BQP and the Polynomial Hierarchy

BQP and the P olynomia l Hierarc h y Scott Aaronson ∗ Abstract The relationship b etw een BQP and PH has been an open pr o blem since the earliest days of quantum co mputing. W e pr esent evidence that q uantum computers can solve problems outside the entire po lynomial hierar ch y , b y relating this ques tion to topics in cir cuit complexity , pseudorandomnes s, a nd F ourier a na lysis. First, we show that there exis ts a n oracle relatio n pro blem (i.e., a problem with many v alid outputs) that is solv able in BQP , but not in PH . This also yields a non-orac le rela tion problem that is s olv able in quantum lo garithmic time, but not in AC 0 . Second, we show that a n ora cle de cision pro blem separ ating BQP from PH would follow from the Gener alize d Linial-Nisan Conje ctur e , which we for mulate here a nd which is likely of independent interest. The origina l Linial-Nisa n Conjecture (abo ut pseudora ndomness a gainst constant-depth circ uits) was re c ently prov ed by B rav er man, after b eing op en for tw ent y years. Con ten ts 1 In t ro duction 2 1.1 Motiv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Our Resu lts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 In Defense of Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Preliminaries 8 2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Complexit y Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Quan tum Algorithms 11 3.1 Quant um Algorithm f or F ourier Fishing . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Quant um Algorithm f or F ourier Checking . . . . . . . . . . . . . . . . . . . . . . 13 4 The Classical Complexit y of F ourier Fishing 16 4.1 Constan t-Depth Circu it Lo wer Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Secretly Biased F ourier Co eﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Putting It All T ogether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5 The Classical Complexit y of F ourier Chec king 23 5.1 Almost k -Wise Indep endence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 Oracle S eparation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ∗ MIT. Email: aaronson@csail.mit.e du. Supp orted by an NSF CAREER Awa rd, a DARP A YF A grant, MIT CSAIL, and the Keck F ound ation. 1 6 The Generalized Linial-Nisan Conjecture 29 6.1 Lo w-F at P olynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7 Discussion 34 8 Op en Problems 36 9 Ac kno wledgmen t s 37 1 In tro duction A central task of qu antum computing theory is to un derstand how BQP —meaning Bounded-Error Quant um Pol ynomial-Time, th e class of all problems feasible for a quan tum compu ter—ﬁts in with classical complexit y classes. In their original 1993 pap er deﬁnin g B QP , Bernstein and V azirani [11] sh o wed that BPP ⊆ BQP ⊆ P # P . 1 Informally , this sa ys that quan tum compu ters are at least as fast as classical probab ilistic computers and no more than exp onen tially faster (ind eed, they can b e sim ulated using an oracle for counting). B ernstein and V azirani also ga v e evidence that BPP 6 = BQP , b y exhib iting an oracle p roblem called R ecursive Fourier Sampling th at requires n Ω(log n ) queries on a classical computer bu t only n qu eries on a quantum compu ter. 2 The evidence for the p o wer of qu an tu m computers b ecame dramatically stronger a yea r later, when Shor [33] (b uilding on w ork of Simon [34]) show ed that F actoring and Discrete Logar ithm are in BQP . On the other hand, Bennett et al. [9] ga ve oracle evidence that NP 6⊂ BQP , and while no one r egards su c h eviden ce as decisive , to d a y it seems extremely unlikely that quan tum compu ters can solve NP -complete problems in p olynomial time. A v ast b o dy of researc h, conti n uing to the present , has sought to map out the detailed b oun dary b et ween th ose NP problems that are feasible for quantum compu ters and those that are n ot. Ho wev er, there is a complemen tary question that—despite b eing un iv ers ally r ecognized as one of the “grand c hallenges” of the ﬁeld—h as had essentia lly zero pr ogress o ver the last sixteen y ears: Is BQP in NP ? Mor e gener al ly, is BQP c ontaine d anywher e in the p olynomial hier ar- chy PH = NP ∪ NP NP ∪ NP NP NP ∪ · · · ? The “default” conjecture is presumab ly B QP 6⊂ PH , since no one kno ws wh at a sim ulation of BQP in PH would lo ok lik e. Before this wo r k, ho wev er, there was no formal evidence for or against that conjecture. Almost all th e problems for wh ic h we h av e quantum algorithms— including F actoring and Discrete Logarithm —are easily seen to b e in NP ∩ coNP . 3 One notable exception is Recur sive F ourier Sampling , the p r oblem that Bernstein and V azirani [11] originally u sed to construct an oracle A relativ e to wh ic h BPP A 6 = BQP A . On e can show, without to o m u c h diﬃculty , that Recursive Fourier S ampling yiel ds oracles A relativ e to whic h BQP A 6⊂ NP A and ind eed BQP A 6⊂ M A A . Ho wev er, wh ile it is reasonable to conjecture that Recursive Fourier Sampling (as an oracle pr oblem) is n ot in PH , it is op en ev en to sho w that this problem (or any other BQP oracle prob lem) is not in AM ! Recall that AM = NP under 1 The up p er b ound was later impro ved to BQP ⊆ PP by Adleman, DeMarrais, an d Huang [3]. 2 F or more ab out Recursive Fourier Sampling see Aaronson [2]. 3 Here we exclude BQP -complete problems such as approximati ng the Jones p olynomial [5], which, by the very fact of b eing BQP -complete, seem hard to interpret as “evidence” for BQP 6⊂ PH . 2 plausible d erandomization assumptions [26 ]. Thus, until we solv e th e problem of constructing an oracle A su c h that BQP A 6⊂ AM A , w e cannot ev en claim to ha ve oracle evidence (wh ich is itself, of course, a weak form of evidence) that BQP 6⊂ NP . Before going fur ther, we sh ould clarify that there are t w o questions h ere: wh ether BQP ⊆ PH and whether PromiseBQP ⊆ PromisePH . In the unr elativized w orld , it is entirely p ossible that quan tum computers can solv e p romise p roblems outside th e p olynomial hierarc hy , b ut that all languages in BQ P are nev ertheless in PH . Ho wev er, for the sp eciﬁc p u rp ose of constru cting an oracle A suc h that B QP A 6⊂ PH A , th e tw o questions are equiv alen t, b asically b ecause one can alw ays “oﬄoad” a p romise in to the constru ction of the oracle A . 4 1.1 Motiv ation There are at least four reasons why the BQP versus PH question is so interesting. A t a basic lev el, it is b oth theoretically and pr actically imp ortan t to und erstand wh at classical r esources are needed to simulate qu an tu m physics. F or example, when a quant um system evo lv es to a giv en state, is there alw ays a short classical pr o of that it do es so? Can one estimate quan tum amplitudes using appro ximate counting (which w ou ld imply B Q P ⊆ BPP NP )? If s omethin g like this w ere tr ue, then w hile the exp onentia l sp eedup of S h or’s f actoring algorithm migh t s tand, quant um computing w ould nev ertheless seem muc h less diﬀerent fr om classical compu ting than pr eviously though t. Second, if BQP 6⊂ PH , then man y p ossibilities for new quan tum algorithms migh t op en up to us. O n e often hears the complaint that there are to o few quantum algorithms, or that progress on quan tum algorithms h as slo w ed since th e mid-1990s. In our opinion, the real issue here h as nothing to do with quan tum computing, and is simply that th ere are to o few natural NP -inte rmediate problems for whic h there plausib ly c ould b e quantum algorithms! In other words, instead of fo cussing on Graph Isomo rphism and a small num b er of other NP -in termediate problems, it migh t b e fruitful to lo ok for quant um algorithms solving completely diﬀerent types of problems— problems that are not necessarily ev en in PH . I n this pap er, w e will see a new example of such a quan tum algorithm, wh ich solv es a pr ob lem called F o urier Checking . Third, it is natural to ask whether the P ? = BQP qu estion is related to that other f undamental question of complexit y theory , P ? = NP . More concretely , is it p ossible that qu an tu m computers could provide exp onent ial sp eedups even if P = NP ? If BQP ⊆ PH , then certainly the answer to that question is no (sin ce P = NP = ⇒ P = PH ). Th erefore, if w e w an t evidence that quant um computing could s u rvive a collapse of P and NP , we m ust also seek evidence that BQP 6⊂ PH . F our th , a ma j or c hallenge for quantum compu ting researc h is to get b e tter evidenc e that quantum c omputers c annot solve NP - c omplete pr oblems in p olynomia l time . As an example, could w e sho w that if NP ⊆ BQP , then the p olynomial hierarc hy collapses? A t ﬁ rst glance, this seems like a w ild hop e; certainly w e ha v e no idea at present how to pro v e an ything of the kind. Ho w ever, notice 4 Here is a simple pro of: let Π = (Π YES , Π NO ) b e a promise problem in PromiseBQP A \ PromisePH A , for some oracle A . Then clearly , every PromisePH A mac hine M fa ils to solve Π on inﬁnitely many inp uts x in Π YES ∪ Π NO . This means that we can pro duce an inﬁnite sequence of in p uts x 1 , x 2 , . . . in Π YES ∪ Π NO , whose lengths n 1 , n 2 , . . . are spaced arbitrarily far apart, such that every PromisePH A mac hine M fails to solv e Π on at least one x i . N ow let B b e an oracle that is identical to A , ex cept that for each inp ut length n , it reveals (i) whether n = n i for some i and ( ii) if so, what th e corresp onding x i is. Also, let L b e th e unary language that contains 0 n if and only if (i) n = n i for some i and ( ii) x i ∈ Π YES . Then L is in BQP B but not PH B . 3 that if BQP ⊆ AM , then the desired implication w ould follo w immediately! F or in that case, NP ⊆ B QP = ⇒ coNP ⊆ BQP = ⇒ coNP ⊆ AM = ⇒ PH = Σ P 2 where the last implication wa s sho wn by Boppana, H ˚ astad, and Zac hos [12]. S imilar remarks apply to the questions of whether NP ⊆ BQP wo uld imply PH ⊆ B QP , and whether the folklore result NP BPP ⊆ BPP NP has the qu an tum analogue NP BQP ⊆ BQP NP . In eac h of these cases, we ﬁ nd that understand ing some other issue in quan tum complexit y theory requires ﬁrst coming to grips with whether BQP is con tained in some lev el of the p olynomial hierarc h y . 1.2 Our R esults This pap er presents the ﬁ rst formal evidence for the p ossibilit y th at BQP 6⊂ PH . P erhaps more imp ortantl y , it p laces th e r elativized BQP v ersu s PH question at the fr on tier of (classical) circuit lo wer b ound s. The heart of the pr oblem, w e will ﬁnd, is to extend Bra v erman ’s sp ectacular recent pro of [13] of the L inial-Nisan Conjecture, in w ays that w ould reve al a great d eal of information ab out small-depth circuits indep endent of the implications for quantum compu ting. W e hav e tw o main contributions. First, w e ac hiev e an oracle separation b et ween BQP and PH for the case of r elation pr oblems . A relation problem is simply a p r oblem wh ere the desir ed output is an n -bit string (rather than a single b it), and any string from some non emp t y set S is acceptable. Relation problems arise often in th eoretical compu ter science; one well- kno wn example is ﬁndin g a Nash equilibrium (shown to b e PP AD -complete by Dask alakis et al. [15]). Within quantum computing, there is considerable p receden t f or studying relation problems as a wa r m up to the harder case of decision problems. F or example, in 2004 Bar-Y ossef, Ja yram, and K er en idis [6] ga ve a relation pr ob lem with qu an tu m on e-wa y comm un ication complexit y O (log n ) and randomized one-w ay communicatio n complexity Ω ( √ n ). It took s everal m ore y ears f or Ga vinsky et al. [20] to ac hiev e the same separation for decision problems, and the pro of wa s m uc h more complicate d. The same phen omenon has arisen m an y times in qu an tu m comm u nication complexit y [17, 18, 19, 21, 22], though to our kno wledge, this is the ﬁrst time it has arisen in quant um query complexit y . F ormally , our r esult is as f ollo ws: Theorem 1 Ther e exists an or acle A r elative to which FBQP A 6⊂ FBPP PH A , wher e FBQP and FBPP ar e the r elation v ersions of BQP and BPP r esp e ctively. 5 Underlying Th eorem 1 is a new lo wer b ound against AC 0 circuits (constan t-depth circuits com- p osed of AND, OR, an d NOT gates). The close connection b etw een AC 0 and the p olynomial hierarc hy that we exp loit is n ot new. In the early 198 0s, F ur s t-Saxe-Sipser [16] and Y ao [39] no- ticed that, if w e ha v e a PH mac hine M that computes (sa y) the P ar ity of a 2 n -bit oracle string, then b y simply rein terpreting th e existen tial qu an tiﬁers of M as OR gates and the universal quan- tiﬁers as AND gates, w e obtain an AC 0 circuit of size 2 poly ( n ) solving the same problem. It follo ws that, if we can p ro v e a 2 ω (p olylog n ) lo wer b ound on the size of A C 0 circuits computing P arity , w e 5 Confusingly , the F stands for “function”; we are simply follow ing the stand ard naming con ven tion for classes of relation problems ( FP , FNP , etc). 4 can construct an oracle A relativ e to whic h ⊕ P A 6⊂ PH A . The idea is the s ame for constructing an A relativ e to w hic h C A 6⊂ PH A , wher e C is an y complexit y class. Indeed, the relation b et w een PH and A C 0 is so direct that w e get the follo wing as a more-or-less immediate coun terpart to Theorem 1: Theorem 2 In the unr elativize d world (with no or acle), ther e exists a r elation pr oblem solvable in quantum lo garithmic time but not i n nonuniform AC 0 . The relation pr ob lem that w e us e to separate B Q P fr om PH , an d BQLOGT IM E from A C 0 , is called F ou rier Fishing . The pr oblem can b e informally stated as follo ws. W e are giv en oracle access to n Bo olean fu nctions f 1 , . . . , f n : { 0 , 1 } n → {− 1 , 1 } , whic h we think of as c h osen uniformly at r andom. The task is to output n strings, z 1 , . . . , z n ∈ { 0 , 1 } n , su c h that the corresp ond ing squared F ourier co eﬃcien ts b f 1 ( z 1 ) 2 , . . . , b f n ( z n ) 2 are “often m u c h larger than a verage .” Notice that if f i is a random Bo olean f u nction, th en eac h of its F ourier coeﬃcients b f i ( z ) follo ws a n or- mal distribu tion—meanin g that with o v erwh elming prob ab ility , a constant f raction of the F ourier co eﬃcien ts will b e a constant factor larger than th e mean. F urthermore, it is s traigh tforward to create a qu an tum algorithm that samples eac h z w ith probabilit y prop ortional to b f i ( z ) 2 , so that larger F ourier co eﬃcients are more likel y to b e s ampled than smaller ones. On the other hand, computing any sp e ciﬁc b f i ( z ) is easily seen to b e equiv alen t to su mming 2 n bits. By we ll-kno wn low er b ounds on the size of AC 0 circuits computing the Majority f unction (see H ˚ astad [36] f or example), it follo ws that, for any ﬁxed z , computing b f i ( z ) cannot b e in PH as an oracle problem. Unfortunately , th is do es not directly imply any separation b et w een BQP and PH , s in ce the quant um algorithm do es not compute b f i ( z ) either: it ju st samples a z with probabilit y prop ortional to b f i ( z ) 2 . Ho w eve r, we will sho w that, if there exists a BPP PH mac hine M that eve n appr oximately sim ulates the b eha vior of the quant um algorithm, then one can solv e Majority by m eans of a nondeterministic reduction—whic h us es appro ximate coun ting to estimate Pr [ M outpu ts z ], and adds a constan t n um b er of la yers to the AC 0 circuit. T h e central diﬃculty is that, if M knew the sp eciﬁc z for whic h we w ere in terested in estimating b f i ( z ), then it could c ho ose adversarially never to outp u t that z . T o solve this, w e w ill sho w that we can “smuggl e” a Majority instance into th e estimation of a r andom F ourier coeﬃcient b f i ( z ), in suc h a w a y that it is inform ation-theoretically imp ossible for M to d etermine which z we care ab out. Our second con trib ution is to deﬁne and stud y a new blac k-b o x decision problem, called F o urier Check ing . Informally , in this p roblem we are giv en oracle access to two Bo olean functions f , g : { 0 , 1 } n → {− 1 , 1 } , and are promised that either (i) f and g are b oth u niformly random, or (ii) f is un iformly random, while g is extremely well correlated with f ’s F ourier transform o v er Z n 2 (whic h we call “forr elated”). The problem is to decide wh ether (i) or (ii) is the case. It is n ot hard to show that F ourier Checking is in BQP : basically , on e can prepare a u niform sup erp osition ov er all x ∈ { 0 , 1 } n , then query f , apply a quant um F our ier transform, qu ery g , and c heck whether one has reco vered s omething close to the u niform sup erp osition. On the other hand, b eing forr elated seems like an extremely “global” p rop erty of f and g : one that would not b e apparent f r om querying any small n um b er of f ( x ) and g ( y ) v alues, regardless of the outcomes 5 of those qu eries. And th u s, one migh t conjecture that Fourier Che cking (as an oracle problem) is not in PH . In this pap er, we addu ce str on g evidence for that conjecture. Sp eciﬁcally , w e sho w that for ev ery k ≤ 2 n/ 4 , the forrelated distribution o ver h f , g i pairs is O  k 2 / 2 n/ 2  -almost k -wise indep e ndent . By this we mean that, if one h ad 1 / 2 prior probability th at f and g w ere un iformly rand om, an d 1 / 2 prior probability that f and g w ere forrelated, then ev en conditioned on an y k v alues of f and g , the p osterior probabilit y that f and g were forrelated wo uld still b e 1 2 ± O  k 2 2 n/ 2  . W e conjecture th at this almost k -wise in dep en dence prop erty is enough, by itself, to imply that an oracle problem is not in PH . W e call this the Gener alize d Li ni al-N isan Conje ctur e . Without the ± O  k 2 / 2 n/ 2  error term, our conjecture would b e equiv alen t 6 to a famous con- jecture in circuit complexit y made by Linial an d Nisan [28] in 1990. Their conjecture stated that p olylo garithmic indep endenc e fo ols A C 0 : in other words, eve ry probabilit y distribution o ver N -bit strings that is unif orm on ev er y small sub set of bits, is indistingu ish able from the truly u niform distribution b y AC 0 circuits. When we b egan in v estigating th is topic a y ear ago, ev en the original Linial-Nisan Conjecture w as still op en. Sin ce then, Bra verman [13] (bu ilding on earlier w ork b y Bazzi [7] and Razb oro v [30]) has given a b eautiful p ro of of that conjecture. In other words, to construct an oracle r elativ e to whic h BQP 6⊂ PH , it now s u ﬃces to generalize Bra v erman’s Theorem from k -wise indep enden t distr ibutions to almost k -wise indep en d en t ones. W e b eliev e that this is b y far the m ost promisin g approac h to the BQ P v ersu s PH p roblem. Alas, generalizing Bra verman’s pro of is muc h harder than one migh t ha ve hop ed. T o pro ve th e original L in ial-Nisan Conjecture, Bra verman sh o wed that eve ry A C 0 function f : { 0 , 1 } n → { 0 , 1 } can b e w ell-approximat ed, in the L 1 -norm, by low-de gr e e sandwiching p olynomials : real p olynomials p ℓ , p u : R n → R , of degree O ( p olylog n ), su c h th at p ℓ ( x ) ≤ f ( x ) ≤ p u ( x ) for all x ∈ { 0 , 1 } n . Since p ℓ and p u trivially ha v e the same exp ectation on an y k -wise ind ep end en t distrib ution that they ha ve on the uniform distribu tion, one can sho w that f m ust ha ve almost the same exp ectation as w ell. T o generalize Brav erman’s result from k -wise indep endence to almost k -w ise ind ep end ence, w e will sho w that it suﬃces to construct lo w-degree sandwic h p olynomials that satisfy a certain additional condition. This n ew condition (whic h w e call “lo w-fat”) basically sa ys that p ℓ and p u m u st b e r epresent able as linear com b in ations of terms (that is, pro ducts of x i ’s and (1 − x i )’s), in suc h a w ay that the sum of the absolute v alues of the co eﬃcien ts is b ound ed—thereby prev en ting “massiv e cancellations” b et ween p ositive and negativ e terms. Un fortunately , while we kn o w t wo tec hniqu es for appro ximating A C 0 functions by lo w-degree p olynomials—that of Linial-Mansour- Nisan [27] and th at of Razb oro v [29] and S molensky [35]—neither tec hnique pro v id es anyt hing like the control o ve r co eﬃcien ts that we need. T o construct low-fa t sandwic hing p olynomials, it seems necessary to r epro ve the LMN and Razb oro v-S molensky theorems in a more “conserv ativ e,” less “proﬂigate” w a y . And such an adv ance seems lik ely to lead to breakthroughs in circuit complexit y and computational learning theory having nothing to do with quantum compu ting. Let us m en tion t wo further applications of F ourier Chec king : (1) I f the Generalized L in ial-Nisan C onjecture holds, then just like with F o urier Fishing , w e can “scale do wn by an exp onentia l,” to obtain a promise prob lem that is in B QLOGTIME but not in AC 0 . 6 Up to unimp ortant v ariations in the parameters 6 (2) Without an y assumptions, we can p ro ve the new results that there exist oracles relativ e to whic h BQP 6⊂ BPP path and BQP 6⊂ S ZK . W e can also r epro ve all previous oracle separations b et w een BQP and classical complexit y classes in a un iﬁed fashion. T o summarize our conclusions: Theorem 3 Assuming the Gener alize d Linial-N isan Conje ctur e, ther e exists an or acle A r elative to which BQP A 6⊂ PH A , and ther e also exists a pr omise pr oblem in BQLOGTIME \ AC 0 . Unc ondi- tional ly, ther e exists an or acle A r elative to which BQP A 6⊂ BPP A path and BQP A 6⊂ SZK A . As a candidate p roblem, Fourier Checking has at least ﬁve adv an tages o ver the Re cursive F o urier Sampl ing problem of Bernstein and V azirani [11]. Firs t, it is m uc h simp ler to deﬁn e and reason ab out. Second, F our ier Checking has the almost k -wise indep end ence prop ert y , which is not shared by Re cursive F ourier Sampl ing , and whic h immed iately connects the former to general questions ab out pseu d orandomness against constan t-depth circuits. Third, F ou rier Checking can yield exp onential separations b et ween qu an tu m and classical mo dels, rather than just qu asip olynomial ones. F ourth, one can h op e to u se Fourier Checking to giv e an oracle relativ e to whic h BQP is not in PH [ n c ] (or PH w ith n c alternations) for any ﬁxed c ; by contrast, Recursive F ourier Sampl ing is in PH [lo g n ]. Fin ally , it is at least conceiv able that the quan tum algorithm for Fourier Checking is g o o d for something. W e lea ve the challe nge of ﬁnd ing an explicit compu tational problem that “instant iates” F ourier Chec king , in the same wa y that F actoring and Discrete Logarithm instan tiated Shor’s p erio d-ﬁnd ing pr oblem. 1.3 In Defense of Oracles This pap er is concerned w ith ﬁn ding oracles relativ e to whic h B QP outp erforms classical complexit y classes. As such, it is op en to the usual ob jections: “But don’t oracle results m islead us ab out the ‘real’ world? What ab out n on-relativizing r esults lik e IP = PSP A CE [32]?” In our view, it is most helpful to think of oracle separations, n ot as stran ge metamathematica l claims, b u t as lower b ounds in a c oncr ete c omputationa l mo del that is natur al and wel l-motivate d in its own right. The mo d el in qu estion is query c omplexity , wh ere the resour ce to b e minimized is the n um b er of accesses to a very long input string. When someone giv es an oracle A r elativ e to whic h C A 6⊂ D A , what they really mean is simp ly that they ha ve foun d a problem th at C mac hines can solv e using sup erp olynomially few er queries than D mac hines. In other words, C has has “cleared the ﬁrst p ossible obstacle”—the query complexit y obstacle—to having capabilities b ey ond those of D . O f cours e, it could b e (and sometimes is) that C ⊆ D for other reasons, but if we do n ot even ha ve a query complexit y lo w er b ound, then p r o ving one is in some s ense the ob vious place to start. Oracle separations hav e play ed a role in many of the cen tr al dev elopments of b oth classical and quan tum complexity theory . As mentio ned earlier, p ro vin g query complexit y lo we r b oun ds for PH mac h in es is essentiall y equ iv alen t to pro ving size lo wer b ounds for AC 0 circuits—and indeed, the pioneering AC 0 lo wer b ounds of the early 1980s w er e explicitly motiv ated by the goal of pro v- ing oracle separations for PH . 7 Within qu an tum computing, oracle results hav e p la ye d an ev en more decisiv e role: the ﬁrst evidence for the p ow er of qu an tu m computers came fr om the oracle 7 Y ao’s p ap er [39] w as entitled “S ep arating the p olynomial-time hierarc hy by oracles”; the F u rst-Saxe-Sipser pap er [16] was entitled “P arity , circuits, and th e p olyn omial time h ierarch y .” 7 separations of Bernstein-V azirani [11] an d Simon [34], and Sh or’s algorithm [33] cont ains an oracle algorithm (for th e Period-Finding pr oblem) at its core. Ha ving said all that, if for some reason one still feels a verse to the language of oracles, then (as menti oned b efore) one is free to scale ev erything do wn by an exp onentia l, and to reint erpret a relativized separation b et wee n BQP and PH as an un relativized separation b et we en BQLOGTIM E and AC 0 . 2 Preliminaries It will b e con v enient to consider Bo olean functions of the f orm f : { 0 , 1 } n → {− 1 , 1 } . Th roughout this pap er, w e let N = 2 n ; w e will often view the truth table of a Bo olean fun ction as an “input” of size N . Giv en a Bo olean function f : { 0 , 1 } n → {− 1 , 1 } , the F ourier trans form of f is deﬁ ned as b f ( z ) := 1 √ N X x ∈{ 0 , 1 } n ( − 1) x · z f ( x ) . Recall Pa rsev al’s identit y: X x ∈{ 0 , 1 } n f ( x ) 2 = X z ∈{ 0 , 1 } n b f ( z ) 2 = N . 2.1 Pr oblems W e ﬁ r st deﬁne the F our ier Fishing problem, in b oth “distrib utional” and “promise” v ersions. In the distrib utional ve rsion, w e are giv en oracle access to n Bo olean f unctions f 1 , . . . , f n : { 0 , 1 } n → {− 1 , 1 } , which are c hosen uniformly and ind ep endently at random. The task is to output n str ings, z 1 , . . . , z n ∈ { 0 , 1 } n , at least 75% of w hic h satisfy    b f i ( z i )    ≥ 1 and at least 25% of whic h satisfy    b f i ( z i )    ≥ 2. (Note that these thresholds are not arb itrary , but we re carefully chosen to pro duce a separation b et ween the quantum and classical mo dels!) W e n o w w ant a version of Fourier Fishing that r emo ve s the n eed to assume the f i ’s are uniformly random, replacing it with a worst-c ase p romise on the f i ’s. Call an n -tuple h f 1 , . . . , f n i of Bo olean fu nctions go o d if n X i =1 X z i : | b f i ( z i ) | ≥ 1 b f i ( z i ) 2 ≥ 0 . 8 N n, n X i =1 X z i : | b f i ( z i ) | ≥ 2 b f i ( z i ) 2 ≥ 0 . 26 N n. (W e will show in Lemma 8 that the v ast m a jorit y of h f 1 , . . . , f n i are go o d .) In Promise Fourier Fishing , w e are giv en oracle access to Boolean f u nctions f 1 , . . . , f n : { 0 , 1 } n → { − 1 , 1 } , whic h are promised to b e go o d. Th e task, again, is to output s tr ings z 1 , . . . , z n ∈ { 0 , 1 } n , at least 75% of whic h satisfy    b f i ( z i )    ≥ 1 and at least 25% of whic h satisfy    b f i ( z i )    ≥ 2. Next w e d eﬁ ne a decision problem called Fourier Checking . Here w e are giv en oracle access to t wo Bo olean functions f , g : { 0 , 1 } n → {− 1 , 1 } . W e are pr omised that either 8 (i) h f , g i was drawn fr om the u niform distr ibution U , which s ets ev ery f ( x ) and g ( y ) b y a fair, indep en den t coin toss. (ii) h f , g i w as dra w n fr om the “forrelated” distribution F , whic h is d eﬁned as follo ws. First c ho ose a r andom real vec tor v = ( v x ) x ∈{ 0 , 1 } n ∈ R N , b y dra w ing eac h entry in dep end en tly from a Gaussian distribu tion with mean 0 and v ariance 1. Then set f ( x ) := sgn ( v x ) and g ( x ) := sgn ( b v x ) for all x . Here sgn ( α ) :=  1 if α ≥ 0 − 1 if α < 0 and b v is the F ourier transform of v ov er Z n 2 : b v y := 1 √ N X x ∈{ 0 , 1 } n ( − 1) x · y v x . In other words, f and g individual ly are still uniform ly random, b ut they are no longer indep en den t: no w g is now extremely well correlated with the F our ier transform of f (hence “forrelated”). The problem is to accept if h f , g i was dr a wn fr om F , and to reject if h f , g i was d r a wn from U . Note that, since F and U o v erlap sligh tly , we can only hop e to succeed with o verwhelming pr obabilit y o ve r the c h oice of h f , g i , n ot for ev ery h f , g i p air. W e can also deﬁne a pr omise-pr oblem v ers ion of Fourier Che cking . In Pr omise F ou rier Checking , we are promised that th e quan tit y p ( f , g ) := 1 N 3   X x,y ∈{ 0 , 1 } n f ( x ) ( − 1) x · y g ( y )   2 is either at least 0 . 05 or at most 0 . 01. The problem is to accept in the former case and r eject in the latter case. 2.2 Complexity Classes See the Complexit y Zo o 8 for the deﬁ n itions of standard complexit y classes, s u c h as BQP , AM , and PH . When we write C PH (i.e., a complexit y class C with an oracle for th e p olynomial hierarch y), w e mean ∪ k ≥ 1 C Σ P k . W e will consider n ot only d ecision problems, but also r elation pr oblems (also called function pr oblems ). In a relation problem, the output is n ot a single bit but a p oly ( n )-bit string y . There could b e many v alid y ’s for a giv en instance, and the algorithm’s task is to output an y one of them. The deﬁn itions of FP and FNP (the relation versions of P and NP ) are s tandard. W e n o w d eﬁne FBPP and F BQP , the relation v ersions of BPP and BQP . Deﬁnition 4 FBPP is the class of r elations R ⊆ { 0 , 1 } ∗ × { 0 , 1 } ∗ for which ther e exists a pr ob a- bilistic p olynom ial-time algorithm A that, given any input x ∈ { 0 , 1 } n , pr o duc es an output y such that Pr [( x, y ) ∈ R ] = 1 − o (1) , 8 www.complexit yzo o.com 9 wher e the pr ob ability is over A ’s internal r andomness. (In p articular, this implies that for every x , ther e exists at le ast one y such that ( x, y ) ∈ R .) F BQP is deﬁne d the same way, exc ept that A is a quantum algorithm r ather than a classic al one. An imp ortant p oin t ab out FBPP and FBQP is that, as far as w e kn o w, these classes d o not admit ampliﬁcation. In other words, the v alue of an algorithm’s success probabilit y might actually matter, not ju st the fact that the pr obabilit y is b ound ed ab o ve 1 / 2. T his is why w e adopt the con ven tion that an algorithm “succeeds” if it outp u ts ( x, y ) ∈ R with probabilit y 1 − o (1). In practice, we will giv e oracle p roblems for whic h the FB QP algorithm succeeds with probability 1 − 1 / exp ( n ), w hile any FB PP PH algorithm succeeds w ith probabilit y at most (sa y) 0 . 99. Ho w far the constant in this separation can b e impr o ved is an op en problem. Another imp ortan t p oin t is that, w hile BPP PH = P PH (whic h follo ws f rom B PP ⊆ Σ P 2 ), th e class FBPP PH is strictly larger than FP PH . T o see this, consider the relation R = { (0 n , y ) : K ( y ) ≥ n } , where w e are giv en n , and aske d to outp ut any string of Kolmogoro v complexit y at least n . Clearly this p roblem is in FBPP : just output a random 2 n -b it string. On the other h and, j ust as ob viously the problem is not in FP PH . This is why w e need to constru ct an oracle A su c h that FBQP A 6⊂ FBPP PH A : b ecause constructing an oracle A su c h th at FBQP A 6⊂ FP PH A is trivial and not ev en related to quantum computing. W e n o w d iscuss some “lo w-lev el” complexit y classes. A C 0 is the class of prob lems solv able by a nonuniform family of AND/OR/NOT circu its, with depth O (1), size p oly ( n ), and u n b ound ed fanin. When we sa y “ A C 0 circuit,” w e mean a constan t-depth circuit of AND/OR/NOT gates, n ot necessarily of p olynomial size. An y suc h circuit can b e made into a formula (i.e., a circuit of fanout 1) with only a p olynomial increase in size. T he circuit has depth d if it consists of d alternating la ye rs of AND and OR gates (without loss of generalit y , th e NOT gate s can all b e push ed to the b ottom, and w e do n ot coun t them to w ard s the depth). F or examp le, a DNF (Disjunctiv e Normal F orm) formula is just an AC 0 circuit of depth 2. W e will also b e in terested in qu an tu m lo garithmic time, which can b e deﬁned naturally as follo ws: Deﬁnition 5 BQLOGTIME is the class of languages L ⊆ { 0 , 1 } ∗ that ar e de cidable, with b ounde d pr ob ability of err or, by a LOG TIME -uniform family of qu antum cir cuits { C n } n such that e ach C n has O (log n ) gates, and c an include gates that make r andom-ac c ess queries to the input string x = x 1 . . . x n (i.e., that map | i i | z i to | i i | z ⊕ x i i f or every i ∈ [ n ] ). One other complexit y class th at arises in this pap er, w h ic h is less well kn own than it should b e, is BPP path . Lo osely sp eaking, BPP path can b e deﬁned as the class of pr oblems th at are solv able in pr obabilistic p olynomial time, giv en the ability to “p ostselect” (that is, discard all runs of the computation that do n ot pro duce a desired resu lt, even if suc h run s are the o verwhelming ma j orit y). F ormally: Deﬁnition 6 BPP path is the class of languages L ⊆ { 0 , 1 } ∗ for which ther e exists a BPP machine M , which c an either “suc c e e d” or “fail” and c onditione d on suc c e e ding either “ ac c e pt” or “r eje ct,” such that for al l inputs x : 10 (i) P r [ M ( x ) suc c e e ds ] > 0 . (ii) x ∈ L = ⇒ Pr [ M ( x ) ac c epts | M ( x ) suc c e e ds ] ≥ 2 3 . (iii) x / ∈ L = ⇒ P r [ M ( x ) ac c epts | M ( x ) suc c e e ds ] ≤ 1 3 . BPP path w as deﬁned b y Han, Hemaspaandra, and Th ierauf [25], who also sho wed that MA ⊆ BPP path and P NP || ⊆ BPP path ⊆ BPP NP || . Usin g F ourier Checking , we will construct an oracle A r elativ e to wh ic h BQP A 6⊂ BPP A path . This result might not sound amazing, but (i) it is new, (ii) it do es not follo w fr om the “standard” quan tu m algo rithms, suc h as those of Simon [34] and Shor [33], and (iii) it sup ersedes almost all pr evious oracle results placing BQP outside classical complexit y classes. 9 As another illustration of the v ersatilit y of Fourier Che cking , we u se it to giv e an A such that BQ P A 6⊂ SZ K A , where SZK is Statistical Zero Kno w ledge. The opp osite direction—an A su ch th at SZK A 6⊂ BQP A —w as sho wn by Aaronson [1] in 2002 . 3 Quan tum Algorithms In this section, w e sho w that F ourier Fishing and F ourier Checking b oth admit simp le quan tum algorithms. 3.1 Quan tum Algorit hm for F ourier Fishing Here is a quantum algorithm, FF-A LG , that solv es F our ier Fishing with o v erw helming probability in O  n 2  time and n qu an tum queries (one to eac h f i ). F or i := 1 to n , ﬁrst p repare the state 1 √ N X x ∈{ 0 , 1 } n f i ( x ) | x i , then apply Hadamard gates to all n qub its, then measur e in th e compu tational basis and output the result as z i . In tuitiv ely , FF-ALG samples the F ourier coeﬃcient s of eac h f i under a distribu tion that is sk ewed to wa rds larger co eﬃcients; the algorithm’s b eha vior is illustrated pictorially in Figure 1. W e now giv e a formal analysis. Recall the d eﬁnition of a “go o d” tuple h f 1 , . . . , f n i from Section 2.1. Assuming h f 1 , . . . , f n i is goo d, it is easy to analyze FF-ALG ’s success p robabilit y . Lemma 7 A ssuming h f 1 , . . . , f n i i s go o d, FF-ALG suc c e e ds with pr ob ability 1 − 1 / exp ( n ) . Pro of. Let h z 1 , . . . , z n i b e the algorithm’s output. F or eac h i , let X i b e the ev ent that    b f i ( z i )    ≥ 1 and let Y i b e the ev en t that    b f i ( z i )    ≥ 2. Also let p i := Pr [ X i ] and q i := Pr [ Y i ], where the 9 The one exception is t h e result of Green and Pruim [24] that there exists an A relativ e to which BQP A 6⊂ P NP A , but that can also b e easily reprodu ced using Fourier Checking . 11 Figure 1: T he F our ier co eﬃcien ts of a rand om Bo olean function follo w a Gaussian distribu tion, with mean 0 and v ariance 1. Ho w ever, larger F ourier co eﬃcients are more likel y to b e obs er ved b y the quantum algorithm. probabilit y is o v er FF -ALG ’s in ternal (quantum) r an d omness. Then clearly p i = 1 N X z i : | b f i ( z i ) | ≥ 1 b f i ( z i ) 2 , q i = 1 N X z i : | b f i ( z i ) | ≥ 2 b f i ( z i ) 2 . So by assu mption, p 1 + · · · + p n ≥ 0 . 8 n, q 1 + · · · + q n ≥ 0 . 26 n. By a Chern oﬀ/Hoeﬀdin g b ound, it follo ws that Pr [ X 1 + · · · + X n ≥ 0 . 75 n ] > 1 − 1 exp ( n ) , Pr [ Y 1 + · · · + Y n ≥ 0 . 25 n ] > 1 − 1 exp ( n ) . Hence FF-ALG su cceeds with 1 − 1 / exp ( n ) p robabilit y by the union b oun d. W e also hav e the follo wing: Lemma 8 h f 1 , . . . , f n i is go o d with pr ob ability 1 − 1 / e xp ( n ) , if the f i ’s ar e chosen uniformly at r andom. 12 Pro of. Ch o ose f : { 0 , 1 } n → {− 1 , 1 } u n iformly at random. Then for eac h z , the F ourier co eﬃcien t b f ( z ) follo ws a n ormal distribution, w ith mean 0 and v ariance 1. S o in the limit of large N , E f    X z : | b f ( z ) | ≥ 1 b f ( z ) 2    = X z ∈{ 0 , 1 } n Pr h    b f ( z )    ≥ 1 i E h b f ( z ) 2 |    b f ( z )    ≥ 1 i ≈ 2 N √ 2 π Z ∞ 1 e − x 2 / 2 x 2 dx ≈ 0 . 801 N . Lik ewise, E f    X z : | b f ( z ) | ≥ 2 b f ( z ) 2    ≈ 2 N √ 2 π Z ∞ 2 e − x 2 / 2 x 2 dx ≈ 0 . 261 N . Since the f i ’s are chose n in dep end en tly of one another, it follo ws by a C hernoﬀ b oun d that n X i =1 X z i : | b f i ( z i ) | ≥ 1 b f i ( z i ) 2 ≥ 0 . 8 N n, n X i =1 X z i : | b f i ( z i ) | ≥ 2 b f i ( z i ) 2 ≥ 0 . 26 N n with probability 1 − 1 / exp ( n ) ov er the c hoice of h f 1 , . . . , f n i . Com b ining Lemmas 7 and 8, we ﬁn d that FF-ALG succeeds with probabilit y 1 − 1 / exp ( n ), w here the pr obabilit y is o ver b oth h f 1 , . . . , f n i and FF-A LG ’s in tern al randomn ess. 3.2 Quan tum Algorit hm for F ourier Checking W e no w tur n to F ourier Chec king , the p roblem of d eciding whether tw o Bo olean f u nctions f , g are indep enden t or forrelated. Here is a quan tum algorithm, FC- ALG , th at solve s Fourier Check- ing w ith constan t error pr obabilit y usin g O (1) queries. First pr ep are a uniform su p erp osition o ver all x ∈ { 0 , 1 } n . Then qu ery f in sup erp osition, to create the state 1 √ N X x ∈{ 0 , 1 } n f ( x ) | x i Then apply Hadamard gates to all n qubits, to create the state 1 N X x,y ∈{ 0 , 1 } n f ( x ) ( − 1) x · y | y i . Then query g in sup erp osition, to create the state 1 N X x,y ∈{ 0 , 1 } n f ( x ) ( − 1) x · y g ( y ) | y i . 13 Then apply Hadamard gates to all n qubits again, to create the state 1 N 3 / 2 X x,y ,z ∈{ 0 , 1 } n f ( x ) ( − 1) x · y g ( y ) ( − 1) y · z | z i . Finally , measur e in the computational basis, and “accept” if and only if the outcome | 0 i ⊗ n is observ ed . If needed, rep eat the whole algorithm O (1) times to b o ost the su ccess probabilit y . It is clear th at the p robabilit y of observing | 0 i ⊗ n (in a single run of FC-A LG ) equals p ( f , g ) := 1 N 3   X x,y ∈{ 0 , 1 } n f ( x ) ( − 1) x · y g ( y )   2 . Recall that Promise F ourier Checking wa s the pr oblem of deciding whether p ( f , g ) ≥ 0 . 05 or p ( f , g ) ≤ 0 . 01, promised that one of these is the case. Thus, we immediately get a quan tum algorithm to solv e Promise Fourier Check ing , with constan t error pr ob ab ility , u sing O (1) queries to f and g . F or the d istributional v ersion of Fourier Checking , w e also need the follo wing theorem. Theorem 9 If h f , g i i s dr awn fr om the uniform distribution U , then E U [ p ( f , g )] = 1 N . If h f , g i is dr awn f r om the forr elate d distribution F , then E F [ p ( f , g )] > 0 . 07 . Pro of. Th e ﬁrst part follo ws immediately by sym m etry (i.e., the fact that all N = 2 n measuremen t outcomes of the quant um algorithm are equally lik ely). F or the second p art, let v ∈ R N b e the vect or of indep endent Gaussians used to generate f and g , let w = v / k v k 2 b e v scaled to hav e un it norm, and let H b e the n -qubit Hadamard matrix. Also let ﬂat ( w ) b e the unit ve ctor whose x th en try is sgn ( w x ) / √ N = f ( x ) / √ N , and let ﬂat ( H w ) b e the un it vect or whose x th en try is sgn ( b v x ) / √ N = g ( x ) / √ N . Then p ( f , g ) equ als  ﬂat ( w ) T H ﬂat ( H w )  2 , or the squ ared inner pro du ct b et ween the v ectors ﬂat ( w ) and H ﬂat ( H w ) . Not e th at w T · H H w = w T w = 1. So the whole problem is to understand the “discretization err or” incurred in rep lacing w T b y ﬂat ( w ) T and H H w b y H ﬂat ( H w ) . By the tr iangle inequalit y , the angle b et w een ﬂat ( w ) and H ﬂat ( H w ) is at most the angle b et wee n ﬂ at ( w ) and w , plu s the angle b et we en w and H ﬂat ( H w ). In other words: arccos  ﬂat ( w ) T H ﬂat ( H w )  ≤ arccos  ﬂat ( w ) T w  + arccos  w T H ﬂat ( H w )  . 14 No w, ﬂat ( w ) T w = X x ∈{ 0 , 1 } n w x · 1 √ N | w x | w x = 1 √ N X x ∈{ 0 , 1 } n | w x | = P x ∈{ 0 , 1 } n | v x | √ N k v k 2 . Recall that eac h v x is an indep en d en t real Gaussian with m ean 0 and v ariance 1, meanin g that eac h | v x | is an ind ep end ent nonnegativ e random v ariable w ith exp ectation p 2 /π . So b y standard tail b ounds, for all constan ts ε > 0 we ha ve Pr v   X x ∈{ 0 , 1 } n | v x | ≤ r 2 π − ε ! N   ≤ 1 exp ( N ) , Pr h k v k 2 2 ≥ (1 + ε ) N i ≤ 1 exp ( N ) . So by th e union b ound, Pr v " ﬂat ( w ) T w ≤ r 2 π − ε # ≤ 1 exp ( N ) . Since H is u nitary , the same analysis applies to w T H ﬂ at ( H w ). Therefore, for all constants ε > 0, with 1 − 1 / exp ( N ) probabilit y we hav e arccos  ﬂat ( w ) T w  ≤ arccos r 2 π ! + ε, arccos  w T H ﬂ at ( H w )  ≤ arccos r 2 π ! + ε. So setting ε = 0 . 0001 , arccos  ﬂat ( w ) T H ﬂat ( H w )  ≤ arccos  ﬂat ( w ) T w  + arccos  w T H ﬂat ( H w )  ≤ 2 arccos r 2 π ! + 2 ε ≤ 1 . 3 Therefore, with 1 − 1 / exp ( N ) probabilit y o ver h f , g i d ra w n from F , ﬂat ( w ) T H ﬂat ( H w ) ≥ cos 1 . 3 , in which case p ( f , g ) ≥ (cos 1 . 3) 2 ≈ 0 . 072. Com b ining Theorem 9 with Mark o v’s inequalit y , we immediately get the follo wing: 15 Corollary 10 Pr h f ,g i∼U [ p ( f , g ) ≥ 0 . 01] ≤ 100 N , Pr h f ,g i∼ D [ p ( f , g ) ≥ 0 . 05] ≥ 1 50 . 4 The Classical Complexit y of F ourier Fishing In S ection 3.1, we ga ve a qu an tum algorithm for F ourier F ishing that made only one query to eac h f i . By con trast, it is not h ard to sho w that an y classical algorithm for F o urier Fishing requires exp onent ially man y qu eries to the f i ’s. In th is section, we pro v e a muc h stronger result: that F ou rier Fishing is n ot in FBPP PH . Th is r esult do es not rely on any u nprov ed conjectures. 4.1 Constant-Depth Circuit Low er Bounds Our starting p oint will b e the follo wing AC 0 lo wer b ound , wh ic h can b e found in th e b o ok of H ˚ astad [36] for example. Theorem 11 ([36]) Any depth- d cir c uit that ac c epts al l n -bit strings of H amming weight n/ 2 + 1 , and r eje cts al l strings of Hamming weight n/ 2 , has size exp  Ω  n 1 / ( d − 1)  . W e no w giv e a corollary of T h eorem 11, whic h (though simple) seems to b e new, and migh t b e of indep end en t in terest. Consider the follo wing problem, wh ic h we call ε -Bias Detect ion . W e are giv en a string y = y 1 . . . y m ∈ { 0 , 1 } m , and are promised that eac h bit y i is 1 with in dep en d en t probabilit y p . The task is to decide wh ether p = 1 / 2 or p = 1 / 2 + ε . Corollary 12 L et U [ ε ] b e the distribution over { 0 , 1 } m wher e e ach bit is 1 with indep endent pr ob- ability 1 / 2 + ε . Then any depth- d cir cu i t C such that     Pr U [ ε ] [ C ] − Pr U [0] [ C ]     = Ω (1) has size exp  Ω  1 /ε 1 / ( d +2)  . Pro of. Supp ose such a distinguishin g circuit C exists, with dep th d and size S , for some ε > 0 (the parameter m is actually irrelev ant). Let n = 1 /ε , and assume f or s im p licit y th at n is an in teger. Using C , w e will constru ct a new circuit C ′ with depth d ′ = d + 3 and s ize S ′ = O ( n S ) + p oly ( n ), whic h accepts all strings x ∈ { 0 , 1 } n of Hammin g w eigh t n/ 2 + 1, and rejects all strings of Hamming w eight n/ 2. By Theorem 11, th is will imp ly that the original circuit C must h a ve h ad size S = 1 n exp  Ω  n 1 / ( d ′ − 1)  − p oly ( n ) = exp  Ω  1 /ε 1 / ( d +2)  . So ﬁx an inpu t x ∈ { 0 , 1 } n , and su pp ose w e c ho ose m bits x i 1 , . . . , x i m from x , with eac h index i j c hosen u niformly at rand om w ith replacemen t. C all the r esulting m -bit strin g y . O bserve that if 16 x had Hamming weigh t n/ 2, then y will b e d istributed according to U [0], w hile if x had Hamming w eight n/ 2 + 1, then y will b e distributed according to U [ ε ]. So by assump tion, Pr [ C ( y ) | | x | = n/ 2] = α, Pr [ C ( y ) | | x | = n/ 2 + 1] = α + δ for some constants α and δ 6 = 0 (w e can assume δ > 0 without loss of generalit y). No w sup p ose w e rep eat the ab o ve exp eriment T = k n times, for some constant k = k ( α, δ ) . That is, we create T strings y 1 , . . . , y T b y c ho osing random bits of x , so that eac h y i is distributed indep en den tly according to either U [0] or U [ ε ]. W e then apply C to eac h y i . Let Z = C ( y 1 ) + · · · + C ( y T ) b e the num b er of C inv o cations that accept. Th en b y a Cher n oﬀ b ound , if | x | = n / 2 then Pr  Z > αT + δ 3 T  < exp ( − n ) , while if | x | = n/ 2 + 1 then Pr  Z < αT + 2 δ 3 T  < exp ( − n ) . By taking k large enough, we can mak e b oth of th ese pr obabilities less th an 2 − n . By the union b oun d , this implies that there m ust exist a w ay to choose y 1 , . . . , y T so that | x | = n 2 = ⇒ Z ≤ αT + δ 3 T , | x | = n 2 + 1 = ⇒ Z ≥ αT + 2 δ 3 T for every x with | x | ∈ { n/ 2 , n/ 2 + 1 } simulta neously . In forming the circuit C ′ , w e simply h ardwire that c h oice. The last step is to decide whether Z ≤ αT + δ 3 T or Z ≥ αT + 2 δ 3 T . This can b e don e using an A C 0 circuit f or the Appro xima te Majo rity problem (see Viola [37] for example), wh ich has depth 3 and size p oly ( T ). The end result is a circuit C ′ to distinguish | x | = n/ 2 from | x | = n/ 2 + 1, whic h has depth d + 3 and size T S + p oly ( T ) = O ( nS ) + p oly ( n ). 4.2 Secretly Biased F ourier Co eﬃcien ts In this s ection, we pro v e tw o lemmas indicating that one can slightly bias one of the F our ier co eﬃcien ts of a random Bo olean function f : { 0 , 1 } n → {− 1 , 1 } , and ye t still ha ve f b e information- theoreticall y ind istinguishable f rom a random Bo olean fu nction (so that, in particular, an adv ersary has no wa y of kno wing which F ourier co eﬃcien t was biased). Th ese lemmas will pla y a key role in our reduction fr om ε -Bias Detection to Fourier Fishing . Fix a strin g s ∈ { 0 , 1 } n . Let A [ s ] b e the probability distribution o ver functions f : { 0 , 1 } n → {− 1 , 1 } where eac h f ( x ) is 1 with indep enden t p r obabilit y 1 2 + ( − 1) s · x 1 2 √ N , and let B [ s ] b e the distribution where eac h f ( x ) is 1 with ind ep endent p robabilit y 1 2 − ( − 1) s · x 1 2 √ N . Then let D [ s ] = 1 2 ( A [ s ] + B [ s ]) (that is, an equal mixture of A [ s ] and B [ s ]). 17 Lemma 13 Supp ose Alic e cho oses s ∈ { 0 , 1 } n uniformly at r andom, then dr aws f ac c or ding to D [ s ] . She ke eps s se cr et, but se nds the truth table of f to Bob. A f ter examining f , Bob outputs a string z such that    b f ( z )    ≥ β . Then Pr [ s = z ] ≥ e β + e − β 2 √ eN . wher e the pr ob ability is over al l runs of the pr oto c ol. Pro of. By Y ao’s principle, w e can assume without loss of generalit y that Bob’s strategy is deter- ministic. F or eac h z , let F [ z ] b e the set of all f ’s that cause Bob to output z . Then the ﬁrst step is to low er-b oun d Pr D [ z ] [ f ], for some ﬁxed z and f ∈ F [ z ] . Let N f [ z ] b e the num b er of inp uts x ∈ { 0 , 1 } n suc h that f ( x ) = ( − 1) z · x . It is not hard to see that N f [ z ] = N 2 + √ N b f ( z ) 2 . So Pr D [ z ] [ f ] = 1 2  Pr A [ z ] [ f ] + Pr B [ z ] [ f ]  = 1 2   Y x ∈{ 0 , 1 } n  1 2 + ( − 1) z · x f ( x ) 2 √ N  + Y x ∈{ 0 , 1 } n  1 2 − ( − 1) z · x f ( x ) 2 √ N    = 1 2 N +1  1 + 1 √ N  N f [ z ]  1 − 1 √ N  N − N f [ z ] +  1 − 1 √ N  N f [ z ]  1 + 1 √ N  N − N f [ z ] ! = 1 2 N +1      1 + 1 / √ N  ( √ N b f ( z )+ N ) / 2  1 − 1 / √ N  ( √ N b f ( z ) − N ) / 2 +  1 − 1 / √ N  ( √ N b f ( z )+ N ) / 2  1 + 1 / √ N  ( √ N b f ( z ) − N ) / 2     = 1 2 N +1  1 − 1 N  N/ 2   1 + 1 / √ N 1 − 1 / √ N ! √ N b f ( z ) / 2 + 1 − 1 / √ N 1 + 1 / √ N ! √ N b f ( z ) / 2   = 1 2 √ e 2 N  e b f ( z ) + e − b f ( z )  ≥ e β + e − β 2 √ e 2 N . Here the second-to-last line tak es the limit as N → ∞ , while the last line follo w s from the as- sumption    b f ( z )    ≥ β , together w ith the fact that e y + e − y increases monotonically a w ay from y = 0. 18 Summing o ver all z and f , Pr [ s = z ] = X z ∈{ 0 , 1 } n X f ∈F [ z ] Pr [ f ] · Pr [ s = z | f ] = X z ∈{ 0 , 1 } n X f ∈F [ z ] Pr [ f ] · Pr [ f | s = z ] Pr [ s = z ] Pr [ f ] = 1 N X z ∈{ 0 , 1 } n X f ∈F [ z ] Pr D [ z ] [ f ] ≥ e β + e − β 2 √ eN . No w let D = E s [ D [ s ]] (that is, an equal mixture of all the D [ s ]’s). W e claim that D is extremely close in v ariation distance to U , the uniform distribution o ver all Boolean functions f : { 0 , 1 } n → {− 1 , 1 } . Lemma 14 kD − U k ≤ e − 1 2 √ 2 eN . Pro of. By a calculation from Lemm a 13, for all f and s w e ha ve Pr D [ s ] [ f ] = 1 2 √ e 2 N  e b f ( s ) + e − b f ( s )  in the limit of large N . Hence Pr D [ f ] = E s  Pr D [ s ] [ f ]  = 1 2 √ eN 2 N X s ∈{ 0 , 1 } n  e b f ( s ) + e − b f ( s )  . Clearly E f [Pr D [ f ]] = 1 / 2 N . Our goal is to u pp er-b ound the v ariance V ar f [Pr D [ f ]], wh ic h mea- sures the d istance from D to the un if orm d istribution. In the limit of large N , we h av e E f  Pr D [ f ] 2  = 1 4 eN 2 2 2 N   X s E f   e b f ( s ) + e − b f ( s )  2  + X s 6 = t E f h e b f ( s ) + e − b f ( s )   e b f ( t ) + e − b f ( t ) i   = 1 4 eN 2 2 2 N   P s 1 √ 2 π R ∞ −∞ e − x 2 / 2 ( e x + e − x ) 2 dx + P s 6 = t h 1 √ 2 π R ∞ −∞ e − x 2 / 2 ( e x + e − x ) dx i 2   = 1 4 eN 2 2 2 N  2 e 2 + 2  N + 4 eN ( N − 1)  = 1 2 2 N 1 + ( e − 1) 2 2 eN ! . Hence V ar f  Pr D [ f ]  = E f  Pr D [ f ] 2  − E f  Pr D [ f ]  2 = ( e − 1) 2 2 eN 2 2 N . 19 So by C auc hy-Sc hw arz, E f      Pr D [ f ] − Pr U [ f ]      ≤ s V ar f  Pr D [ f ]  = e − 1 √ 2 eN · 1 2 N and kD − U k ≤ e − 1 2 √ 2 eN . An immediate corollary of Lemm a 14 is that, if a F o urier Fishing algorithm succeeds with probabilit y p on h f 1 , . . . , f n i drawn fr om U n , then it also succeeds with probabilit y at least p − kD n − U n k ≥ p − ( e − 1) n 2 √ 2 eN on h f 1 , . . . , f n i drawn fr om D n . 4.3 Put ting It All T ogether Using the results of S ections 4.1 and 4.2, w e are no w ready to pro ve a low er b ound on the constant - depth circuit complexit y of F ou rier Fishing . Theorem 15 Any depth- d cir cuit that solves the Fourier Fishing pr oblem, with pr ob ability at le ast 0 . 99 over f 1 , . . . , f n chosen uniformly at r andom, has size exp  Ω  N 1 / (2 d +8)  . Pro of. Let C b e a circuit of dep th d and size s . Let G b e the set of all h f 1 , . . . , f n i on whic h C suc c e e ds : that is, for wh ic h it outpu ts z 1 , . . . , z n , at least 75% of which satisfy    b f i ( z i )    ≥ 1 and at least 25% of whic h satisfy    b f i ( z i )    ≥ 2. S u pp ose Pr U n [ h f 1 , . . . , f n i ∈ G ] ≥ 0 . 99 . Then by L emm a 14, we also ha ve Pr D n [ h f 1 , . . . , f n i ∈ G ] ≥ 0 . 99 − ( e − 1) n 2 √ 2 eN ≥ 0 . 98 for suﬃcientl y large n . Using the ab ov e fact, w e w ill con vert C into a n ew circuit C ′ that solv es the ε -Bias Dete ction problem of Corollary 12, with ε := 1 2 √ N . This C ′ will ha v e depth d ′ = d + 2 and size S ′ = O ( N S ). By Corollary 12 , this will imp ly that C itself must h a ve had size S = exp  Ω  1 /ε 1 / ( d ′ +2)  = exp  Ω  N 1 / (2 d +8)  . Let M = N 2 n , and let R = r 1 . . . r M ∈ { 0 , 1 } M b e a string of b its where eac h r j is 1 with indep en den t probabilit y p . W e wan t to decide whether p = 1 / 2 or p = 1 / 2 + ε —t hat is, wh ether R 20 w as dr a wn from U [0] or U [ ε ]. W e can do this as follo ws. First, choose str in gs s 1 , . . . , s n ∈ { 0 , 1 } n , bits b 1 , . . . , b n ∈ { 0 , 1 } , and an inte ger k ∈ [ n ] uniformly at random. Next, deﬁn e Bo olean functions f 1 , . . . , f n : { 0 , 1 } n → {− 1 , 1 } u sing the ﬁr s t N n bits of R , like so: f i ( x ) := ( − 1) r ( i − 1) N + x + s i · x + b i . Finally , feed h f 1 , . . . , f n i as inpu t to C , and consider z k , the k th output of C (discarding the other n − 1 outputs). W e are in terested in Pr [ z k = s k ], wh ere the p robabilit y is o ver R , s 1 , . . . , s n , b 1 , . . . , b n , and k . If p = 1 / 2, notice that f 1 , . . . , f n are in dep end en t and un iformly r andom r egardless of s 1 , . . . , s n . So C gets no in formation ab out s k , and Pr [ z k = s k ] = 1 / N . On the other hand , if p = 1 / 2 + ε , then eac h f i is drawn indep enden tly f rom the distribution D [ s ] stu died in Lemm a 13. So b y the Lemma, for ev ery i ∈ [ n ], if    b f i ( z i )    ≥ β then Pr f i [ z i = s i ] ≥ e β + e − β 2 √ eN . So assuming C succeeds (that is, h f 1 , . . . , f n i ∈ G ), w e ha v e Pr f 1 ,...,f n ,k [ z k = s k ] ≥ 1 4  e 2 + e − 2 2 √ eN  + 1 2  e 1 + e − 1 2 √ eN  ≥ 1 . 038 N . So for a r andom h f 1 , . . . , f n i dra wn according to D n , Pr f 1 ,...,f n ,k [ z k = s k ] ≥ 0 . 98  1 . 038 N  ≥ 1 . 017 N . Notice that this is b ounded ab o ve 1 / N by a multiplicativ e constan t. No w let us rep eat the ab o v e exp erimen t N times. That is, for all j := 1 to N , w e generate Bo olean functions f j 1 , . . . , f j n : { 0 , 1 } n → {− 1 , 1 } b y the same pr ob ab ilistic p ro cedure as b efore, but eac h time using a new N n -bit substrin g of R j of R , as well as new s , b , and k v alues (denoted s j 1 , . . . , s j n , b j 1 , . . . , b j n , and k j ). W e then app ly C to eac h n -tuple h f j 1 , . . . , f j n i . Let z j b e the k th j string that C outpu ts when run on h f j 1 , . . . , f j n i . Then by the ab ov e, for eac h j ∈ [ N ] we h a ve p = 1 2 = ⇒ Pr  z j = s j k j  = 1 N , p = 1 2 + ε = ⇒ Pr  z j = s j k j  ≥ 1 . 017 N . F u r thermore, these probabilities are indep endent across the diﬀeren t j ’s. So let E b e the eve n t that there exists a j ∈ [ N ] suc h that z j = s j k j . Th en if p = 1 / 2 we h a ve Pr [ E ] = 1 −  1 − 1 N  N ≈ 1 − 1 e ≤ 0 . 633 , while if p = 1 / 2 + ε we ha ve Pr [ E ] ≥ 1 −  1 − 1 . 017 N  N ≥ 0 . 638 . 21 It should now b e clear h o w to create the circuit C ′ , whic h distinguishes R ∈ { 0 , 1 } M dra wn from U [0] f r om R dr a wn from U [ ε ] with constant b ias. F or eac h j ∈ [ N ], generate an n -tuple of Bo olean functions h f j 1 , . . . , f j n i fr om R and apply C to it; then chec k whether there exists a j ∈ [ N ] such that z j = s j k j . This chec king step can b e d on e by a d epth-2 circuit of size O ( N n ). Therefore, C ′ will ha v e depth d ′ = d + 2 and size s ′ = O ( N s ). A tec hn icalit y is that our c h oices of the s j i ’s, b j i ’s, and k j ’s were made randomly . Ho we v er, by Y ao’s principle, there clearly exist s j i ’s, b j i ’s, and k j ’s suc h that Pr U [ ε ]  C ′ ( R )  − Pr U [0]  C ′ ( R )  ≥ 0 . 638 − 0 . 633 = 0 . 005 . So in forming C ′ , we s imply hardw ire those c h oices. Com b ining Theorem 15 with standard diagonalization tricks, we can n o w prov e an oracle sep- aration (in f act, a r andom oracle separation) b etw een the complexit y classes FBQP and FBPP PH . Theorem 16 FBQP A 6⊂ FBPP PH A with pr ob ability 1 for a r andom or acle A . Pro of. W e inte rpret the oracle A as enco ding n random Bo olean functions f n 1 , . . . , f nn : { 0 , 1 } n → {− 1 , 1 } for eac h p ositiv e in teger n . Let R b e the r elational problem wh ere we are giv en 0 n as input, and suc c e e d if and only if w e output strin gs z 1 , . . . , z n ∈ { 0 , 1 } n , at least 3 / 4 of which satisfy    b f ni ( z i )    ≥ 1 and at least 1 / 4 of wh ic h satisfy    b f ni ( z i )    ≥ 2. Then by Lemmas 7 and 8, there exists an FBQP A mac hine M suc h that for all n , Pr [ M ( 0 n ) succeeds] ≥ 1 − 1 exp ( n ) , where the p robabilit y is o ver b oth A and the quant um randomness. Hence Pr [ M (0 n ) su cceeds] ≥ 1 − 1 / exp ( n ) on all but ﬁ nitely many n , with probab ility 1 ov er A . Since we can simply hard wire the answers on the n ’s for whic h M fails, it f ollo ws that R ∈ FB QP A with probability 1 o ver A . On the other hand, let M b e an FBPP PH A mac hine. Then b y the standard con v ersion b et wee n PH and A C 0 , for ev er y n there exists a prob ab ilistic A C 0 circuit C M ,n , of size 2 poly ( n ) = 2 polyl og( N ) , that tak es A as inp ut and simulates M (0 n ). By Y ao’s principle, we can assume without loss of generalit y that C M ,n is deterministic, since the oracle A is already random. Then b y Th eorem 15, Pr A [ C M ,n succeeds] < 0 . 99 for all suﬃcien tly large n . By the indep end en ce of the f ni ’s, this is true even if w e condition on C M , 1 , . . . , C M ,n − 1 succeeding. S o as in the stand ard rand om oracle argument of Bennett and Gill [10], for eve ry ﬁ xed M we h av e Pr A [ C M , 1 , C M , 2 , C M , 3 , . . . succeed] = 0 . So by th e union b ound, Pr A [ ∃ M : C M , 1 , C M , 2 , C M , 3 , . . . succeed] = 0 as wel l. It follo ws that FBQP A 6⊂ FBPP PH A with probability 1 o ver A . If we “scale do wn by an exp onen tial,” then w e can eliminate the need for the oracle A , and get a relation pr oblem that is solv ab le in quantum lo garithmic time b ut not in AC 0 . 22 Theorem 17 Ther e exists a r elation pr oblem solvable in BQLOGTIME but not in AC 0 . Pro of. In our relation p roblem R , the inpu t (of size M = 2 n n ) will enco d e the tru th tables of n Bo olean functions, f 1 , . . . , f n : { 0 , 1 } n → {− 1 , 1 } , which are promised to b e “go o d ” as d eﬁ ned in Section 2.1. T he task is to solv e Promise Fourier Fish ing on h f 1 , . . . , f n i . By Lemma 7, there exists a quant um algorithm that runs in O ( n ) = O (log M ) time, making random accesses to the truth tables of f 1 , . . . , f n , that solv es R with probabilit y 1 − 1 / exp ( n ) = 1 − 1 / M Ω(1) . On the other hand, supp ose R is in AC 0 . Then we get a nonuniform circuit family { C n } n , of depth O (1) and size p oly ( M ) = 2 O ( n ) , that solv es Fourier Fish ing on all tuples h f 1 , . . . , f n i that are goo d. Recall th at by Lemma 8, a 1 − 1 / exp ( n ) fraction of h f 1 , . . . , f n i ’s are go o d. Therefore { C n } n actually solv es Fourier Fishing with probabilit y 1 − 1 / e xp ( n ) on h f 1 , . . . , f n i c hosen uniformly at random. But this con tradicts Th eorem 15. Hence R ∈ FBQLOGTIM E \ F AC 0 (where FBQLOG TIME and F AC 0 are the relatio n v ersions of BQLOGTIME and AC 0 resp ectiv ely). 5 The Classical Complexit y of F ourier Chec k ing Section 4 settled the relativized BQP v ersu s PH question, if w e are w illing to talk ab out relation problems. Ultimatel y , though, w e also care ab out decision problems. So in this section we consider the Fourier Check ing pr oblem, of deciding w hether tw o Bo olean fun ctions f , g are ind ep end en t or f orr elated. In Section 3.2, we sa w that Fourier Checking has quantum query complexit y O (1). What is its classical qu ery complexit y? 10 It is not hard to giv e a classica l algorithm that solv es F ou rier Checking using O  √ N  = O  2 n/ 2  queries. The algorithm is as follo ws: for some K = Θ  √ N  , ﬁrst choose sets X = { x 1 , . . . , x K } and Y = { y 1 , . . . , y K } of n -bit strings u n iformly at random. Th en qu ery f ( x i ) and g ( y i ) for all i ∈ [ K ]. Finally , compu te Z := K X i,j =1 f ( x i ) ( − 1) x i · y j g ( y j ) , accept if | Z | is greater than some cutoﬀ cK , and reject otherwise. F or suitable K and c , one can sho w that th is algorithm accepts a forrelated h f , g i pair with probabilit y at least 2 / 3, and accepts a rand om h f , g i pair with prob ab ility at least 1 / 3. W e omit the details of the analysis, as they are tedious and not needed elsewhere in the pap er. In the next section, we will sho w that Fourier Checking has a prop ert y called almost k -wise indep endenc e , whic h imm ediately implies a lo wer b ound of Ω  4 √ N  = Ω  2 n/ 4  on its classical query complexit y (as w ell as exp onential lo w er b ounds on its MA , BPP path , and SZ K qu ery complexities). Indeed, we conjecture that almost k -wise in d ep end ence is enough to im p ly that Fourier Chec king is not in PH . W e discu ss the status of that conjecture in Section 6. 10 So long as we consider t h e distributional versio n of Fourier Checking , the deterministic and randomized q uery complexities are the same (by Y ao’s principle). 23 5.1 Almost k -Wise Indep endence Let Z = z 1 . . . z M ∈ {− 1 , 1 } M b e a string. Then a liter al is a term of the form 1 ± z i 2 , and a k - term is a pro d uct of k literals (eac h in v olving a diﬀerent z i ), wh ic h is 1 if the literals all tak e on prescrib ed v alues and 0 otherwise. Let U b e the un iform d istribution o ve r {− 1 , 1 } M . The follo w ing deﬁn ition will pla y a ma jor role in this w ork. Deﬁnition 18 A distribution D over {− 1 , 1 } M is ε -almost k - wise indep endent if for every k -term C , 1 − ε ≤ Pr D [ C ] Pr U [ C ] ≤ 1 + ε. (Note that Pr U [ C ] is just 2 − k .) No w let M = 2 n +1 = 2 N , and let F b e the forrelated distribution o v er pairs of Bo olean functions f , g : { 0 , 1 } n → {− 1 , 1 } . That is, w e samp le h f , g i ∈ F by ﬁ rst choosing a v ector v = ( v x ) x ∈{− 1 , 1 } n ∈ R N of indep endent N (0 , 1) Gaussians, then setting f ( x ) := sgn ( v x ) for all x and g ( y ) := sgn ( b v y ) for all y . Theorem 19 F or al l k ≤ 4 √ N , the forr elate d distribution F is O  k 2 / √ N  -almost k -wise inde- p endent. Pro of. As a ﬁr st step, we will pro v e an analogous statemen t for the real-v alued fu nctions F ( x ) := v x and G ( y ) := b v y ; then we will generalize to the discrete versions f ( x ) and g ( y ). Let U ′ b e the probabilit y m easur e o ve r h F , G i that corresp onds to case (i) of F our ier Checking : that is, w e c ho ose eac h F ( x ) and G ( y ) in dep end en tly from the Gaussian measure N ( 0 , 1). Let F ′ b e th e probabilit y measure o v er h F , G i that corresp onds to case (ii) of F ourier Check ing : that is, w e c ho ose eac h F ( x ) indep endent ly from N (0 , 1) , then set G ( y ) := b F ( y ) where b F ( y ) = 1 √ N X x ∈{ 0 , 1 } n ( − 1) x · y F ( x ) is the F ourier transform of F . Observ e that s in ce th e F ourier transform is unitary , G has the same marginal distribu tion as F un d er F ′ : namely , a p ro du ct of indep enden t N (0 , 1) Gaussians. Fix inputs x 1 , . . . , x K ∈ { 0 , 1 } n of F and y 1 , . . . , y L ∈ { 0 , 1 } n of G , for some K, L ≤ N 1 / 4 . Then giv en constan ts a 1 , . . . , a K , b 1 , . . . , b L ∈ R , let S b e the set of all h F , G i that satisfy the K + L equations F ( x i ) = a i for all 1 ≤ i ≤ K , (1) G ( y j ) = b j for all 1 ≤ j ≤ L . Clearly S is a (2 N − K − L )-dimensional aﬃne sub space of R 2 N . The me asur e of S , u nder s ome probabilit y measure µ on R 2 N , is deﬁ ned in the usu al w ay as µ ( S ) := Z h F ,G i∈ S µ ( F, G ) d h F , G i . 24 No w let ∆ S := a 2 1 + · · · + a 2 K + b 2 1 + · · · + b 2 L b e the squ ared distance b et w een S and the origin (that is, the minimum s quared 2-norm of an y p oint in S ). Then by the sph erical sym metry of the Gaussian measure, it is not hard to see that S has measure U ′ ( S ) = e − ∆ S / 2 √ 2 π K + L under U ′ . Ou r k ey claim is th at 1 − O  ( K + L ) ∆ S √ N  ≤ F ′ ( S ) U ′ ( S ) ≤ 1 + O  ( K + L ) ∆ S √ N  . T o prov e th is claim: r ecall that the p robabilit y measure o ve r F ind uced b y F ′ is just a sph erical Gaussian G on R N , and that G = b F uniqu ely determines F and vice v ersa. So consider the ( N − K − L )-dimensional aﬃne subspace T of R N deﬁned by the K + L equations F ( x i ) = a i for all 1 ≤ i ≤ K , b F ( y j ) = b j for all 1 ≤ j ≤ L . Then F ′ ( S ) = G ( T ): that is, to compute how m u c h measure F ′ assigns to S , it suﬃces to compute ho w m u c h measure G assigns to T . W e ha ve G ( T ) = e − ∆ T / 2 √ 2 π K + L , where ∆ T is the squared E uclidean distance b etw een T and the origin. Thus, our pr oblem redu ces to minimizing ∆ F := X x ∈{ 0 , 1 } n F ( x ) 2 o ve r all F ∈ T . By a standard fact ab out quadratic optimizati on, th e min imal F ∈ T will ha v e the form F ( x ) = α 1 E 1 ( x ) + · · · + α K E K ( x ) + β 1 χ 1 ( x ) + · · · + β L χ L ( x ) where E i ( x ) :=  1 if x = x i 0 otherwise is an in d icator function, and χ j ( x ) := ( − 1) x · y j √ N is the y th j F our ier c haracter ev aluated at x . F urtherm ore, the co eﬃcien ts { α i } i ∈ [ K ] , { β j } j ∈ [ L ] can 25 b e obtained by solving the linear system           1 0 0 ± 1 / √ N · · · ± 1 / √ N 0 . . . 0 . . . . . . . . . 0 0 1 ± 1 / √ N · · · ± 1 / √ N ± 1 / √ N · · · ± 1 / √ N 1 0 0 . . . . . . . . . 0 . . . 0 ± 1 / √ N · · · ± 1 / √ N 0 0 1           | {z } A           α 1 . . . α K β 1 . . . β L           | {z } u =           a 1 . . . a K b 1 . . . b L           | {z } w Here A is simply a m atrix of co v ariances: the top left blo ck records the inner pro d uct b et w een eac h E i and E j (and h ence is a K × K iden tity matrix), the b ottom righ t blo c k records th e inn er pro du ct b et w een eac h χ i and χ j (and hence is an L × L iden tity matrix), and the r emaining tw o blo c k s of size K × L r ecord the inner pr o duct b etw een eac h E i and χ j . Th us, to get the v ector of co eﬃcien ts u ∈ R K + L , we simply need to calculat e A − 1 w . Deﬁne B := I − A . Then by T a ylor series expansion, A − 1 = ( I − B ) − 1 = I + B + B 2 + B 3 + · · · Notice th at ev ery en tr y of B is at m ost 1 / √ N in absolute v alue. This means that, for all p ositiv e in tegers t , eve ry entry of B t is at most ( K + L ) t − 1 N t/ 2 in abs olute v alue. Since K + L ≪ √ N , th is in turn means that ev ery entry of I − A − 1 has absolute v alue O  1 / √ N  . So A − 1 is exp on entially close to the iden tity matrix. Hence, when we compute the vec tor u = A − 1 w , we ﬁ n d that α i = a i + ε i for all 1 ≤ i ≤ K , β j = b j + δ j for all 1 ≤ j ≤ L , for some small error terms ε i and δ j . Sp eciﬁcally , eac h ε i and δ j is the inner p ro du ct of w , a ( K + L )-dimensional v ector of length √ ∆ S , with a vec tor every entry of which h as absolute v alue O  1 / √ N  . By Cauc hy-Sc hw arz, th is implies th at | ε i | , | δ j | = O p ( K + L ) ∆ S √ N ! 26 for all i, j . So ∆ T = min F ∈ T X x ∈{ 0 , 1 } n F ( x ) 2 = K X i =1 α 2 i + L X j =1 β 2 j + 2 K X i =1 L X j =1 α i β j √ N = K X i =1 ( a i + ε i ) 2 + L X j =1 ( b j + δ j ) 2 + 2 K X i =1 L X j =1 ( a i + ε i ) ( b j + δ j ) √ N = ∆ S ± O ( K + L ) ∆ S √ N + ( K + L ) 2 ∆ S N + ( K + L ) 3 ∆ S N 3 / 2 ! = ∆ S  1 ± O  K + L √ N  , where the fourth line m ade r ep eated use of Cauc hy-Sc hw arz, and the ﬁfth line used the fact that K + L ≪ √ N . Hence F ′ ( S ) U ′ ( S ) = e − ∆ T / 2 / √ 2 π K + L e − ∆ S / 2 / √ 2 π K + L = exp  ∆ S − ∆ T 2  = exp  ± O  ( K + L ) ∆ S √ N  = 1 ± O  ( K + L ) ∆ S √ N  whic h pro v es the claim. T o prov e the theorem, we no w need to generalize to the discrete fu n ctions f and g . Here w e are giv en a term C that is a conjun ction of K + L in equalities: K of th e form F ( x i ) ≤ 0 or F ( x i ) ≥ 0, and L of the form G ( y j ) ≤ 0 or G ( y j ) ≥ 0. If we ﬁx x 1 , . . . , x K and y 1 , . . . , y L , w e can think of C as just a con vex region of R K + L . T hen giv en an aﬃne subsp ace S as deﬁned by equation (1), we will (abusin g notation) write S ∈ C if the vec tor ( α 1 , . . . , α K , β 1 , . . . , β L ) is in C : that is, if S is compatible with the K + L in equ alities that deﬁne C . W e need to sho w that the ratio 27 Pr F [ C ] / Pr U [ C ] is close to 1. W e can do so u sing the previous r esult, as follo ws: Pr F [ C ] Pr U [ C ] = R S ∈ C F ′ ( S ) dS R S ∈ C U ′ ( S ) dS = R S ∈ C U ′ ( S ) h 1 ± O  ( K + L )∆ S √ N i dS R S ∈ C U ′ ( S ) dS = R S ∈ C h e − ∆ S / 2 / √ 2 π K + L i h 1 ± O  ( K + L )∆ S √ N i dS R S ∈ C h e − ∆ S / 2 / √ 2 π K + L i dS = (1 / 2) K + L ± O  R S ∈ C h e − ∆ S / 2 / √ 2 π K + L i ( K + L )∆ S √ N dS  (1 / 2) K + L = 1 ± 2 K + L ( K + L ) √ N O Z S ∈ C e − ∆ S / 2 √ 2 π K + L ∆ S dS ! = 1 ± K + L √ N O Z S e − ∆ S / 2 √ 2 π K + L ∆ S dS ! = 1 ± O ( K + L ) 2 √ N ! . Setting k := K + L , this completes the pro of. 5.2 Oracle Separation Results The follo wing lemma shows th at any almost k -wise ind ep end en t distribu tion is ind istinguishable from the un iform distribu tion by BPP path or SZK machines. Lemma 20 Supp ose a pr ob ability distribution D over or acle strings is 1 /t ( n ) -almost p oly ( n ) -wise indep endent, for some sup erp olynomial function t . Then no B PP path machine or SZ K pr oto c ol c an distinguish D fr om the uniform distribution U with non-ne gligi b le bias. Pro of. Let M b e a BPP path mac hine, and let p D b e the pr obabilit y that M accepts an oracle string dra wn from distr ibution D . Then p D can b e written as a D /s D , where s D is the fraction of M ’s computation paths that are p ostselected, and a D is the fraction of M ’s paths that are b oth p ostselected and acce pting. Since eac h computation p ath can examine at m ost p oly ( n ) b its and D is 1 /t ( n )-almost p oly ( n )-wise indep en den t, w e ha ve 1 − 1 t ( n ) ≤ a D a U ≤ 1 + 1 t ( n ) and 1 − 1 t ( n ) ≤ s D s U ≤ 1 + 1 t ( n ) . Hence  1 − 1 t ( n )  2 ≤ a D /s D a U /s U ≤  1 + 1 t ( n )  2 . No w let P b e an S ZK proto col. Then b y a resu lt of Sahai and V adh an [31], there exist p olynomial-time samplable distrib u tions A and A ′ suc h that if P accepts, th en k A − A ′ k ≤ 1 / 3, 28 while if P rejects, then k A − A ′ k ≥ 2 / 3. But sin ce eac h computation path can examine at most p oly ( n ) oracle bits and D is 1 /t ( n )-almost p oly ( n )-wise indep end en t, we hav e k A D − A U k ≤ 1 /t ( n ) and k A ′ D − A ′ U k ≤ 1 /t ( n ), where the sub script denotes the distrib u tion from w hic h the oracle string w as drawn. Hence     A D − A ′ D   −   A U − A ′ U     ≤ k A D − A U k +   A ′ D − A ′ U   ≤ 2 t ( n ) and no SZ K proto col exists. W e n o w combine Lemma 20 and T heorem 19 w ith standard diagonalization tric ks, to obtain an oracle relativ e to w h ic h BQP 6⊂ B PP path and BQP 6⊂ S ZK . Theorem 21 Ther e exists an or acle A r elative to which BQP A 6⊂ BPP A path and BQP A 6⊂ SZK A . Pro of. The oracle A will enco de th e truth tables of Bo olean fun ctions f 1 , f 2 , . . . and g 1 , g 2 , . . . , where f n , g n : { 0 , 1 } n → {− 1 , 1 } are on n v ariables eac h. F or eac h n , with 1 / 2 probabilit y we dra w h f n , g n i from the u niform d istribution U , and with 1 / 2 probability w e dra w h f n , g n i from the forrelated d istribution F . Let L b e the un ary language consisting of all 0 n for which h f n , g n i was dra wn from F . By Theorem 9, there exists a BQP A mac hine M that decides L on all bu t ﬁnitely many v alues of n , with probabilit y 1 o ver A . Since w e can simply hard wire the v alues of n on which M fails, it follo ws that L ∈ BQP A with probability 1 o ver A . On th e other hand, we sho w ed in Theorem 19 that F is O  p ( n ) 2 / 2 n/ 2  -almost p ( n )-wise indep en den t for all p olynomials p . Hence, by Lemma 20, no BPP path mac hine can d istinguish F from U with non-negligible b ias. Let E n ( M ) b e the even t that the BPP A path mac hine M correctly decides wh ether 0 n ∈ L . Then Pr A [ E n ( M )] ≤ 1 2 + o (1) , and moreov er this is tr ue eve n conditioning on E 1 ( M ) , . . . , E n − 1 ( M ). So as in the standard random oracle argument of Bennett and Gill [10], for ev ery ﬁxed M we h a ve Pr A [ E 1 ( M ) ∧ E 2 ( M ) ∧ · · · ] = 0 . So by th e union b ound, Pr A [ ∃ M : E 1 ( M ) ∧ E 2 ( M ) ∧ · · · ] = 0 as well. It follo ws that BQP A 6⊂ BPP A path with p robabilit y 1 ov er A . By exactly the s ame argumen t, w e also get BQP A 6⊂ SZK A with probability 1 o ver A . Since BPP ⊆ MA ⊆ B PP path , T heorem 21 su p ers edes the pr evious results that there exist oracles A relativ e to w hic h BPP A 6 = BQP A [11] and B QP A 6⊂ MA A [38]. 6 The Generalized Linial-Nisan Conjecture In 1990, Linial and Nisan [28] f amously conjectured that “p olylogarithmic indep endence fo ols A C 0 ”—or lo osely sp eaking, that eve ry probability distrib ution D o v er n -bit str ings that is un iform on all s mall subsets of b its, is indistinguishable from the un iform distribu tion by p olynomial-size, 29 constan t-depth circuits. W e n ow state a v arian t of the Linial-Nisan Conjecture, not with the b est p ossible parameters but with weak er, easier-to-understand p arameters that suﬃce for our applica- tion. Conjecture 22 (Linial-Nisan Conjecture) L et D b e an n Ω(1) -wise indep endent distribution over { 0 , 1 } n , and let f : { 0 , 1 } n → { 0 , 1 } b e c ompute d by an A C 0 cir cuit of size 2 n o (1) and depth O (1) . Then     Pr x ∼D [ f ( x )] − Pr x ∼U [ f ( x )]     = o (1) . After sev ente en years of almost n o pr ogress, in 2007 Bazzi [7] ﬁn ally p ro ved Conj ecture 22 for the sp ecial case of d epth-2 circuits (also called DNF formulas). Ba zzi’s pro of w as ab out 50 p ages, but it w as dr amatically s impliﬁed a y ear later, wh en Razb orov [30] disco v ered a 3-page pro of. Then in 2009, Bra v erman [13] ga ve a breakthrou gh pro of of the f ull Linial-Nisan Conjecture. Theorem 23 (Bra verman’s T he orem [13]) L et f : { 0 , 1 } n → { 0 , 1 } b e c ompute d by an AC 0 cir cuit of size S and depth d , and let D b e a  log S ε  7 d 2 -wise indep e ndent distribution over { 0 , 1 } n . Then for al l suﬃcie ntly lar ge S ,     Pr x ∼D [ f ( x )] − Pr x ∼U [ f ( x )]     ≤ ε. W e conjecture a mo dest-seeming extension of Bra v erman’s Th eorem, wh ic h sa ys (inf orm ally) that almost k -wise in dep end en t distr ib utions fo ol AC 0 as wel l. Conjecture 24 (Generalized Linial-Nisan or GLN Conjecture) L et D b e a 1 /n Ω(1) -almost n Ω(1) -wise i ndep endent distribution over { 0 , 1 } n , and let f : { 0 , 1 } n → { 0 , 1 } b e c ompute d by an A C 0 cir cuit of size 2 n o (1) and depth O (1) . Then     Pr x ∼D [ f ( x )] − Pr x ∼U [ f ( x )]     = o (1) . By the usual corresp ondence b et wee n A C 0 and PH , the GLN Conjecture immediately implies the follo w in g counte rpart of Lemma 20. Supp ose a pr ob ability distribution D over or acle strings is 1 /t ( n ) -almost p oly ( n ) -wise indep endent, for some sup erp olynomial function t . Then no PH machine c an distin- guish D fr om the uniform distribution U with non-ne gligible bias. And thus w e get the follo w ing imp lication: Theorem 25 Assuming the GLN Conje ctur e, ther e e xi sts an or acle A r e lative to which BQP A 6⊂ PH A . Pro of. Th e pro of is th e same as th at of Theorem 21; the only diﬀerence is that the GLN Conjecture no w pla ys the r ole of Lemma 20. Lik ewise: 30 Theorem 26 Assuming the GLN Conje ctur e for the sp e c ial c ase of depth- 2 cir cuits (i.e. , DN F formulas), ther e exists an or acle A r elative to which B QP A 6⊂ AM A . Pro of. Just like in Th eorem 21, deﬁn e an oracle A and an asso ciated language L using the F ourier Checking p roblem. Then L ∈ BQP A , with p r obabilit y 1 o v er the choic es made in constructing A . On the other hand, supp ose L ∈ A M A with pr obabilit y 1 o v er A . T hen we claim that F ou rier Checking can also b e solv ed by a family of DNF formulas { ϕ n } n ≥ 1 of size 2 poly ( n ) :     Pr h f ,g i∼F [ ϕ n ( f , g ) ] − Pr h f ,g i∼U [ ϕ n ( f , g ) ]     = Ω (1) . But since F is O  k 2 / 2 n/ 2  -almost k -wise indep en d en t (b y Theorem 19), suc h a family ϕ n w ould violate the d epth-2 case of the GLN Conj ecture. W e now pro ve the claim. F or simplicit y , ﬁx an input length n , and let A refer to a single instance h f , g i of F ou rier Che cking . 11 Let P b e an AM proto col that su ccessfully distinguish es the forrelated distribution F ov er h f , g i p airs from the un iform distribution U . W e can assume without loss of generalit y that P is public- c oin [23]. In other words, Arth u r ﬁ rst sends a rand om c hallenge r ∈ { 0 , 1 } poly ( n ) to Merlin, then Merlin r esp ond s with a witness w ∈ { 0 , 1 } poly ( n ) , then Arth ur runs a deterministic p olynomial-time ve riﬁcation pro cedure V A ( r , w ) to decide whether to accept. By th e assump tion that P succeeds,     Pr A ∼D ,r  ∃ w : V A ( r , w )  − Pr A ∼D ,r  ∃ w : V A ( r , w )      = Ω (1) . So by Y ao’s pr inciple, there exists a ﬁxe d c hallenge r ∗ suc h that     Pr A ∼D  ∃ w : V A ( r ∗ , w )  − Pr A ∼D  ∃ w : V A ( r ∗ , w )      = Ω (1) . No w let Q A,w b e the set of all quer ies that V A ( r ∗ , w ) mak es to A , and let C A,w ( A ′ ) b e a term (i.e., a conjun ction of 1’s and 0’s) that r eturns T R UE if and only if A ′ agrees with A on all queries in Q A,w . T hen we can assume without loss of generalit y that C w := C A,w dep end s only on w , not on A —since Merlin can simp ly tel l Arth u r what queries V is going to mak e and what their outcomes will b e, and Arthur can r eject if Merlin is lying. Let W b e th e set of all w itnesses w suc h that Arth ur accepts if C w ( A ) returns TR UE. Consider the DNF formula ϕ ( A ) := _ w ∈ W C w ( A ) , whic h expresses that there exists a w causing V A ( r ∗ , w ) to acce pt. Then ϕ con tains at m ost 2 poly ( n ) terms with p oly ( n ) literals eac h, and     Pr A ∼D [ ϕ ( A )] − Pr A ∼D [ ϕ ( A )]     = Ω (1) . 11 It is straigh tforwa rd to generalize to th e case where Art hur can query oth er instances, b esides the one he is trying to solve. 31 As a side note, it is conceiv able that one could pro v e Pr x ∼D [ ϕ ( x )] − Pr x ∼U [ ϕ ( x )] = o (1 ) for ev ery almost k -wise in dep end en t distrib u tion D and sm all CNF form ula ϕ , w ithout getting the same resu lt for DNF form ulas (or vice v ersa). How ev er, since BQP is closed un der complemen t, ev en suc h an asymmetric result would imply an oracle A relativ e to wh ic h B QP A 6⊂ AM A . If the GLN Conjecture holds, then we can also “scale down by an exp onential, ” to obtain an unr elativize d decision problem that is solv able in quan tu m logarithmic time but not in A C 0 . Theorem 27 Assuming the GLN Conje ctur e, ther e exists a pr omise pr oblem in BQLOGTIME that is not in AC 0 . Pro of. In our pr omise p roblem Π = (Π YES , Π NO ), the inpu ts (of size M = 2 n +1 ) will enco de pairs of Bo olean fu nctions f , g : { 0 , 1 } n → {− 1 , 1 } , su c h that p ( f , g ) := 1 N 3   X x,y ∈{ 0 , 1 } n f ( x ) ( − 1) x · y g ( y )   2 is either at least 0 . 05 or at most 0 . 01. The problem is to accept in the former case and r eject in the latter case. Using the algorithm FC-ALG from Section 3.2, it is immediate that Π ∈ BQLOGTIME . On the other hand, supp ose Π ∈ A C 0 . Then w e get a nonuniform circuit family { C n } n , of depth O (1) and size p oly ( M ) = 2 O ( n ) , that solv es F o urier Checking on all pairs h f , g i suc h that (i) p ( f , g ) ≤ 0 . 01 or (ii) p ( f , g ) ≥ 0 . 05. By Corollary 10 , the class (i) includes the o ve rwhelming ma jorit y of h f , g i ’s d ra wn f rom the u niform distribu tion U , while the class (ii) in clud es a constant fraction of h f , g i ’s drawn fr om the forr elated distribution F . Therefore, w e actually obtain an AC 0 circuit family that distinguishes U from F w ith constan t bias. But this con tradicts Theorem 19 together with th e GLN Conjecture. 6.1 Low-F at Polynomials Giv en that the GLN Conj ecture would ha v e su c h remark able imp lications f or quantum complexit y theory , the question arises of how w e can go ab out p ro vin g it. As w e are indebted to Louay Bazzi for p ointing out to u s, the GLN Con j ecture is e qu ivalent to th e follo wing conjecture, ab out appro ximating AC 0 functions by low-deg ree p olynomials. Conjecture 28 (Lo w- F at Sandwich Conjecture) F or every function f : { 0 , 1 } n → { 0 , 1 } c om- putable by an AC 0 cir cuit, ther e exist p olynomials p ℓ , p u : R n → R of de gr e e k = n o (1) that satisfy the fol lowing thr e e c onditions. (i) Sa ndwiching: p ℓ ( x ) ≤ f ( x ) ≤ p u ( x ) for al l x ∈ { 0 , 1 } n . (ii) L 1 -Appr oximation: E x ∼U [ p u ( x ) − p ℓ ( x )] = o (1) . (iii) L ow-F at: p ℓ ( x ) and p u ( x ) c an b e written as line ar c ombinations of terms, p ℓ ( x ) = P C α C C ( x ) and p u ( x ) = P C β C C ( x ) r esp e ctively, such that P C | α C | 2 −| C | = n o (1) and P C | β C | 2 −| C | = n o (1) . (Her e a term is a pr o duct of liter als of the form x i and 1 − x i .) 32 If we tak e out condition (iii), then Conjecture 28 b ecomes equiv alen t to the original Linial-Nisan Conjecture (see Bazzi [7] for a pro of ). And indeed, all progress so far on “Linial-Nisan problems” has crucially relied on this connection with p olynomials. Bazzi [7] and Razb oro v [30] p r o ved the depth-2 case of the LN Conjecture by constr u cting lo w-degree, appr oximati ng, sandwic hing p olynomials for ev ery DNF, while Brav erman [13] pr ov ed the full LN Conjecture b y constructing suc h p olynomials for every AC 0 circuit. 12 Giv en this history , p ro vin g Conjecture 28 wo uld seem lik e the “ob vious” app roac h to pr o ving the GLN Conjecture. Belo w w e pro ve one direction of the equiv alence: that to p ro ve the GLN Conjecture, it su ﬃces to constru ct lo w-fat sandwiching p olynomials for every AC 0 circuit. The other direction—that the GLN Conjecture imp lies Conjecture 28, and hence, there is no loss of generalit y in wo rking with p olynomials instead of p robabilit y distribu tions—follo ws from a linear p rogramming du alit y calculatio n that w e omit. Theorem 29 The L ow-F at Sandwich Conje ctur e implies the GLN Conje ctur e. Pro of. Giv en an AC 0 function f , let p ℓ , p u b e the lo w-fat sandwiching p olynomials of degree k that are guarantee d b y Conjecture 28. Also, let D b e an ε -almost k -wise indep endent distribution o ve r { 0 , 1 } n , for some ε = 1 /n Ω(1) . Th en Pr x ∼D [ f ( x )] − Pr x ∼U [ f ( x )] ≤ E D [ p u ] − E U [ p ℓ ] = X C β C E D [ C ] − E U [ p ℓ ] ≤ X C β C + | β C | ε 2 | C | − E U [ p ℓ ] = E U [ p u − p ℓ ] + ε X C | β C | 2 | C | = o (1) + n o (1) n Ω(1) = o (1) . Lik ewise, Pr x ∼U [ f ( x )] − Pr x ∼D [ f ( x )] ≤ E U [ p u ] − E D [ p ℓ ] = E U [ p u ] − X C α C E D [ C ] ≤ E U [ p u ] − X C α C − | α C | ε 2 | C | = E U [ p u − p ℓ ] + ε X C | α C | 2 | C | = o (1) . 12 Strictly sp eaking, Bra verma n constru ct ed approximating p olynomials with slightly diﬀerent (though still suﬃ- cien t) prop erties. W e kn o w from Bazzi [7] t h at it must b e p ossible to get sandwic hing p olynomials as w ell. 33 7 Discussion W e no w tak e a step bac k, and use our r esults to add r ess s ome conceptual questions ab out the relativized BQP versus PH question, the GLN Conj ecture, and what m akes them so diﬃcu lt. The ﬁrst question is an ob vious one. Complexit y theorists ha ve kno wn for d ecades how to pro v e constant-depth circuit lo w er b ounds, and how to use those lo wer b ounds to giv e oracles A relativ e to wh ic h (for example) PP A 6⊂ PH A and ⊕ P A 6⊂ PH A . So why s h ould it b e so m uc h harder to giv e an A relativ e to whic h BQP A 6⊂ PH A ? What mak es this AC 0 lo wer b ound diﬀerent from other AC 0 lo wer b ound s? The answer seems to b e that, while we ha ve p o werful tec hniqu es for pro ving th at a function f is not in A C 0 , al l of those te chniques, in one way or another, involve ar gu ing that f is not appr oximate d by a low-de gr e e p olynomial. T he Razb orov-Smolensky technique [29, 35] argues this explicitly , while even the random r estriction tec hn ique [16, 39, 36] argues it “implicitly ,” as shown b y Lin ial, Mansou r , and Nisan [27]. And this is a problem, if f is also computed b y an eﬃcient quan tum algorithm. F or Beals et al. [8] pr ov ed the follo wing in 1998: Lemma 30 ([8]) Su pp ose a quantum algorithm Q makes T queries to a Bo ole an input X ∈ { 0 , 1 } N . Then Q ’s ac c eptanc e pr ob ability is a r e al multiline ar p olynomial p ( X ) , of de gr e e at most 2 T . In other words, if a fu nction f is in BQP , then for that ve ry reason, f has a lo w -degree appro x- imating p olynomial! As an example, we already sa w th at the follo win g p olynomial p , of degree 4, successfully distinguish es the forrelated d istribution F from the uniform distribu tion U : p ( f , g ) := 1 N 3   X x,y ∈{ 0 , 1 } n f ( x ) ( − 1) x · y g ( y )   2 . (2) Therefore, w e cannot hop e to prov e a low er b ound for F ou rier Checking , by an y argument th at w ould also imply that suc h a p cannot exist. This br ings us to a second question. If (i) ev ery kno wn tec h n ique for proving f / ∈ A C 0 in volv es sh o win g th at f is not appro ximated by a lo w -d egree p olynomial, but (ii) ev ery function f with lo w quan tu m query complexit y is appro ximated by a lo w-degree p oly- nomial, do es that mean there is no h op e of s olving the relativized BQP v ersus PH problem u sing p olynomial- based tec hn iques? W e b eliev e the answe r is n o. Th e essen tial p oint here is that an AC 0 function can b e appro xi- mated by d iﬀeren t kinds of lo w -degree p olynomials. F or example, Linial, Mansour, and Nisan [27] sho wed that, if f : { 0 , 1 } n → { 0 , 1 } is in AC 0 , then there exists a real p olynomial p : R n → R , of degree p olylog n , s u c h that E x ∈{ 0 , 1 } n h ( p ( x ) − f ( x ) ) 2 i = o (1) . 34 By comparison, Razb oro v [29] and Smolensky [35] show ed th at if f ∈ AC 0 , then there exists a p olynomial p : F n → F o v er any ﬁeld F (ﬁn ite or inﬁ n ite), of degree p olylog N , suc h that Pr x ∈{ 0 , 1 } n [ p ( x ) 6 = f ( x )] = o (1) . F u r thermore, to sh o w that f / ∈ AC 0 , it suﬃces to sho w that f is not appro x im ated by a low- degree p olynomial in any one of these sens es. F or example, even though the P ar ity function has degree 1 o ve r the ﬁ nite ﬁeld F 2 , Razb orov and Smolensky sh o wed that ov er other ﬁelds (such as F 3 ), an y degree- o ( √ n ) p olynomial disagrees with P arity on a large fraction of inputs—and that is enough to imply that P arity / ∈ AC 0 . In other w ords, we simply need to ﬁ nd a typ e of p olynomial appr o ximation th at w orks for A C 0 circuits, bu t do es not w ork for the Fourier Checking problem. If true, Conjecture 28 (the Lo w-F at Sandw ich Conj ecture) pro vides exactly suc h a type of appro ximation. But this raises another question: what is the s igniﬁcance of the “lo w-fat” requiremen t in Con- jecture 28? Wh y , of all things, do we wa n t our app ro ximating p olynomial p to b e expressible as a linear com bination of terms, p ( x ) = P C α C C ( x ), such that P C | α C | 2 −| C | = n o (1) ? The answer take s u s to the heart of wh at an oracle separation b etw een BQP and PH wo uld ha ve to accomplish. Notice that, although the p olynomial p from equation (2) solve d the Fourier Checking problem, it did so only by c anc el ling massive numb ers of p ositive and ne gative terms, then r epresen ting the ans w er by the tiny residue left o ver. Not coinciden tally , th is sort of cancella - tion is a cen tral feature of quan tum algorithms. By con trast, Th eorem 29 essentiall y sa ys that, if a p olynomial p do es not in volv e su c h massive cancellations, but is instead m ore “conserv ativ e” an d “reasonable” (lik e the p olynomials that arise fr om classical decision trees), then p cannot distin- guish almost k -wise ind ep end en t distributions from the uniform distribution, and therefore cannot solv e F ou rier Checking . If Conjecture 28 h olds, then ev ery small-depth circuit can b e appr o x- imated, not just b y an y lo w-degree p olynomial, but b y a “conserv ativ e,” “reasonable” lo w-degree p olynomial—one with a b ound on the coeﬃcients that prev en ts massiv e cancellatio ns. Th is w ould pro v e th at F o urier Checking h as no small constant-depth circuits, and h ence that there exists an oracle separating BQP from PH . This brin gs us to the fourth and ﬁ nal question: ho w might one pro v e Conjecture 28? In particular, is it p ossible that s ome tr ivial mo diﬁcation of Brav erman’s pr o of [13] would giv e lo w-fat sandwic hing p olynomials, thereb y establishing the GLN Conjecture? While we cannot rule this out, w e b eliev e that the answ er is no. F or examining Bra v erman’s pro of, we ﬁnd th at it com b in es tw o kinds of p olynomial appr o ximations of A C 0 circuits: that of Linial-Mansour-Nisan [27], and that of Razb oro v [29] and Smolensky [35 ]. Un f ortunately , neither LMN nor R azb or ov-Smolensky gives anything like the c ontr ol over the appr oximating p olynomial’s c o eﬃcie nts that Conje ctur e 28 demands. LMN simply tak es the F our ier transform of an AC 0 function and deletes th e h igh-order coeﬃcients; w hile R azb oro v-Smolensky appro x im ates eac h OR gate b y a pr o duct of r andomly-c h osen linear fun ctions. Both tec hniqu es pr o duce approxima ting p olynomials with a huge num b er of monomials, and n o r easonable b ound on their coeﬃcients. While it is conceiv able that those p olynomials satisfy the lo w-fat condition an ywa y—b ecause of some non-obvio us representat ion as a linear com bination of terms—certainly neither LMN nor Razb oro v-S molensky giv es any id ea what that representat ion w ould lo ok like. T h us, w e susp ect that, to get the d esired con trol o ver the co eﬃcien ts, one will n eed more “constructiv e” p r o ofs of b oth the LMN and R azb oro v-Smolensky theorems. Suc h pro ofs w ould lik ely b e of great inte rest to circuit complexit y and computational learnin g theory for indep endent reasons. 35 8 Op en Problems First, of cours e, prov e the GLN Conjecture, or pro ve the existence of an oracle A relativ e to w hic h BQP A 6⊂ PH A b y some other means. A natural ﬁrst s tep w ould b e to p ro ve th e GLN Conjecture for the sp ecial case of DNFs: as shown in Theorem 26, this would imply an oracle A relativ e to whic h BQP A 6⊂ AM A . W e ha ve oﬀered a $200 pr ize for the PH case and a $100 pr ize for th e AM case. 13 Second, it would b e of in terest to pro v e the GLN Conj ecture for classes of f unctions w eak er than (or incomparable with) DNFs: for example, monotone DNFs, read-once formulas, and read- k -times form ulas. Third, can we giv e an example of a Bo olean f unction f : { 0 , 1 } n → { − 1 , 1 } that is w ell- appro ximated b y a lo w-degree p olynomial, but not by a lo w-degree lo w-fat p olynomial? Here is a more concrete version of the chall enge: let k f − p k := E x ∈{ 0 , 1 } n h ( f ( x ) − p ( x )) 2 i . Then ﬁnd a Boolean fun ction f f or whic h (i) there exists a degree- n o (1) p olynomial p : R n → R such th at k f − p k = o (1), but (ii) there do es not exist a degree- n o (1) p olynomial q : R n → R suc h that k f − q k = o (1) and q can b e written as a linear com bination of terms, q ( x ) = P C α C C ( x ), with P C | α C | 2 −| C | = n o (1) . F our th , can w e give an oracle relativ e to wh ic h BQP 6⊂ IP ? What ab ou t an oracle relativ e to whic h BQP 6 = IP BQP , where IP BQP is the class of problems that admit an in teractiv e proto col with a BPP veriﬁer and a BQP pro ver? 14 Fifth, what other implications do es the GLN Conjecture ha ve? If we assu me it, can w e ad d ress other longstanding op en questions in quantum complexit y th eory , s uc h as those d iscussed in Section 1.1? F or example, can we giv e an oracle relativ e to whic h NP ⊆ BQP but PH 6⊂ BQP , or an oracle relativ e to w h ic h NP ⊆ BQP and PH is inﬁ nite? Sixth, ho w m u ch can we sa y ab out th e BQ P ve rsus PH question in the u nrelativized w orld? As one concrete c hallenge, can we ﬁnd a non trivial w ay to “realize ” the F ou rier Checking oracle (in other words, an explicit compu tational problem th at is solv able using Fourier Chec king )? Sev enth, ho w far can th e gap b et wee n the s uccess probabilities of FBQP and FBPP PH algorithms b e impro ved? T heorem 15 ga v e a r elation f or wh ic h a quantum algorithm su cceeds with probabilit y 1 − c − n , whereas an y FBPP PH algorithm succeeds with probabilit y at most 0 . 99. By c h anging the success criterion for Fourier Fishing —basically , b y requirin g the classical algorithm to output z 1 , . . . , z n suc h that b f 1 ( z 1 ) 2 , . . . , b f n ( z n ) 2 are distr ibuted “almost exactly as they w ould b e in the quan tum algorithm”—one can improv e the 0 . 99 to 1 / 2 + ε for any ε > 0. How ev er, impr o ving the constan t further might require a direct pro du ct th eorem f or A C 0 circuits solving F our ier Fishing . 13 See http://scottaaronson.com/ blog/?p=381 14 If w e let the veriﬁer transmit un entangle d q ubits to the prov er, then the resulting class IP | θ i BQP actually equals BQP , as recently sho wn by Broadb ent, Fitzsimons, and Kasheﬁ [14] (see also Aharono v, Ben-Or, and Eban [4]). It is not known wheth er this IP | θ i BQP = BQP result relativizes; we conjecture that it do es not. 36 9 Ac kno wledgmen ts I thank Lou ay Bazzi for r eform u lating the GLN Conjecture as the Lo w-F at Sand wic h Conjecture; and Andy Druck er, Lance F ortno w, and Sash a Razb oro v for helpfu l discussions. References [1] S. Aaronson. Quan tum lo w er b ound for the collision prob lem. In Pr o c. ACM STOC , pages 635–6 42, 2002. quant- ph/0111102 . [2] S. Aaronson. Q u an tum lo we r b ound f or recursive Fourier samp ling. 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