee derived through as similar proof techique exted to the average expected ad we have see how this ca be accomplished i the proof of Theorem 5.1. The followig corollary for -approximate cout i the average expected case is immediate. Corollary 5.1. Cosider -approximate cout. Let I {0,..., } be a set of cosecutive idices, such that its cardiality I, as a fuctio of, is ω(1), ad k( k) is Θ( ) for every k I. Assume that µ is a probability measure o B such that µ(x) = Ω( I 1 ) 1 ), for every X I, X B. ( X 15 ).

16 Cosider a fixed q [1, ). The for ay > 0 of order o( I ), e ae 1 (q, B, T ) implies that T = Ω(mi(/, )). Ackowledgemets I thak P. Jaksch, J. Traub, A. Werschulz ad H. Woźiakowski for their commets ad suggestios that sigificatly improved this paper. 6 Appedix Lemma 6.1. For N ad 1 c /6 we have ( ) / ± c > e 6c. π Proof: From Stirlig s formula [1, p. 57] we have! = π +1/ e +θ/(1), 0 < θ < 1. Thus, ad Therefore, ad! e π(/e),! π(/e). (/ + c ( ) /+c + c )! e π(/ + c ) e ( ) /+c + c < e π, e (/ c ( ) / c c )! e π(/ c ) e ( ) / c c < e π. e 16

17 From the iequalities above we obtai! (/ + c )!(/ c )! > = = > = e π( + c ) /+c ( c ) / c ( c e π( + c ) / ( c ) / + c ( ) c c e π( 4c ) / + c ( ) c c e π + c ( ) c c e π + c ) c Usig we obtai that Thus, ( ) c ( ) + c 4c c = 1 + < e 4c, c c ( ) c ( ) c + c + c < e 4c e 6c. c c! (/ + c )!(/ c )! > e 6c + π. Refereces [1] Abramowitz, M. ad Stega, I. A. (1965), Hadbook of Mathematical Fuctios, Dover, New York. [] Beals, R., Buhrma, H., Cleve, R., Mosca, R. ad de Wolf, R. (1998), Quatum lower bouds by polyomials, Proceedigs FOCS 98, Also quat-ph/ [3] Boyer, M., Brassard, G., Hoyer, P. ad Tapp (1998), Tight bouds o quatum searchigs, Fortschritte der Physik, 46, Also quat-ph/ [4] Brassard, G., Hoyer, P., Mosca, M., ad Tapp, A. (000), Quatum amplitude amplificatio ad estimatio quat-ph/ [5] Heirich, S. (00), Quatum Summatio with a Applicatio to Itegratio, J. Complexity, 18(1), Also quat-ph/

18 [6] Heirich, S., Kwas, M. ad Woźiakowski (003), Quatum Boolea Summatio with Repetitios i the Worst-Average Settig, Preprit, Computer Sciece Departmet, Columbia Uiversity. [7] Kwas, M. ad Woźiakowski, H. (00), O Quatum Boolea Summatio i Various Error Settigs, Preprit, Computer Sciece Departmet, Columbia Uiversity. [8] Misky, M. ad Papert, S. (1988), Perceptros, MIT Press, Cambridge, MA, d editio. [9] Nielse, M.A. ad Chuag, I.L. (000), Quatum Computatio ad Quatum Iformatio, Cambridge Uiversity Press, Cambridge, UK. [10] Nayak, A. ad Wu, F. (1999), The quatum query complexity of approximatio the media ad related statistics, Proceedigs 31st STOC Also quat-ph/