On Arthur Merlin Games in Communication Complexity

On Arth ur Merlin Games in Comm unication Complexi t y ∗ Hartm ut Klauc k Cen tre for Quan tum T ec hnologies (NUS) and Sc ho ol of Ph ysical and Mathematical Sciences Nan y a ng T ec hnolog ical Unive rsity , Singap ore. Email: hklauck@gma il.com Abstract W e show several results related to interactive pro of modes o f communication co mplexity . Firs t we show low er b ounds for the QMA-communication complexity of the functions Inner P ro duct and Disjoint nes s. W e describ e a genera l metho d to prove low er b ounds for QMA-communication complexity , and sho w ho w one can ’transfer’ hardness under an analogous mea sure in the query complexity mo del to the communication model using Sherstov’s pattern ma trix method. Combin- ing a result by V ereshchagin and the pattern matrix metho d we find a communication pr oblem with AM-communication complexit y O (log n ), PP-communication co mplex it y Ω( n 1 / 3 ), and Q MA- communication complexity Ω( n 1 / 6 ). Hence in the world of communication complexity nonin tera c- tive quantum pro o f sy s tems a re no t able to efficien tly simu la te co-nondeterminism or interaction. These results imply that the rela ted que s tions in T uring ma chine complexity theory ca nnot b e resolved by ’algebriz ing ’ techniques. Fina lly w e show that in MA-pro to cols there is an exp one ntial gap b etw een one- wa y proto co ls and tw o-way proto cols (this r efers to the in teraction b etw een Al- ice and B o b). This is in contrast to nondeter ministic, AM-, and Q MA-proto cols, wher e one-way communication is es sentially optimal. 1 In tro d uction In their seminal 1986 pap er on ’Complexity Classes in Communicatio n C omplexit y’ [BFS86] Babai et al. d efine, among a host of other cla s s es, the comm un ication complexit y analogues of the interact ive pro of classes AM and MA. In this con text, for instance, the class MA consists of a ll comm un ication problems that h a ve MA-comm un ication complexit y at most p oly log n . The MA-comm u nication complexit y is the optimal complexit y of a proto col solving a comm un ication pr oblem with the help of a prov er Merlin, and tw o verifiers Alice and Bob (Alice and Bob eac h see only th eir part of the input, while Merlin sees the whole inp ut, How ev er, Merlin cannot b e trusted). Merlin sends a pr o of follo w ed by a discussion betw een Alic e and Bob. See Section 2.1 for definitions. While the T ur ing m ac hine versions of AM and MA (captur ing interact ive pr o of sys tems with a constan t n umb er of round s b et ween the p ro ver and v erifi er resp . nonin teractiv e, but r andomized pr o of systems [B85, BM88]) play ed a crucial role in the subsequent deve lopment of theoretical computer science, their comm unication complexit y analogues w er e p r obably consid ered too esoteric a topic to merit muc h consideration. One of the few resu lts ab out them is a 2003 lo wer b oun d of Ω( √ n ) f or the MA-comm unication complexit y of the Disjoin tness problem Disj b y this author [K03] ( Di sj is co -NP complete in the w orld of communicatio n complexit y). Th e same pap er also relates these complexit y measures to the p o wer of the rectangle (ak a co r ruption) b ound in communicatio n complexit y . In ∗ Researc h at the Centre for Quantum T echnologies is f u nded by the Singap ore Ministry of Education and the National Researc h F oundation. 1 2004, Raz and Shpilk a [RS 04] prov ed that there is a problem, for whic h its quan tum communicati on complexit y is exp onentially smaller than its MA-comm u nication complexit y , providing another MA- comm unication complexit y lo w er b oun d, as w ell as a problem, for w hic h its Quantum MA (short QMA-) comm u nication complexit y is exp onentiall y smaller than b oth the qu an tum (without prov er) and the MA-comm un ication co m plexities. Raz and Shpilk a left proving lo we r b ound s f or the QMA- comm unication complexit y of an y function as an op en p roblem. Surp r isingly , in 2008 Aaronson and Wigderson [A W09] n ot only show ed that the mentioned low er b ound for Di sj is basically tigh t, bu t also ga ve a new incentiv e to understand the r elations b et w een the complexit y classes in the wo r ld of comm un ication complexit y . Their pap er in vestiga tes a new ’barrier’ in complexit y called algebrization . Without going deep er in to this topic one of their results is that a separation b et we en tw o communicati on complexity classes sho ws that the algebrization barrier applies to th eir T ur ing m achine analogues, i.e., an attempt to show that the t wo classes are the s ame will encoun ter said barrier. A restricted mo d el of MA (and QMA) one-wa y comm u nication complexit y has b een inv estigated b y Aaronson [A06]. V ery rece ntly Ga vinsky and Shers tov [GS10] ha ve sho wn that co-NP is not a subset of MA in the mo del of m u ltipart y comm unication complexit y . One motiv ation for in v estigating pro of systems in comm unication co mp lexit y is to u nderstand the p ow er of pro ofs. The results in [RS04] sho w th at in a setup wher e w e can actually p ro ve s uc h statemen ts, quan tum pro ofs (of type QMA) are more p o we r ful than classical pr o ofs. How ev er, man y questions ab out this topic remained op en: is AM larger than MA? Is co-NP a subset of QMA? Is QMA a subset of AM? In this pap er w e resolve some of these questions. In our fi rst result w e sho w a lo w er b ound of Ω( n 1 / 3 ) for the QMA-comm u nication complexit y of Disj . This means that co-NP is in fact not a su bset of QMA in th e wo r ld of comm unication complexit y , and that trying to pu t co-NP into QMA in the ’real w orld’ (besides the in clusion probably b eing false) would require a nonalg eb rizing tec hnique. W e also sho w ho w a lo wer b oun d method we call ’one-sided discrepancy’ giv es lo w er b oun ds for QMA-comm unication complexit y 1 . F urtherm ore w e sho w a (basica lly tight ) lo we r b ound of Ω ( √ n ) for the QMA-comm u nication complexity of th e function IP 2 , the f u nction that compu tes the inn er pro du ct mo dulo 2. I t is an interesting question, w hether the b ound for Disj is tight, b ecause we ha ve t wo v ery d ifferen t u pp er b ounds of the order e O ( √ n ) f or this prob lem: the MA-proto col f r om [A W09], and the Gro ver based quan tum proto col fr om e.g. [AA03]. Is it p ossible to com bine these pr otocols in s ome w a y to get a more efficien t QMA protocol? Our second m ain result shows that there is a partial function f , for whic h the w eakly u nb ounded error comm un ication co mp lexit y P P ( f ) is Ω( n 1 / 3 ), whereas the AM-comm unication complexit y is only O (log n ). The PP lo wer b ound immediately implies a Ω( n 1 / 6 ) lo we r b ound for the QMA- comm unication complexit y of th e same p roblem. Hence here the tiniest amount of in teraction in classical pro of systems (the v erifiers may challe n ge the prov er with a public coin message) cannot b e simulated by a nonin teractiv e pro of system even with the h elp of quan tu m ness. In terms of algebrizatio n this result shows th at p utting AM in to PP (or ev en in to MA) needs nonalge b rizing tec hniques. On the other hand it is widely b eliev ed that AM=NP b ecause it might b e p ossible to derandomize AM. The result has another imp lication: Since PP-complexit y coincides with the discr ep ancy b ound [K07], it turns out to b e p ossible for a function to ha ve p olynomial size rectangle co vers of its 1-inputs with small err or u nder ev ery distribu tion on the inputs, y et for some distribu tion on the inpu ts ev ery individual rectangle is either exp onentiall y small, or has error exp onentia lly close to 1/2. Th at means an y attempt to pr ov e lo wer b ounds for AM-proto cols by considering the prop erties (error and size) 1 Essen tially the same metho d has b een used by Gavinsky and Sherstov for the multipart y version of MA- comm u nication complexity [GS10 ]. 2 of ind ividual r ectangles alone m u st fail. W e b eliev e that it is imp ortant to sh o w lo w er b ounds for AM-proto cols, b ecause new techniques d ev elop ed for this p roblem n eed to get past this ’rectangle barrier’. F ur thermore such a pro of w ould b e a fi rst step to wards r esolving the Π 2 6 = Σ 2 problem in comm unication complexit y , one of the bigge s t problems left op en b y [BFS86]. Finally , our thir d result considers the str ucture of MA-proto cols. In general, n ondeterministic proto cols require no nontrivial in teraction b et wee n Alice and Bob, i.e., after seeing the pro of, Alice can just send one message to Bob, who accepts or rejects. The same is tr ivially true f or AM-proto cols (with our defin ition). In terestingly , and somewhat co u n terintuitiv ely , Raz and Sh pilk a sh o w that (within a p olynomial increase in comm u nication) one-w a y comm un ication is al s o optimal for QMA- proto cols. W e show that there is a p roblem, for which one-w ay MA-comm u nication is exp onen tially w orse than t wo w ay randomized communicati on. This highlights the difference b et ween quant u m and classical pro ofs, and is somewh at reminiscent of the fact, that in the ’real world’, quantum pro of systems in the class QIP can b e parallelized to on ly 3 rounds [KW00], whereas a similar parallelizat ion of classical pr o ofs w ould collapse the p olynomial h ierarc hy (here p aralleliza tion refers to th e in teraction b etw een the pr o v er and the ve r ifier). 2 Definitions and Preliminaries 2.1 Arth ur Merlin Commu nication Complexit y Definitions F or definitions of more stand ard mo d es of communicat ion complexit y w e refer to Kushilevitz and Nisan’s exce llent monog r aph [KN97]. In Arthur Merlin communicatio n games, there are 3 parties Merlin, Alice, Bob. All of them are computationally un b ounded. Alice sees her inpu t x ∈ { 0 , 1 } n , Bob his input y ∈ { 0 , 1 } n , and Merlin sees b oth inputs. Merlin is the pr over , who wan ts to con vince the verifier , consisting of Alice and Bob to gether, that f ( x, y ) = 1. Definition 1 In a M erlin-Arthur pr oto c ol (sho rt M A-pr oto c ol) for a Bo ole an function f Alic e i ni - tial ly r e c eives a message (also c al le d the p ro of ) fr om M e rlin. After this Alic e and Bob c ommunic ate until they c ompute an output, using public key r andomness (the pr o of c annot dep end on the r andom- ness). The cost of an MA-pr oto c ol is the sum of th e length a of the pr o of, and the length c of the over al l c ommunic ation b etwe en Alic e and Bob. The pr oto c ol c omputes f , if for al l inputs x, y with f ( x, y ) = 1 ther e exists a pr o of such that x, y is ac c epte d with pr ob ability p and for al l inputs x, y with f ( x, y ) = 0 and al l pr o ofs the pr ob ability that x, y is ac c epte d is at most q . p must b e at le ast a c onstant factor lar ger than q . p is the completeness , q the soundness of the pr oto c ols. We wil l c al l max { 1 − p, q } the error of the pr oto c ol, and fr e quently c onsider pr oto c ols with very smal l err or. If not mentione d otherwise we assume p = 2 / 3 and q = 1 / 3 . The Merlin- Arth ur c omplexity of f , denote d M A ( f ) , i s the smal lest c ost of an MA- pr oto c ol f or f . The MA-c omplexity with b ounde d pr o of length a is denote d M A ( a ) ( f ) . Note that the error probabilities (resp. the soundness and completeness) of MA-proto cols ca n b e impro ved arbitrarily b y using stand ard b o osting tec hniques. F or this the pro of itself do es not need to b e rep eated, so the pro of length is not increased. Also note that including the pro of length in the cost is crucial, b ecause otherwise Merlin could pro vid e Alice with a copy of y , wh ose correctness could b e c hec ked with a standard fingerp rin ting proto col, decreasing the complexit y of all functions to O (1) (due to pub lic coins b eing a v ailable) . Definition 2 In an Art hur Merlin (shor t AM-) pr oto c ol, Merlin, Alic e, and Bob shar e a sour c e of r andom bits. First a r andom chal lenge is dr awn f r om this sour c e (of a pr e define d length). Merlin then 3 pr o duc es a message (c al le d the pro of ), which is sent to Alic e. After this Alic e and Bob c ommunic ate until they either ac c ept or r e j e ct. They may not use fr esh r andom bits at this stage, i.e., al l r andom bits ar e known to Me rlin. The c ost of an AM-pr oto c ol is the sum of the leng th of the pr o of and the length of the c ommunic ation b etwe en Alic e and Bob. The pr oto c ol c omputes f , if for al l inputs x, y with f ( x, y ) = 1 with pr ob ability at le ast 2 / 3 ther e exists a pr o of such that x, y is ac c epte d, and for al l inputs x, y with f ( x, y ) = 0 with pr ob ability at most 1 / 3 ther e exists a pr o of suc h that x, y is ac c epte d. The Arth u r Merlin c omplexity of f , denote d AM ( f ) , is the smal lest c ost of an AM -pr oto c ol for f . Note th at a more ge n erous definition is p ossible, in whic h Alice and Bob still h a ve access to pr iv ate random bits after receiving the pro of. W e prefer to call such proto cols AMA-proto cols, b ecause our definition of AM-proto cols is com binatorially cleaner and str ong enough for our separation result. Note that by standard tec hn iques from the theory of Arthur Merlin games [B85, BM88] b oth MA- and AMA-proto cols can b e at most qu adratically c heap er than AM-proto cols, while w e later sho w that AM- p roto cols can in d eed b e exp onen tially more efficient than MA-protocols. Also note that AM-pr oto cols n eed only one round of comm unication b et ween Alic e and Bob: to sim ulate any more complex proto col, Merlin can includ e the whole conv ersation b et w een Alice and Bob in his pro of, w h o just need to c hec k if their p art was rep r esen ted prop erly . W e no w define a quant u m v ersion of MA-proto cols. Definition 3 In a quantum M erlin-Arthur (short Q MA-)pr oto c ol, Merlin, pr o duc es a quantum stat e ρ (the pro of ) on some a qubits, which he sends to A lic e. Alic e and Bob then c ommunic ate using a quantum pr oto c ol, and either ac c ept or r eje ct the inputs x, y . We say that a QMA-pr oto c ol c omputes a Bo ole an function f , if for al l inputs x , y with f ( x, y ) = 1 , ther e exists a (quantum) pr o of, such that the pr oto c ol ac c epts with pr ob ability at le ast p , and for al l inputs x, y with f ( x, y ) = 0 , and al l (quantum) pr o ofs, the pr oto c ol ac c epts with pr ob ability at most q . A gain, we r e qui r e p ≫ q , and we set them to 2/3 r esp. 1/3 if not mentione d otherwise. The c ost of a Q MA-pr oto c ol is the sum of a and the length of the c ommunic ation b etwe en Alic e and Bob. The c ost of the c he ap est pr oto c ol that c omputes f defines Q M A ( f ) . The QMA -c ommunic ation c omplexity with b ounde d pr o of length a is denote d by QM A ( a ) ( f ) . Let us fir st note that, su rprisin gly , the error prob ab ility of QMA-proto cols can b e reduced without rep eating the qu an tum p ro of, du e to a clev er pro cedu re in tro du ced by Marriott and W atrous [MW05] in the con text of standard QMA-games. Since the pro of of [MW05 ] u ses the v erifi er s im p ly as a b lac k b o x , their construction carries o v er to the comm un ication complexit y scenario. Note how ev er, that this b o osting tec hnique increases the n umb er of roun ds b et we en Alice and Bob, b ecause their message sequences are computed and uncompu ted in a sequential mann er. F act 1 If ther e is a QMA -pr oto c ol with pr o of length a , c ommunic ation c and err or 1 / 3 , then ther e is a QM A-pr oto c ol with pr o of length a , c ommunic ation O ( c · k ) and err or 1 / 2 k . W e no w turn to proto cols with (w eakly) un b ounded error. Definition 4 In a we akly unb ounde d err or pr oto c ol Alic e and Bob h ave ac c ess to a private sour c e of r andom bits e ach. The pr oto c ol c omputes a Bo ole an function f if for al l inputs x , y th e pr ob ability p x,y of c omputing the c orr e ct output f ( x, y ) exc e e ds 1/2 . The gap on input x, y is g x,y = p x,y − 1 / 2 , the gap g = min x,y g x,y . The c ost of a we akly unb ounde d err or pr oto c ol with worst c ase c ommunic ation c is c − lo g g , and the we akly unb ounde d e rr or c omplexity of a function f is P P ( f ) , the minimum c ost of any pr oto c ol that c omputes f in the describ e d manner. 4 There is another type of unb ounded err or p roto cols, in w hic h the gap is not consider ed , but only the communicatio n n ecessary to ac hiev e correctness p robabilit y exceeding 1/2. W e d o not consid er this m o del h ere. See e.g. [BFS86] a n d [RR10, S08a, BVdW07] for more. 2.2 In teger Polynomia ls In this section we consider the r epresen tation of Bo olean fun ctions by p olynomials with in teger co efficien ts. Let f : { 0 , 1 } n → { 0 , 1 } b e a Bo olean function. A (partial) assignment A : S → { 0 , 1 } m is an assignmen t of v alues to some sub set S ⊆ { x 1 , . . . , x n } of v ariables. W e sa y that A is c onsistent with x ∈ { 0 , 1 } n if x i = A ( i ) for all i ∈ S . W e write x ∈ A as shorthand for ‘ A is co n sisten t w ith x ’. W e write | A | to r epresen t the card inalit y of S (not to b e confused with the num b er of consisten t inp uts). F ur thermore w e say that an index i app e ars in A , iff i ∈ S w here S is the subset of [ n ] corresp onding to A . W e define a function κ A : { 0 , 1 } n → { 0 , 1 } suc h that κ A ( x ) = 1 iff A is consisten t with x Ev ery f : { 0 , 1 } n → { 0 , 1 } can b e w ritten as s ign( P A : | A |≤ d w A · κ A ( x )), wher e d ≤ n is an in teger, the sum is o ver all p artial assignmen ts, and the w A are integ ers. W e call the minimum of P A : | A |≤ d | w A | that ac hieves this th e thr eshold weight W ( f , d ) o f f with de gr e e d . If d is too small to allo w repr esen tation of f , we set W ( f , d ) = ∞ . W e sa y that the int eger p olynomial sign( P A : | A |≤ d w A · κ A ( x )) sign-r epr esents the f u nction f . F requently in the literature (a n d imp ortan tly for us in Shersto v’s pap er [S08b]) the threshold w eight is defined n ot with partial assignmen ts, but with charac ters χ S of the F ourier transform ov er the Bo olean cub e, i.e., parit y f unctions on subsets S . Note that this changes the v alue of W ( f , d ) at most by a factor of 2 d : in order to r epresen t the function χ S with w eight w S w e can assign weigh t w S to all partial assignmen ts that fix all v ariables in S suc h that the p arit y of the v ariables in S is 1, and − w S to all partial assignmen ts that fix all v ariables in S s uc h that their parity is 0. Hence w e get a represent ation using partial assignmen ts instead of the χ S with total threshold weig ht increased by a factor of at most 2 d . Conv ersely , one can show ho w giv en a partial assignmen t A w ith w eigh t w A one can find a represen tation using a sum of χ S , so that the o v erall threshold w eigh t is in cr eased b y at m ost 2 d . W e omit this since it is not imp ortan t for this pap er. 2.3 Real Pol ynomials In Sectio n 3 we will also use the repr esen tation of Boolean fun ctions by p olynomials with real p oly- nomials. F or definitions co n cerning this topic w e refer to [BdW02] 2.4 P att er n Matrices In [S08b] Sh ersto v in tro d uced a metho d to turn Boolean functions f : { 0 , 1 } n → { 0 , 1 } into co m mu- nication p roblems that are hard, whenever f is hard under certai n measur es of co m p lexit y . Here w e define pattern matrices. Definition 5 F or a function f : { 0 , 1 } n → { 0 , 1 } the p attern matrix P f is the c ommunic ation matrix of the fol lowing pr oblem: A lic e r e c eives a bit string x of length 2 n , B ob r e c eives two bit strings y , z of length n e ach. The output of the function describ e d by P f on inputs x, y , z is f ( x ( y ) ⊕ z ) , wher e ⊕ is the bi twise xor, and x ( y ) denotes the n bit string that c ontains x 2 i − y i in p osition i = 1 , . . . , n . 3 QMA-complexit y of Disjoin tness In this section w e pro ve that th e Disjoin tness problem Disj requir es Q MA-communicat ion complexit y Ω( n 1 / 3 ). 5 Let us first define the problem. Definition 6 The Disjointness pr oblem Disj has two n - bit strings x, y as inputs. Disj ( x, y ) = 1 ⇐ ⇒ V i =1 ,...,n ( ¬ x i ∨ ¬ y i ) . Previous result ab out this problem are: [BFS86] pro ve R ( Disj ) = Ω( √ n ) (and observe that Disj is complete for the comm u nication complexity ve r sion of co-NP). [KS92] and lat er [Raz92] prov e the tigh t Ω( n ) b ound. [Raz03] s ho ws the tigh t Ω ( √ n ) lo wer b ound f or quantum proto cols. This result w as repr ov ed in a simpler wa y in [S08b]. [K03] giv es a Ω( √ n ) lo wer b ound for MA-proto cols. Finall y , [A W09] sho w a O ( √ n log n ) upp er b oun d for MA-protocols, and [AA03] gi ve a O ( √ n ) u pp er b ound for qu an tum proto cols. Here w e pro v e: Theorem 1 QM A ( Disj ) = Ω( n 1 / 3 ) . W e are going to giv e t wo pr o ofs of this. The first here us es Razb orov’s metho d. F ollo wing this (in S ection 4) we describ e a general metho d to pro ve QMA lo we r b ound s, and sho w h ow Shertov’s tec hnique can b e u sed to yield an ov erall sim p ler pro of (wh en taking the p ro of of Razb orov’s metho d in to accoun t). Razb oro v’s metho d can b e s u mmarized as follo ws [Raz03], s ee also [K S W07]. F act 2 Consider a c -qub i t quantum c ommunic ation pr oto c ol on n - bit inputs x and y , with ac c eptanc e pr ob abilities denote d by p ( x, y ) . Define p ( i ) = E | x | = | y | = n / 4 , | x ∧ y | = i | [ p ( x, y )] , wher e the exp e ctation is taken uniformly over al l x, y that e ach have weight n/ 4 and that have interse c tion i . F or ev e ry d ≤ n/ 4 ther e exists a de gr e e- d p olyno mial q such that | p ( i ) − q ( i ) | ≤ 2 − d/ 4+2 c for al l i ∈ { 0 , . . . , n/ 8 } . Pro of of T he orem 1. Su pp ose w e are giv en a QMA-proto col for Disj with comm unication c and pro of length a ≥ 1 and error 1 / 3. In what will b ecome a recurr in g theme, we first amplify the su ccess probabilit y to 1 / 2 10 a b y emplo ying Marriott-W atrous b o osting (F act 1). W e end up with a proto col that stil l h as pro of length a , but no w the co m m un ication is c ′ = O ( ac ). W e will sho w that this proto col needs comm u n ication at least Ω( √ na ), wh ic h implies the theorem. A t this p oin t w e simply replace Merlin’s pro of with the to tally m ixed s tate. W e end up with an ordinary quan tum protocol, that h as the follo wing prop erties: 1. All 1-inputs of Disj are accepted with p robabilit y at le ast (1 − 2 − 10 a ) / 2 a . 2. No 0-input of Disj is accepted with p robabilit y larger than 1 / 2 10 a . No w w e can simply inv ok e F act 2. W e set d = 12 c ′ . Then we receiv e a p olynomial q , suc h that 1. the degree of q is d . 2. 1 + 2 − c ′ ≥ q (0) ≥ (1 − 2 − 10 a ) / 2 a − 2 − c ′ . 3. − 2 − c ′ ≤ q ( i ) ≤ 2 − 10 a + 2 − c ′ for all 1 ≤ i ≤ n/ 8. No w w e define a r escaled p olynomial q ′ = 1 − q /q (0). 1. Th e d egree of q ′ is d . 2. q ′ (0) = 0. 3. 1 + 2 − c ′ / (1 + 2 − c ′ ) ≥ q ′ ( i ) ≥ 1 − (2 − 10 a + 2 − c ′ ) / ((1 − 2 − 10 a ) / 2 a − 2 − c ′ ) ≥ 1 − 2 − 8 a for all 1 ≤ i ≤ n/ 8. 6 The resulting p olynomial must rise v ery steeply b et wee n q ′ (0) and q ′ (1). W e can apply a result b y Buhrman et al. [BCWZ99], their Theorem 17. F act 3 Every p olynomial s of de gr e e d ≤ M − 1 such that s (0) = 0 and 1 − ǫ ≤ s ( x ) ≤ 1 for al l inte gers i ∈ [1 , M ] has ǫ ≥ 1 u e − vd 2 / ( M − 1) − 8 d/ √ M , wher e u, v ar e c onstants. Setting M = n / 8, and rescal in g q ′ sligh tly , we can use this fact to see that d ≥ Ω( √ na ) in o r der to en ab le ǫ ≤ 2 − Ω( a ) . Hence c ′ ≥ Ω( √ na ), and √ ac ≥ Ω ( √ n ). This implies that a + c ≥ Ω ( n 1 / 3 ), whic h is our th eorem.  4 A Lo w er Bound Metho d for QMA -proto c ols In this section w e dev elop a general method to pro v e low er b ounds for QMA proto cols, and we sh o w ho w to use the patte r n matrix metho d [S08b] for QMA-proto cols. 4.1 A Discrepancy Measure Let us start w ith the familiar notion of the d iscrepancy b ound in comm u n ication complexit y (see [KN97]). Definition 7 The (r e ctangle) discr ep ancy of a Bo ole an function f u nder a distribution µ is disc µ ( f ) = 1 max R | µ ( f − 1 (0) ∩ R ) − µ ( f − 1 (1) ∩ R ) | , wher e the maximum is over al l r e ctangles R in the c ommunic ation matrix. The discr ep ancy of f is disc ( f ) = max µ disc µ ( f ) . The follo wing linear pr ogram (see [JK10]) c haracterizes discrepancy . In th e follo wing R denotes the set of all rectangles in the comm unication matrix. Primal min X R ∈R w R + v R ∀ ( x, y ) ∈ f − 1 (1) : X R :( x,y ) ∈ R w R − v R ≥ 1 , ∀ ( x, y ) ∈ f − 1 (0) : X R :( x,y ) ∈ R v R − w R ≥ 1 , ∀ R : w R , v R ≥ 0 . Dual max X ( x,y ) µ x,y ∀ R : X ( x,y ) ∈ f − 1 (1) ∩ R µ x,y − X ( x,y ) ∈ R ∩ f − 1 (0) µ x,y ≤ 1 , ∀ R : X ( x,y ) ∈ f − 1 (0) ∩ R µ x,y − X ( x,y ) ∈ R ∩ f − 1 (1) µ x,y ≤ 1 , ∀ ( x, y ) : µ x,y ≥ 0 . The rectangle discrepancy c haracterizes the we akly unb ounded error co m munication complexit y P P ( f ), and serv es as a low er b oun d for b ound ed error quantum and r andomized comm unication. F or the b ounded error mo des it often yields only ve r y p o or results. Here is the relation to PP- comm unication complexit y [K07]. F act 4 F or al l Bo ole an f u nctions f : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } we have P P ( f ) ≥ Ω(log disc ( f )) and P P ( f ) ≤ O (log di sc ( f ) + log n ) . 7 No w we defi ne a lo w er b oun d method that w e will use for QMA-c ommunication complexit y , the one-sided smooth discrepancy . It is similar to the smooth discrepancy [K07, S08b], in whic h the primal linear program for discrepan cy is augmente d with additional upp er and lo w er b ound s . Here w e augment the pr ogram only f or the 0-inpu ts. Essentiall y the same metho d (in wh at w e later defin e as its ’natural’ v ersion) w as used r ecen tly b y Ga vinsky and Shersto v in the setting o f multipart y proto cols and MA-comm un ication [GS10]. Definition 8 (One- Side d Smo oth Discrepancy) L et f : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } b e a Bo ole an function. The one-side d smo oth discr ep ancy of f , denote d sdisc 1 ǫ ( f ) , is given by the optimal v alue of the fol lowing line ar pr o gr am. Primal min X R ∈R w R + v R ∀ ( x, y ) ∈ f − 1 (1) : X R :( x,y ) ∈ R w R − v R ≥ 1 , ∀ ( x, y ) ∈ f − 1 (0) : 1 + ǫ ≥ X R :( x,y ) ∈ R v R − w R ≥ 1 , ∀ R : w R , v R ≥ 0 . Dual max X ( x,y ) µ x,y − (1 + ǫ ) φ x,y ∀ R : X ( x,y ) ∈ f − 1 (1) ∩ R µ x,y − X ( x,y ) ∈ R ∩ f − 1 (0) ( µ x,y − φ x,y ) ≤ 1 , ∀ R : X ( x,y ) ∈ f − 1 (0) ∩ R ( µ x,y − φ x,y ) − X ( x,y ) ∈ R ∩ f − 1 (1) µ x,y ≤ 1 , ∀ ( x, y ) : µ x,y ≥ 0; φ x,y ≥ 0 . Note that for all ( x, y ) ∈ f − 1 (1) : φ x,y = 0 in an optimal solution. W e are n o w lo oking f or a ’natural’ definition of one-sided smooth discrepancy , i.e., a d efinition in whic h the one-sided sm o oth discrepancy of a function f is related to the discrepancy of a fun ction g that is similar to f . The v alue of one-sided smo oth discrepancy will b e the d iscrepancy of g under a distrib ution ν . The ab o ve dual sho ws us that we should h a v e f − 1 (1) ⊆ g − 1 (1). F urthermore, not to o man y 0-inpu ts of f should b e 1-inputs of g . It is also quite easy to see that for no inpu t φ x,y > 0 and µ x,y > 0 sim ultaneously in an optimal solution to the du al. Belo w w e present the natural definition of one-sided smo oth discrepancy . Definition 9 (One- sided Smo ot h Discrepancy , Natural Definition) L et f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } b e a Bo ole an function. The δ - one-side d smo oth discr ep ancy of f , denote d g sdisc 1 δ ( f ) , is define d as fol lows: g sdisc 1 δ ( f ) def = max { g sdisc λ, 1 δ ( f ) : λ distribution on { 0 , 1 } n × { 0 , 1 } n } . g sdisc λ, 1 δ ( f ) def = m ax { disc λ ( g ) suc h that g : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , f − 1 (1) ⊆ g − 1 (1) and λ ( f − 1 (1)) ≥ δ · λ ( g − 1 (1)) } . The follo w ing lemma sh o ws the equiv alence of the tw o definitions of one-sided smo oth discrepancy . Lemma 1 L et f : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } b e a function and let δ > 0 . Then 1. g sdisc 1 δ/ 3 ( f ) ≥ sdisc 1 δ ( f ) . 2. δ · g sdisc 1 2 δ ( f ) ≤ sdisc 1 δ ( f ) . Pro of. 8 1. S upp ose sdisc 1 δ ( f ) ≥ K . Then there is a solution µ, φ to the dual program that achiev es v alue K . W e ha ve to define a fun ction g and a distribution λ . W.l.o .g. we can assu me that f or ev ery x, y e ith er µ x,y = 0 or φ x,y = 0. W e d efine g ( x, y ) = f ( x, y ) when φ x,y = 0, else g ( x, y ) = 1. Clearly , f − 1 (1) ⊆ g − 1 (1). F urthermore d efi ne L = P ( x,y ) µ x,y + φ x,y . Set λ x,y = ( µ x,y + φ x,y ) /L . The co n strain ts of the d ual n ow imply that disc λ ( g ) ≥ L ≥ K . Denote µ 0 = P x,y : f ( x,y )=0 µ x,y /L, and µ 1 = P x,y : f ( x,y )=1 µ x,y /L and φ = P x,y φ x,y /L . S ince the set of all inpu ts is a rectangle, we ha ve that µ 0 ≤ 1 / 2 + 1 /L , and 1 / 2 + 1 /L ≥ µ 1 + φ ≥ 1 / 2 − 1 /L . Assume that λ ( f − 1 (1)) ≤ ( δ / 3) · λ ( g − 1 (1)). This m eans that µ 1 ≤ ( δ / 3) · ( µ 1 + φ ) ≤ ( δ / 6) + 1 /L . Then µ 1 + µ 0 − (1 + δ ) φ ≤ δ / 6 + 1 / 2 − (1 + δ )(1 / 2 − δ/ 6) + 3 / L < 0 , and the ob jectiv e fu nction would b e negativ e, w h ic h is imp ossible. 2. S upp ose g sdisc 1 2 δ ( f ) ≥ K . Then there are a fun ction g and a distr ibution λ as in the definition of g sdisc 1 2 δ . F or all x, y with g ( x, y ) = 1 bu t f ( x, y ) = 0 we set φ x,y = λ x,y · K , for all other x, y w e set µ x,y = λ x,y · K . All other v ariables are set to 0. Clearly , all constrain ts of the d ual are satisfied. D efin e µ 0 = P x,y : f ( x,y )=0 µ x,y , and µ 1 = P x,y : f ( x,y )=1 µ x,y and φ = P x,y φ x,y . Because the set of all inputs is a rectangle, w e h av e that µ 0 ≥ K/ 2 − 1, and K / 2 + 1 ≥ µ 1 + φ ≥ K / 2 − 1. The ob j ectiv e function is µ 1 + µ 0 − (1 + δ ) φ ≥ δ K − 1 + K/ 2 − 1 − (1 + δ )( K/ 2 + 1 − δK ) ≥ δ K.  W e can n o w sho w that the one-sided smo oth discrepancy yields lo w er b ounds for QMA-c ommunication complexit y . Theorem 2 L et QM A ( a ) ( f ) ≤ c . Then log sdisc 1 2 − 10 a ( f ) ≤ O (( a + 1) c ) . QM A ( f ) ≥ Ω  q log sdisc 1 ( f )  . One immediate corollary is a low er b ound for the fun ction Inn er Pro d uct mo d 2 ( IP 2 ), b ecause it is well kn o wn that ev en th e discrep ancy of IP 2 is at most 2 − Ω( n ) [CG85]. Corollary 1 QM A ( IP 2 ) ≥ Ω( √ n ) . Note that th is lo wer b ound is tigh t within a log fac tor d u e to the MA-proto col of Aaronson an d Wigderson [A W 09]. Pro of of Theorem 2. Supp ose we h a ve a QMA-pr otocol with p ro of length a ≥ 1, communicatio n c , and error 1/3, and a + c optimal. W e b o ost the success p r obabilit y us ing the Marriott-W atrous tec hnique F act 1. This giv es us a QMA-proto col with pro of length a , comm u nication c ′ ≤ O ( ca ), and error 2 − 13 a . W e replace the p ro of at this p oin t with the totally m ixed state, lea v in g us with a quantum proto col that acce p ts all 1-inputs with probability at least (1 − 2 − 13 a ) / 2 a , and accepts 0-inputs with probability at most 2 − 13 a . Our goal is to sho w, that th is give s u s a solution to the lin ear pr ogram f or one-sided smo oth discrepancy . W e consider the matrix of acceptance probabilities. 9 Definition 10 F or a matrix M with r e al entries, denote by µ ( M ) the minimum of P R | u R | (wher e R r anges over al l r e ctangles in the matrix M ) such that M = P R u R · f R , wher e f R is the char acteristic function of r e ctangle R . Linial and Shr aibman [LS09] ha v e sho w n the follo wing. F act 5 If M is the matrix of ac c eptanc e pr ob abilities of a quantum pr oto c ol (with shar e d entangle- ment) with c ommunic ation c , then µ ( M ) ≤ O (2 c ) . So let M b e the mat r ix w e constructed b efore (i.e ., for 1-inputs x, y the en tr y at p osition x , y is b et ween 2 − a (1 − 2 − 13 a ) and 1, and for 0-inpu ts b et ween 0 and 2 − 13 a ). Consider the system of w eights u R that ac h iev e µ ( M ) ≤ O (2 c ′ ). T o turn this into a solution for our primal program for the one-sided smo oth discrepancy b ound w e can simply multiply all the u R b y a facto r of 2 a +2 and subtract 1 + 2 − 12 a +2 from the u R for the rectangle R = { 0 , 1 } n × { 0 , 1 } n . Th en, for all R , when u R < 0 w e set v R = − u R and w R = 0, otherwise we s et w R = u R and v R = 0. The result is a f easible solution with the parameter ǫ ≤ 2 − 12 a +2 ≤ 2 − 10 a , and th e co st of the linear program is at most O (2 a · 2 c ′ ) ≤ O (2 ( a +1) c ). Hence log sdisc 1 2 − 10 a ( f ) ≤ O (( a + 1) c ). Also, QM A ( f ) ≥ a + c ≥ Ω  q log sdisc 1 2 − 10 a ( f )  .  4.2 Pro ving Lo w er Bounds for P att ern Matrices In this section w e follo w th e approac h of Sherstov [S08b], w hic h can b e su mmarized as follo ws: for a Bo olean fun ction f : { 0 , 1 } n → { 0 , 1 } , w e ca n d efine a comm un ication pr oblem P f , the p attern matrix (see Section 2.4), and then lo wer b ou n d the communicati on complexit y of P f in terms of s ome parameter of the function f . Sherstov u ses m ainly the appro ximate degree of f to get lo w er b ounds on the smo oth discrepancy of the fu nction P f (and hence its quan tum comm unication complexit y). Our goal is to rela te the one-sided smooth discrepancy of f , redefin ed for query problems, to the one-sided sm o oth d iscrepancy of P f . Thanks to the natural definition of sdisc 1 (Definition 9) it is actually sufficien t to r elate the discrepancies of f and P f . In the n ext sect ion we defin e d iscrepancy measures for query complexit y . 4.3 Another Notion of Discrepancy In this section w e defi n e a notion of discrepancy for Bo olean fu nctions (which can b e used as a lo wer b ound for qu er y complexit y , and in fact c haracterizes PP-query complexit y). Here su b cub es defi n ed b y partial assignments (see Section 2.2) tak e the role of the r ectangles. W e define the query complexit y v ersion of d iscrepancy as follo ws. Definition 11 (Discrepancy) L et f : { 0 , 1 } n → { 0 , 1 } b e a func tion. The (p olynomia l) discr ep ancy of f , denote d p dis c ( f ) , is given by the optimal value of the fol lowing line ar pr o gr am. Primal min X A ( w A + v A ) · 2 | A | ∀ x ∈ f − 1 (1) : X A : x ∈ A w A − v A ≥ 1 , ∀ x ∈ f − 1 (0) : X A : x ∈ A v A − w A ≥ 1 , ∀ A : w A , v A ≥ 0 . Dual max X x µ x ∀ A : X x ∈ f − 1 (1) ∩ A ( µ x ) − X x ∈ A,x 6∈ f − 1 (1) ( µ x ) ≤ 2 | A | , ∀ A : X x ∈ A,x 6∈ f − 1 (1) ( µ x ) − X x ∈ A ∩ f − 1 (1) ( µ x ) ≤ 2 | A | , ∀ x : µ x ≥ 0 . 10 W e only give the ’natural’ d efinition of th e corresp ond ing notion of one-sided smo oth d iscr ep ancy . The linea r pr ograms are easy to state and analogous to the co m m un ication complexit y v ersions. Definition 12 L et f : { 0 , 1 } n → { 0 , 1 } b e a Bo ole an function. The δ - one-side d smo oth (p olynomia l) discr ep ancy of f , denote d sp dis c 1 δ ( f ) , is define d as fol lows: sp disc 1 δ ( f ) def = max { sp disc λ, 1 δ ( f ) : λ distribution on { 0 , 1 } n } . sp disc λ, 1 δ ( f ) def = m ax { p disc λ ( g ) suc h that g : { 0 , 1 } n → { 0 , 1 } , f − 1 (1) ⊆ g − 1 (1) and λ ( f − 1 (1)) ≥ δ · λ ( g − 1 (1)) } . Essen tially this measure lo oks at a com bination of threshold weigh t and degree of p olynomials that sign-represent a function w ith a large enough gap, but without requir ing in teger coefficien ts. W e now wa nt to r elate the qu ery complexit y v ersion and the comm u nication complexit y version of discrepancy . Shersto v pro ve d the follo wing statemen t [S08b]. F act 6 L et P f b e the p attern matrix of a function f , and d a p ositive inte ger. Then disc ( P f ) 2 ≥ min  W ( f , d − 1) 2 d n , 2 d  , wher e W ( f , d ) denot e s the thr eshold weight of f with de gr e e d , i.e., the minimum thr eshold weight of a p olyno mial with inte ger c o efficients and de gr e e d , that sign- r epr esents f . Refer to s ection 2.2 for an explanation of th reshold weigh t as used here (wh ich is sligh tly differen t from Shersto v’s usage, leading to an extra f actor of 2 d ). Insp ecting the pro of of F act 6 it b ecomes clear, that the inte gralit y of the p olynomial co efficien ts in the definition of W ( f , d ) is only used to ensure that the gap b et ween th e v alue of a p olynomial on 1-inpu ts and 0-inputs is at least 1. I t turns out w e ca n r eplace W ( f , d − 1) b y p disc . F urtherm ore, since the linear program for p disc already incorp orates the factors 2 | A | the min im um w ith 2 d is u nnecessary . An additional adv ant age of this is that the program f or p disc (b eing linear) has a prop er d u al, wh er eas Shersto v’s pro of works with an appro ximate ’du al’ of W ( f , d ) pro ved in a combinato r ial w ay (leading to the square on the lhs and the factor 1 /n on the rhs, see his Th eorem 3.4 in [S08b]). Mo difying h is pro of this w a y yields th e follo wing theorem. Theorem 3 L et P f b e the p attern matrix of a function f . Then disc µ λ ( P f ) ≥ Ω ( p disc λ ( f )) , wher e µ λ is a distribution, in which the inputs y , z to the c ommunic ation pr oblem P f (se e Se ction 2.4) ar e chosen uniformly, and the input bits x ( y ) ar e c hosen such that x ( y ) ⊕ z is distribute d as in λ . The r emaining bits in x ar e uniform. Thanks to th e natural definitions of one-sided smo oth discrep an cy we get an analogous result for one-sided smo oth discrepancy . Shersto v’s te chnique allo ws to trans fer a discrepancy lo wer b oun d for the fun ction g (from the natural definition of one-sided smo oth discrepancy of f ) to a lo wer b oun d on the discrepancy of P g . P g (toget h er with the d istribution µ λ ) can then b e used as a witness for the hardness of P f . Theorem 4 L et P f b e the p attern matrix of a function f . Then sdisc 1 δ ( P f ) ≥ Ω( δ · sp disc 1 2 δ ( f )) . 11 Pro of. When sp disc 1 2 δ ( f ) = K , then there are a f u nction g and a distrib ution λ as in the definition of sp disc 1 . Then disc µ λ ( P g ) ≥ Ω ( p disc λ ( g )) by Theorem 3, and by th e definition of µ λ and the pr op erties of λ, f , g we h a v e µ λ ( P − 1 f (1)) ≥ 2 δ µ λ ( P − 1 g (1)). Then g sdisc 2 δ ( P f ) ≥ Ω ( K ), and consequen tly sdisc δ ( P f ) ≥ Ω ( δ K ) with L emm a 1.  Hence it is enough to analyze the one-sided smo oth discrepancy of f to lo wer b oun d the QMA- comm unication complexit y of P f . 4.4 The One-sided Smo oth Discrepancy of AND The AND fu nction is defined by AN D ( x 1 , . . . , x n ) = x 1 ∧ · · · ∧ x n . P AN D con tains the comm un ication matrix of Disj as a submatrix [S08b]. Lemma 2 log s p disc 1 1 / 3 ( AN D ) = Ω ( √ n ) . log sp disc 1 2 − a ( AN D ) = Ω √ an ) . The pro of of this lemma is analogous to the corresp ondin g part of the pro of of Theorem 1 (and follo ws from r esults in [BCWZ99]). Corollary 2 QM A ( P AN D ) = Ω ( n 1 / 3 ) . 5 AM- vs. PP-comm unication In this section w e describ e a p roblem, for wh ich its PP-comm un ication complexit y is exp onen tially larger than b oth its AM-communicat ion complexit y and its co-AM-co m m un ication complexit y . Then w e can easily show that also th e QMA-communicat ion complexit y of the problem must b e large. T he lo w er b ound for P P-comm unication complexit y also implies that the discrepancy metho d cannot b e applied to get AM-comm u n ication complexit y lo w er b ounds. This essentia lly means that to lo wer b ound AM-co m m un ication complexit y , it is not sufficien t to study the pr op erties (size and err or) of individual r ectangles. Es s en tially we describ e a p r oblem, for which for all distributions on the inpu ts there is an O (log n ) nondeterministic p roto col (i.e., a p oly ( n ) size co ve r of the 1-inputs) with constant error, whereas there is a distrib ution under w hic h eac h rectangle has exp onen tially small discrep ancy , i.e., all rectangles are either exp onentia lly small, or they ha v e error exp onenti ally close to 1/2. 5.1 The Problem In [V95] V ereshchag in d escrib es a similar separation for query complexit y . W e start with h is Bo olean function, whic h is a relaxed ve r sion of the Minsky-P ap ert function [MP88]. Definition 13 L et M b e a matrix in { 0 , 1 } n × m . M is go o d , if eve ry r ow of M c ontains a 1. M i s δ -b ad, if at le ast δ n of its r ows c ontain only zer os. The function AppMP takes such matric es M as inputs, ac c epts go o d matric es, r eje cts δ - b ad ma- tric es, and is undefine d on al l other matric es. We wil l fix m = 4 n 2 and δ = 1 / 2 in this p ap er. So the input size of the pr oblem is N = 4 n 3 . Since the complemen t of the function Ap pMP is not necessarily easy to compute by an AM-query algorithm, V ereshc h agin defines a function f or whic h also its complemen t is easy . 12 Definition 14 The function AppMPC takes p airs of Bo ole an n × n matric es M , M ′ as inputs. If M is go o d, and M ′ is 2/3-b ad, then AppM PC ( M , M ′ ) = 1 , an d if M is 2/3-b ad and M ′ is go o d, then AppMPC ( M , M ′ ) = 0 . In al l other c ases the function i s undefine d. W e can no w state the main result from [V95] in o u r terminolog y . F act 7 1. F or any p olynomial with inte ger c o efficients with de gr e e d ≤ n/ 2 that sign-r epr esents the function AppMP , the thr eshold weight is at le ast 0 . 5 e n/ 15 , i.e., W ( A ppMP , n/ 2) ≥ 0 . 5 e n/ 15 . 2. Ther e is a c onstant ζ such that for any p olynom ial with inte ger c o efficients with de gr e e ζ √ n that sign-r epr esents AppMPC , the thr eshold weight is at le ast 2 ζ n , i.e., W ( AppMPC , ζ √ n ) ≥ 2 ζ n . W e no w define comm unication complexit y v ersions of th ese problems via patt er n matrices. Definition 15 The function P A ppMP is the c ommunic ation pr oblem define d by the p attern matrix of the Bo ole an function AppMP . The function P A ppMPC is the c ommunic ation pr oblem define d by the p attern matrix of the Bo ole an fu nction AppMPC . W e no w state the ob vious fact that AM-comm un ication is small for P AppMPC and P AppMP . Lemma 3 1. AM ( P AppM P ) = O (log n ) . 2. AM ( P A ppMPC ) , AM ( ¬ P AppMPC ) = O (log n ) . Pro of. In an AM-pr otocol for P Ap pMP , the p ublic co in random n umb er i repr esents a random ro w of the matrix M which is the inpu t to the function A ppMP enco d ed in the pattern matrix. Merlin replies w ith a p osition j suc h that M ( i, j ) = 1, if su ch a j exists. Ali ce and Bob can easily v erify with logarithmic comm unication whether M ( i, j ) = 1. Th ey accept if this is the case. Clearly , if the i -th ro w of M does not con tain a 1 they will not accept an y pr o of. On δ -bad matrices this happ ens with probabilit y at least δ . Go o d matrices M on the other h and ha ve their inp uts to their comm unication problem ac cepted with probabilit y 1. The remaining proto cols are along the same lines.  W e no w turn to the lo wer b oun d . As it happ en s, all w e really need to do is to (again) app eal to a result by S hersto v [S08b] (r estated here for con venience). F act 8 L et P f b e the p attern matrix of a function f , and d a p ositive inte ger. Then disc ( P f ) 2 ≥ min  W ( f , d − 1) 2 d n , 2 d  , wher e W ( f , d ) denot e s the thr eshold weight of f with de gr e e d , i.e., the minimum thr eshold weight of a p olyno mial with inte ger c o efficients and de gr e e d , that sign- r epr esents f . See section 2.2 for an explanation of threshold weigh t. P u tting these results together w e find that for d = n/ 100 the f u nction A ppMP has threshold w eigh t at least 2 Ω( n ) , and then the pattern matrix has d iscrepancy at least 2 Ω( n ) . T his readily implies that P P ( P A ppMP ) = Ω( n ) with F act 4. Similarly , c ho osing d = ζ √ n w e ge t P P ( P A ppMPC ) ≥ Ω( √ n ). Theorem 5 1. P P ( P AppMP ) = Ω( N 1 / 3 ) . 2. P P ( P AppMPC ) = Ω( N 1 / 4 ) . 3. AM ( P A ppMP ) = O (log N ) . 4. AM ( P A ppMPC ) , AM ( ¬ P AppMPC ) = O (log N ) . 13 5.2 QMA vs. AM In this subsection w e note the foll owing consequence of the lo w er b ound in th e previous subsection, whic h follo ws from the fact that P P-proto cols can sim ulate Q MA-pr otocols within a qu adratic increase in communication (this can b e p ro ved by fi rst b o osting with F act 1, then remo ving the pro of, wh ic h lea v es a quantum proto col with a large enough gap. This can b e turned int o a wea kly unb ou n ded error quantum proto col, and suc h proto cols are exactly as p o we r ful as classical w eakly unb ounded error protocols [K07]). Corollary 3 1. QM A ( P AppM P ) = Ω( N 1 / 6 ) . 2. AM ( P A ppMP ) = O lo g N ) . 6 Rounds in M A-comm unication F or man y ’realistic’ mo des of communicatio n complexit y there are p r oblems that require the pla y ers Alice and Bob to in teract by using many round s of comm u nication in order to ac h iev e go o d proto- cols. Th is is not altogether su rprising, since one would exp ect conv ersations with many rou n ds of in teraction to b e more p ow erful than monologues. Examp les of this ph enomenon are deterministic, randomized (see [NW93]), and quantum comm u nication complexit y (see [KNTZ 07, J R S 02]). Ho w ever, in the n ondeterministic mo de of communicati on , monologues are in fact optimal: the pro ver can p ro vide the w hole con ve r sation to the pla ye r s, who now just need to verify that their role in th e co nv ersation is represen ted correctly . In this section we sho w that there is a p artial fu nction, for wh ic h ev ery one-w a y MA-protocol with comm u nication going from Alice to Bob is exp onen tially more exp en siv e than a randomized one-w a y pr otocol with communicatio n going from Bob to Alice. Note that suc h p roblems trivially do n ot exist when w e replace MA- w ith n ondeterministic, or AM-comm unication complexit y . Raz and Shpilk a [RS04] prov e that one-w ay comm u nication (in any d irection) is also optimal (within a p olynomial increase in comm unication) for QMA protocols 2 . Round s in MA- communicatio n do not seem to ha ve b een considered b efore, although Aaronson [A06] considers a w eake r v ariant of one-wa y MA-proto cols: Merlin sends the pro of to Bob only , so that Alice has to send her message w ithout ha ving seen the p ro of. This mo del is m u c h wea ker than the standard one-w ay MA-communicat ion mo del, in fact at most quadratically m ore efficien t than randomized one-wa y communication. Let us d efine the fu nction. A usual susp ect for this k in d of s ep aration is the Ind ex f u nction Ix , for which Alice r eceiv es a string x ∈ { 0 , 1 } n , and Bob a n index i ∈ { 1 , . . . , n } and the goal is to compute x i (see [KNR99]). But due to Bob’s input b eing short, the nond eterministic complexit y of this problem is small, and h ence also th e one-wa y MA-comm unication: Th e pro v er can simply pro vid e Alice with Bob’s inp u t i . T h e problem w e use instead gives Bob man y ind ices, and w e are trying to determine whether for man y of th em x i = 1. Definition 16 The func tion MajIx ( x, I ) , wher e I = { i 1 , . . . , i √ n } , e ach i j ∈ { 1 , . . . , n } , and x ∈ { 0 , 1 } n is define d as fol lows: 1. i f |{ j : x i j = 1 }| = √ n then MajIx ( x, I ) = 1 , 2. i f |{ j : x i j = 1 }| ≤ 0 . 9 √ n then MajIx ( x, I ) = 0 , 3. otherwise MajIx ( x, I ) is undefine d. 2 They prov e that every problem with QMA- comm u nication complexity c can b e reduced to a p roblem called LS D of size 2 poly log c , for which a logarithmic QMA one-wa y proto col exists ( they call this a tw o round p rotocol). 14 It is easy to s ee that R B → A ( MajIx ) = O (log n ), b ecause Bob can just pic k 100 indices i j from I randomly , and send them to Alice, wh o accepts if and only if all x i j = 1. In AM-proto cols a single round of comm u nication from Alic e to Bob is alw ays optimal (and find ing such a proto col for Ma jIx is an easy exercise) . In tu itively , Merlin’s pr ob lem with MajIx is that he cannot pro vid e information ab out man y ind ices in I , unless his pro of is ve r y long. Ho we ver, it is n ot clear at all that this is necessary , and ind eed, the same in tuition would app ly to the quantum case, in whic h there is a one-w a y p roto col with p oly log( n ) comm unication and p ro of length for the p roblem. W e pr o v e the follo wing lo w er b ound, whic h is close to optimal. Theorem 6 M A A → B ( MajIx ) ≥ Ω( √ n ) . Pro of. Before starting let u s d efine the notion of a one-wa y r ectangle, w hic h is a basic ob ject when considering randomized one-wa y communication co mp lexit y . Definition 17 F or a (p artia l) function f : X × Y → { 0 , 1 } a one-w a y rectangle i s a subset R ⊆ X , c ouple d with a fu nction d R : Y → { 0 , 1 , − 1 } . The o ne - way r e ctangle acce p ts al l inputs ( x, y ) ∈ R × Y with d R ( y ) = 1 , rejects al l inputs ( x, y ) ∈ R × Y with d R ( y ) = 0 , and is unde cide d ab out the r emaining inputs in R × Y . The error of a one- way r e ctangle under some distribution is define d in the obvious way. The size of small error one-w a y rectangles under distributions on the inp u ts characte r izes the one-w a y r an d omized communicati on complexity [K04 ],[JK N08]. No w let us b egin with the p ro of. W e are giv en an M A A → B proto col, wh ic h h as pro of length at most a , and comm u nication at most c , and error 1/3. W e assume that a, c ≤ γ √ n for some s mall constan t γ . As usu al w e b o ost the s uccess probabilit y by rep eating the comm unication among Alice and Bob 100 √ n times, s o that the (soundn ess and completeness) error drops to ǫ = 2 − 10 √ n . Note that the p r o of d o es not need to b e rep eated, so after this step the pro of length is still a , a n d the comm unication is c ′ ≤ δ n for some small constan t δ . 3 Our argumen t ca n n o w b e sum marized as follo ws: First we consid er a d istribution µ on 1-inpu ts. W e find and fix a p ro of for wh ic h many 1-inputs are accepted with high probabilit y (w e will iden tify pro ofs w ith the sets of 1-inpu ts th at are accepted with high probabilit y when using th ose pro ofs). After fixing the pro of we are left with a ran d omized one-wa y p roto col, that accepts all 1-inputs in the pro of with probability 1 − ǫ , but accepts 0-inputs with probability at m ost ǫ eac h. No w we d efine a distrib ution σ on 0- in puts. Finally , w e sho w that under the distrib ution, in whic h 1- in puts are c hosen according to µ and 0-inpu ts according to σ an y large one-w ay rectangle must hav e large error. This sho ws that c ′ = Ω( n ), and hen ce a + c = Ω( √ n ). So let us b egin with the distribu tion µ on 1-inputs. F or th is we emplo y a go o d error-correcting co de to generate x ∈ { 0 , 1 } n . It do es not matter wh ether the co de is constructible or not, so a randomized co n struction suffices. A simp le modifi cation of the Gilb ert-V arshamov boun d give s u s a co de C ⊆ { 0 , 1 } n that has d istance n/ 4 and 2 0 . 87 n co dew ord s, eac h of whic h has Hamming weigh t exactly n/ 2. F or th e distr ibution µ w e firs t uniformly c ho ose an elemen t x ∈ C from the cod e. Denote by I the set of all sets I of √ n different indices from { 1 , . . . , n } . W e con tin u e by choosing an ind ex s et I = { i 1 , . . . , i √ n } ∈ I , u n der the cond ition that all of the i j ∈ I satisfy x i j = 1. This fi nishes the description o f µ . 3 In the quantum case this app ears to b e imp ossible, b ecause the Marriott-W atrous b oosting tec hniqu e does n ot w ork for one-wa y proto cols. Hence here is a p oint where a QM A A → B lo wer b oun d along these lines wo u ld fail. 15 In our MA-proto col there are at most 2 a differen t pr o ofs p . W e id entify eac h p ro of p with the set of 1-inpu ts ( x, I ), for wh ic h Alice and Bob accept ( x, I ) with p robabilit y at least 1 − ǫ when giv en pro of p . Since completeness is 1 − ǫ ev ery 1-input is in at least one pro of. No 0-input is in any pro of. No w w e simply fix the large st p r o of p under the distribu tion µ . Then µ ( p ) ≥ 2 − a . Ha ving fixed our pr o of p , we are left with a randomized one-wa y proto col that accepts at least the 1-inputs in p with probabilit y 1 − ǫ , and ac cepts no 0-input with probabilit y larger than ǫ . In order to s ho w that this pr otocol needs communicat ion Ω( n ) we create a hard distribu tion on all inp uts, by mixing µ with a distribu tion on 0-inpu ts: this distribution σ is simply un iform on all 0-inputs. Note that there are more 0-inp uts than 1-inpu ts to the function MajIx , du e to the pr omise defi- nition. Ho w ever, w hen w e denote the to tal n u m b er of 1-inpu ts ( x, I ) with x ∈ C by K and the to tal n u m b er of 0-inputs ( x, I ) with x ∈ C b y L , then a simp le calculat ion rev eals th at L/K ≤ 3 √ n . W e can co n clude that for eac h 0-input ( x, I ) and eac h 1-input ( x ′ , I ′ ) ∈ p w e ha ve µ ( x ′ , I ′ ) ≤ 3 √ n · σ ( x, I ) . (1) Let ν b e the distribu tion on all inpu ts, wh ic h resu lts from mixing µ and σ with probabilit y 1/2 eac h. W e ma y n o w fix the remaining rand omness in our proto col and get a deterministic one-wa y proto col, that has comm unication c ′ , and under ν accepts a set p ′ ⊆ p of 1-inputs with ν ( p ′ ) ≥ 2 − a · (1 − ǫ ) / 2, but acce p ts a set q of 0-inputs with ν ( q ) ≤ ǫ/ 2. Note that ǫ ≪ 2 − a (1 − ǫ ). T o simp lify our argumen t w e w ill r emo ve the 1 -in p uts that ha ve limited co ntribution to the size of p ′ . A string x ∈ C is slim if ν ( { ( x, I ) : ( x, I ) ∈ p ′ } ) ν ( { x } × I ) ≤ 2 − 2 a . Let p ′′ denote the set of inputs ( x, I ) ∈ p ′ suc h that x is n ot slim. Then ν ( p ′′ ) ≥ 2 − a · (1 − ǫ ) / 2 − 2 2 a ≥ 2 − a − 2 . Our goal is to sho w that u nder the distribution ν ev ery one-w a y proto col with the ab o ve prop erties m us t ha ve comm un ication Ω( n ). A one-w a y proto col partitions the inp uts in to one-w ay rectangles. Let us consider a one-wa y rectangle R, d R , su ch that there are x 6 = y with x, y ∈ C ∩ R , and b oth x, y are not slim. Recall that x and y ha ve Hamming distance n/ 4. Lemma 4 If x, y ar e b oth in the same one-wa y r e ctangle, then the err or (r estricte d to r ows that ar e not slim) is ν (( R × d − 1 R (1)) ∩ MajIx − 1 (0)) ν ( R × I ) ≥ 2 − 2 √ n . Since the p r otocol accepts only a set of 0-inpu ts with size at m ost ǫ/ 2 und er ν , and accepts all inputs in p ′′ , at most half of all inputs in p ′′ (under ν ) can b e in one- wa y rectangles that con tain at least 2 differen t co dewo r d s as ro w s, else the error exceeds 2 − a − 3 · 2 − 2 √ n ≫ ǫ/ 2. This means that there m ust b e 2 Ω( n ) one-w a y rectangles in the proto col, and hence the co m m u- nication c ′ ≥ Ω( n ), whic h in turn implies c + a ≥ Ω( √ n ), finishing our pro of.  Pro of of Lemma 4. Consider a one-wa y rectangle R, d R . R con tains at least t wo d ifferen t co dew ord s x ∈ C and y ∈ C (that are b oth not slim). W e are in terested in the restrictions that this places on d R . W e identify x, y with s ubsets of { 0 , 1 } n . Then x ∩ y ≤ n (1 / 2 − 1 / 8) b ecause x and y ha ve Hamming d istance at least n / 4. Let S ⊆ I b e the set of I suc h that ( x, I ) ∈ p ′′ . T hen for all I ∈ S w e ha v e that all of the i ∈ I m u st satisfy x i = 1. 16 First let T ⊆ S b e the set of all I ∈ S , s uc h that | I ∩ x ∩ y | ≥ 0 . 8 √ n . Then | T | ≤ √ n ·  3 n 8 0 . 8 √ n  ·  n 8 0 . 2 √ n  ≤ 2 − α √ n ·  n 2 √ n  , for some constan t α > 0. Note that the bin omial co efficient on the right hand side is j ust the num b er of I ∈ I suc h that ( x, I ) is a 1-inpu t. He n ce ν (( { x } × T ) ∩ p ′′ ) P I ∈I : M ajIx ( x,I )=1 ν ( x, I ) ≤ 2 − α √ n . Since w e can limit a suc h that 2 − 2 a ≥ 2 · 2 − α √ n the set T contributes little to the set of I ∈ I with ( x, I ) ∈ p ′′ : ν ( { ( x, I ) : ( x, I ) ∈ p ′′ and I 6∈ T } ) ≥ 2 − 2 a − 1 · X I ∈I : M ajIx ( x,I )=1 ν ( x, I ) ≥ 2 − 2 a − 2 · ν ( { x } × I ) . So let us examine the set of all I ∈ S suc h that | I ∩ x ∩ y | < 0 . 8 √ n . F or all such I we hav e either | x ∩ I | ≤ . 9 √ n or | y ∩ I | ≤ . 9 √ n , i.e., either ( x, I ) or ( y , I ) is a 0-input. Since we h av e assumed that ( x, I ) ∈ p ′′ , we get that ( y , I ) is a 0-inpu t. Now 0-inputs h a ve a smaller probabilit y eac h than 1-inputs, but on the other hands the allo w ed err or ǫ is v ery small. W e can no w calc u late the error of the one-w a y r ectangle in the ro ws x and y . ν ( { ( y , I ) : Ma jIx ( y , I ) = 0 and d R ( I ) = 1 } ) ≥ ν ( { ( x, I ) : I ∈ S − T } ) 3 √ n ≥ 2 − 2 a − 2 / 3 √ n · ν ( { x } × I } ) . W e can pla y this game also with x and y exc hanged and h ence the tw o r o ws x, y together ha ve substanti al error. W e can con tin ue with more pairs of ro ws x, y of the one-w a y rectangle, until only one (o r no) ro w is left. Hence the o verall error is at least 2 − 2 √ n .  The pro of tec hn ique used ab ov e can also b e used to give a simple p ro of of a go o d low er b ound for the r an d omized one-w ay comm u nication complexit y of the Ind ex f u nction Ix . 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