QMA variants with polynomially many provers

QMA variants with po l y nomially many provers Sevag Gharibian ∗ Jamie Sikora † Sarvagya Upadhyay ‡ September 3, 2012 Abstract W e study three var ia nts of multi-pro ver quantum Merlin-Arthur proo f systems. W e first show that the class of pr oblems that can be effi ciently verified usin g polynomially many quan- tum pro ofs, each of logarithmic-size, is exactly MQA (also known as QCMA), the class of prob- lems which can be efficiently verified via a classica l proof and a quantum verifier . W e then study the class BellQMA ( poly ) , characteriz ed by a verifier who first a pplies unentangled, non- adaptive measurements to each of the polynomially many proofs, followed by a n arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. W e show that if the number of outcomes per nonadap tive measurement is a polynomially-bounded f unc- tion, then the expressive power o f the proof system is exactly QMA. Finally , we study a class equivalent to QMA( m ), denoted S e pQMA ( m ) , where the verifier ’s measurement operator cor- responding to outcome accept is a fully separable operator a cross the m quantum proofs. Using cone pr ogramming duality , we give a n alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633 –642 (2 010) ] that sho ws a perfect para llel repetition theorem for Se pQMA ( m ) for any m . 1 Introduction and summary of results The study of classical pr oof s ystems has yielde d some of the greatest achievements in the oretica l computer science, from the Cook-Levin theorem [Coo71, Lev73], which formally ushered in the age of NP verifica tion s ystems and the no w ubiquitous notion of NP-har dness, to the more modern PCP the o rem [AS98, ALM + 98], which has led to significant advancements in our und erstanding of hardness of approximation. A natural generalization of t h e class NP, o r more accurately its probab ilistic cousin Merlin-Arthur (MA), to t he quantu m se tting is the class quantum Me r lin- Arthur (QMA) [KSV 02 ], where a computationally powe r ful but untrustwort hy prover , Me rlin, sends a quantum pr oof to convinc e an efficient quantum verifier , Arthur , t h at a g iven input string x ∈ { 0, 1 } n is a YES-instance for a s pecified promise problem. More s pecifically , a Q MA proof sys tem for a given promise probl em A is characterized by the following properties (see S ection 2.1 for formal d efinitions): • For every YES-instance x of A , there exists a p olynomial-size q u antum proof which can convince Art hur of this fact with high probabi lity , with the smallest such success p robability over all YE S-instances called t he completeness of the protocol. ∗ Department of Computer Science, University of Illinois. Email: sggharib@cs.u waterloo.ca . † Laboratoire d ’ Informatique Algorithmique: Fondements et Applications, Universit ´ e Paris Diderot. Email: jwjsikor@uwat erloo.ca . ‡ Centre for Quantum T echnologies, National University of Singapore. Email: sarvagy a@nus.edu.sg . 1 • For ever y NO-instance x o f A and for any purporte d quantum proof, A rthur rejects w ith high probabili ty , with the maximum s uccess probabili ty over all NO-instances called the soundness of th e protocol. It is easy to see that QMA proof syste ms ar e at least as powerful as NP or MA, since th e abil- ity to process and exchange quantum information does not prevent A r t hur from choosing to act classical ly . Much atte ntion has been devote d to QMA over recent ye ars. W e now have a number of prob- lems which ar e complete for QMA (see e. g. [Bra06, Liu06, BS07, LCV07, SV09, JGL10, WMN10, Ros11]), with the quantum analogue of classical constraint s atisfaction, the phys icall y motiva ted k -local H amiltonian problem [KSV02, KR03, KKR 06 , OT08, AGIK09], being the canonical QMA- complete pr oblem. In analogy with NP-complete pr oblems, it is temp t ing to think of QMA- complete problems as hard even fo r a quantum compute r to so lve, though this is somew hat of a misno mer as even N P -complet e problems are gene rally believed to be intractab le for quantum computers. QMA is an extremely robust complexity class that satisfies strong error- reduction properties, and using these properties one can, e. g., give a very eleg ant and simple proof that MA ⊆ QMA ⊆ PP (the first containment follows trivially from the d efinition) [MW05]. Howeve r , there still remai n important open questions — for example, despite the fact that MA is contained in the polyno mial hierarchy (PH) [AB09], we do not even know whe ther BQP ⊆ PH. An appr oach for unde r s tanding a complexity class is to con s ider how introducing variations to its definition changes its properties. In t his paper , we thus ask : How does allowing multiple unen- tangled pr overs affect the expr essive power of QMA ? In particular , we ar e interested in variants of the class QMA ( poly ) , a.k.a. quantum Merlin-Arthur proof sy stems with polynomially many Merlins, where the verifier receives a polynomial number of quantum proofs, which are p romi sed to be unentangled with each other . N ote that t he classical vers ion of t his class collapses trivially to MA, as the se t of potential strateg ies o f a single Merlin and the set o f potential strategies of multiple Merlins coincide. This logic fails, howeve r , in the quantum case, as a single Merlin simulating t he action of multiple Merlins can try to cheat by entangling t he multiple proofs. Despite much e f fort, very little is known (more details und er Prev ious Work below) about the structural properties of QMA ( p oly ) , except for the obvious containments QMA ⊆ QMA ( poly ) ⊆ NEXP. Our results: W e show th e following three results regar ding variants of QMA ( po ly ) . 1. A complete characteriz ation in the logarithmic-siz e mes sage setting. Let QMA log ( poly ) de- note the restriction o f the class QMA ( po ly ) to the set t ing where each prover ’s proof is at most a logari thmic number of qu antum bits, or qubits . W e show (for MQA defined below): Theorem 1.1. QMA log ( poly ) = MQA . Here, MQA, also known as QCMA in t he literature [AN02, JW06, Aar06, AK07, B e i08, ABOBS08, WY08] (the name MQA was s ugges ted by W atrous [W at09]), is define d as QMA except Merlin’s proof is a polynomial-size classical string. Theorem 1.1 says that if e ach prover is restricted to sending short quantum proofs, then one can no t o nly d o away with multi ple provers, but also o f the need for qu antum proofs altogethe r . 2. T owards a non -trivial upp er bou n d on BellQMA ( po ly ) . Another approach to study ing t he question of w h e ther QMA = QMA(poly) is to unde r s tand the properties of restricted versions of QMA ( p oly ) , and this is precisely where t h e class BellQMA(poly) comes into play . BellQMA ( poly ) 2 is defined [Bra08, A BD + 09, CD10] analogou s ly t o QMA ( poly ) , e xcept that be fore applying his quantum verification circ uit to the polynomially many unentangled qu antum pr oofs, Arthur must measure each proof using a nonadaptive and unentangled (acr oss all proofs) measurement (we call this Stage 1 of t h e verification). He the n feeds the resulting classi cal out comes induced by thes e measurements into h is arbitrary effici ent q u antum cir cuit (we call this S tage 2 ). This quantum cir cuit implements a t wo-outcome measurement op eration corresponding t o outcomes accept and r eject . The significance of B ellQMA ( p o ly ) here is th at if QMA 6 = BellQMA ( poly ) , t h e n it follows that QMA 6 = QMA ( po ly ) , since QMA ⊆ BellQMA ( po ly ) ⊆ QMA ( poly ) . T o this end, Brand ˜ ao has shown the ne gative result that for consta nt m , QMA = BellQMA ( m ) [Bra08]. Where the class B ellQMA ( po ly ) lies, however , r emains ope n. Fo r example, the techniques used to show QMA ( 2 ) = QMA ( poly ) [HM10] do not straightfo r w ar dly extend to s how the analogous result BellQMA ( 2 ) = B e llQMA ( po ly ) as the y requir e e ntangled measurements (i.e. SW A P te s t mea- surements) across multiple proofs, which violate the definition of B ellQMA. T o make progr ess on BellQMA ( p oly ) , we introduce t he class BellQMA [ r , m ] , which is d efined as Be llQMA ( m ) with m provers and the additional restriction that in Stage 1 above, the num- ber of outcomes p er proof in Arthur ’s no n adapt ive measurements is u p per bounded by r . Our contribution is the following: Theorem 1.2. For any polynomial ly bounded functions r , m : N → N , we have the containment BellQMA [ r , m ] ⊆ QMA (wher e the containment holds with equality when r ≥ 2 ) . In other words, BellQMA ( poly ) cannot be used to show QMA 6 = QMA ( p o ly ) if the verifier in the BellQMA ( p oly ) protocol is restricted to have a polynomially bounde d number of measurement outcomes pe r p roof in S tage 1. W e remark that, in general, the number of s uch measurement out- comes can be e x p onential in the input length — t he restriction t h at r be a polyno mially bounded function is crucial for the proof of Theorem 1.2. For this reason, o u r r esult complements , rather than subsumes Brand ˜ a o’s result [Bra08]. In other words, in our no tation, Br and ˜ ao has s hown t hat BellQMA [ ex p , const ] = QMA, and we show Be llQMA [ poly, poly ] = QMA. Note t hat we allow the second s tage of the verification procedur e above to be qu antum , as per the definition sugg ested by Chen and Dr ucker [CD1 0], as opposed to classical , as studied by Brand ˜ ao [Bra08]. The conclusion of Theorem 1.2 holds even if t he s econd stage o f verification is completely classical. Finally , it is worth no ting that by combining Theorems 1.1 and 1.2, we conclude that in the setting of B ellQMA ( p o ly ) , if MQA 6 = QMA, t hen having the Merlins sen d logarithmic-size proofs without any r estriction on the n u mber of local measurement out comes of Arth u r in Stage 1 has less expressive powe r than se n d ing polyno mial-siz e p roofs but restricting the number of outcomes, even t hough the number of me asu rement ou t comes in Stage 1 pe r Merlin in both cases is the same, i.e. polynomial in the input length. 3. Perfect parallel repetition for S epQMA ( m ) . A key ques tion in designing proof s ystems is how to improve the completene s s and soun d ness parameter s of a verification protocol without incr easing t h e required number o f rounds of communi cation. A natural approach for do ing s o is to repeat t he protocol multiple times in parallel. W ith QMA, however , t his raises the concern that Merlin might try to che at by entangling his proofs across thes e parallel runs. I f, though, perfect parall el r epetiti on holds, it means t h at for any input string x , if t he verification procedure V acc epts with probabi lity p ( | x | ) , then if we run V k times in parallel, the probabi lity of accepting in all k runs of V is precisely p ( | x | ) k . Note that we do not put any restriction on the quantum p roof, 3 which can be entangled acr oss the k e xecutions of the protocol. I n ot her words, if perfect parallel repetition holds, the re is no incentive for Merlin to che at — an ho nest proof which is a product state across all k runs achieves the maximum s uccess probabil ity . Our final contribution is an alternate pr oof of a perfect parallel r epetition t heorem for a class which is equivalent [HM10] t o QMA( m ), namely SepQMA ( m ) . T h e theorem was first pr oved in Harrow and Mo ntanar o [HM10] in conne ction with an error reduction technique for QMA ( poly ) . However , our p roof is significantly d iffer ent fr om t h e irs and uses the cone programmi ng char - acterization of QMA ( poly ) . Here SepQMA ( m ) is defined as QMA( m ) w ith the restriction that Arthur ’s measur ement o perator correspond ing to acceptance is a separabl e op erator across the m unentangled proofs. (Note that this does not imply that Arth u r ’s measurement o p erator corre- sponding to rejection is also sep arable.) W e show: Theorem 1.3 (se e [HM10 ] for al ternate pr oof) . The cla ss SepQMA ( m ) admits perf ect paralle l repeti - tion. Our alternate proof of Theo rem 1.3 is significant in t hat, to the best of our know led ge, it is the first u se o f d uality th e ory for a cone program other than a semidefinite program to establish a par- allel repetition result (note that cone programming generalizes semidefin ite programmi ng). W e rema rk that semidefinite pr ograms have been previously used to sho w perfect or strong parallel repetition t h e orems for various ot her models of (single o r two-prover) quantum inte ractive proof syste ms [CSUU08, KR T10, Gut09], and that the alternate proof of Theorem 1.3 of Harrow and Montanaro is not based on cone programming. Perfect paral lel repetition for SepQMA ( m ) in it- self is inter esting, as it has been used to show that error reduction is possible for QMA ( m ) proof syste ms [HM10]. Proof ideas and tools: The proof of our first result, The orem 1.1, is simple, and is an application of the facts that (1) quantum states of a logarithmi c number of qubits can be des cribed to within inverse expone ntial precision us ing a polynomial number of classical bits, and conversely that (2) given such a classical des cription, a log arithmic-size q u antum s tate can be efficiently p repar ed by a qu antu m circuit. Hence, roughly speaking, o ne can replace a polynomial number of logarithmic- size quantum p roofs with a single p o lynomial size classical proof, thereby avoiding the dange r of a cheating Merlin using ent ang leme nt. Although the proof is simple, one cannot hope for a bette r characterization using othe r t echniques because the reverse containment MQA ⊆ QMA log ( poly ) holds using similar ideas. More technically challenging is our second result, Theorem 1.2. T o show the cont ainment BellQMA [ po ly, poly ] ⊆ QMA (note t hat t h e reverse containment QMA ⊆ BellQMA [ p oly, poly ] is trivia l since QMA = BellQMA [ 2, 1 ] ), we demons trate a QMA pr otocol which s imulates an ar- bitrary BellQMA [ poly, po ly ] protocol using the fo llowing observation: Althoug h consolidating m quantum pr oofs into a single q u antum pr oof raises the pos sibility of cheating using entanglemen t , if A rthur is also sent an appropriate classical “consiste ncy-check” s tring, then a d ishonest Merlin can be caught with no n-negligible probabi lity . Specifically , in our QMA protocol, we ask a single Merlin t o se nd t h e m quantum proofs of the original BellQMA p rotocol (denoted by a single s tate | ψ i ), accompanied by a “consistency-check” string p which is a classical description of the probabili ty distributions obtained as the o utput of Stage 1. One can think of this as having the QMA verifier delegat e Stage 1 of t he BellQMA verification to Merlin. Arthur then performs a consistency check betw een | ψ i and p based on the premi se that if Merlin is honest, then p should arise from running Stage 1 of the original verification on | ψ i . If this che ck passes, then Arthur runs S tage 2 of the Be llQMA verification 4 on p . If Me rlin tries to cheat, howe ver , we sho w that the check d etects this with no n-negligible probab ility . N ote that the accuracy of the consistency check crucially uses the fact t hat there are at most polynomially many out comes t o check for each local me asu rement of Stage 1. Finally , our last result, The orem 1.3, is shown using du ality th e ory for a class of cone pro- grams that captures the s u ccess probabil ity of a QMA ( poly ) p rotocol. In particular , we phrase the maximum acceptance proba bility of a (pos sibly cheating) pr over for the two-fold r epetition of a SepQMA ( m ) verification p rotocol as a cone program. W e then demons t rate a feasible so lution for its dual yielding an upper bound on the maximum acceptance probabili ty . The o bjective value of t his d ual s olution is p reci sely t h e p roduct of the optimum values of t h e two instances of the SepQMA ( m ) verification protocols. W e conclude that one of the optimal s trategies of the p rovers is to be faithful in the following sense: Each pr over elects not to entangle his/her two quantum proofs for the t w o instances of the SepQMA ( m ) p rotocol and instead se n d s a te nsor product of optimal proofs for both the instances. Previous work. Th e expressive powe r of multiple Merlins was first s t udied by Kobayashi, Mat- sumoto and Y amakami [KMY03], who showed t hat QMA ( 2 ) = QMA ( po ly ) if and only if the class of QMA(2) protocols with complete ness c and s oundnes s s (with at least invers e polynomial gap) is exactly equal to QMA ( 2 ) protocols with completene ss 2/ 3 and so undness 1/ 3. A subst an- tial amount o f resear ch has s ince been de vot ed to unde rstanding the properties of multi-prover quantum Merlin-Arthur p roof sys tems. Re cently , Harrow and Mont anaro [HM10] de monstrated a product state test , wherein given two copies of a pur e quantum state o n multiple sy stems, the test distinguishe s betwee n t h e cases wh e n the quantum state is a fu lly product state acr oss all the sy stems or far fr om any such state. Using this t est, they answe red a few important qu estions regar ding QMA ( po ly ) . I n particular , t hey showed that QMA ( 2 ) = QMA ( poly ) and that e rror r eduction is po ssible for s uch proof syst ems. Prior to t heir result, the answers to both the questions were k nown to be affirmati ve assuming a weak version o f the Ad ditivity Con- jecture [ABD + 09]. One o f t he crucial properties of the product st ate tes t is that it can be converte d into a QMA ( 2 ) protocol, whe re Arthur ’s measu rement ope rator corresponding t o outcome accept is a sep arable o perator acr oss the two proofs. H arr ow and Mont anaro established a pe rfect paral- lel repetition theorem for s uch proof systems , a crucial s tep in o btaining exp onentially small error probab ilities. Blier and T app initiated t he stud y of logarithmic -size unentangled quantu m proofs [BT09]. They showed that two un e ntangled quantum proofs suffice to sho w that a 3-coloring of an input graph exists, implying that NP has succinct unentangled quantum proofs. A drawbac k of their protocol is that althoug h it has perfect completeness , its s oundnes s is only inverse polyno mially bou nded away from 1. Shortly after , Aar onson, Beigi, Drucker , Feff erman and S hor [AB D + 09] s howed that s atisfiabili ty of any 3-SA T formula of size n can be pr oven by e O ( √ n ) unentangled quantum proofs of O ( log n ) qubits with p erfect completenes s and constant sound ness (se e also [CD10]). In a subseque nt paper [Bei08], Beigi improved directly on Blier and T app’s result [BT09] by showing that by sacrificing perfect comp let eness, one can s how t hat NP has two log arithmic-siz e quantum proofs with a better gap between completene ss and soundness probabili ties than in [BT09]. V ery recently , Chiesa and Forbes showed a bette r complet eness and soundne ss gap of Ω  1 n 2  for t he Blier and T app protocol [CF11]. Also, Le Gall, Nakagawa, and N ish imura s howed that 3-SA T has a QMA log ( 2 ) proof syste m with complet eness 1 and sound ness 1 − Ω  1 n polylog ( n )  [GNN12]. 5 Finally , one of the open questions raised in R ef. [AB D + 09] concerns the powe r o f Arthur ’s verification procedur e. In particular , the p aper introduces two d if ferent classes of verification procedur es, B ellQMA and LOCCQMA verification. Rou g hly speaking, LOCCQMA verification corresponds to Arthu r applying a measur ement operation that can be implemented by Local Operations and Classical Communication (LOCC) (with respect t o t he partition induced by the multiple proofs). The authors raised the que stion of whe ther Be llQMA ( poly ) = QMA o r no t. Brand ˜ ao [Bra08] sh o wed t hat Be llQMA ( m ) is equal to QMA for cons tant m . I n a recent d evelop- ment, Brand ˜ ao, Christandl and Y ard [BCY11] showed that LOCCQMA ( m ) is equal to QMA for constant m . Organization o f this paper . W e be g in in Section 2 with background and notation, d efining rel- evant complexity classes in Section 2.1, and reviewing cone programming in Section 2.2. The- orems 1.1, 1.2, and 1.3 are proved in Sections 3, 4, and 5, respectively . W e conclude with op en problems in Section 6 . 2 Preliminaries and Notati on W e begin by setting our no tation, and s ubsequently r eview the background material required for t his p aper . First , t h e notation [ m ] indicates the se t { 1, . . . , m } , and | x | the length of a string x ∈ { 0, 1 } ∗ . W e let uppercase s cript lett ers X , Y , Z denote complex Euclidean spaces. W e de note the sets of linear , He rmitian, positive semidefinite, and de nsity operators acting on vecto r space X by L ( X ) , Herm ( X ) , Pos ( X ) , and D ( X ) , respectively . W e d enote the standard Hilbert-Schmidt inner product of operators A and B as h A , B i : = T r ( A ∗ B ) , where A ∗ denote s the adjoint o f A . The spectral and trace norms of an o p erator A are g iven by k A k ∞ : = max { k A u k : k u k = 1 } and k A k tr : = T r  √ A ∗ A  , respectively , where k u k denote s t he Euclidean norm of a vector u . These can be thought of as the largest singular value and the s um of singular values of A , r espectively . A useful lemma in this paper regar ding the trace norm is the following: Lemma 2.1 ([W at02]) . Let { ρ 1 . . . , ρ k } ⊂ D ( X ) and { σ 1 , . . . , σ k } ⊂ D ( X ) . Then     k O i = 1 ρ i − k O i = 1 σ i     tr ≤ k ∑ i = 1 k ρ i − σ i k tr . Next, we say a (po ssibly unnormalized) operator A ∈ Pos ( X 1 ⊗ · · · ⊗ X m ) is fully separabl e if it can be writt en as A = k ∑ i = 1 P 1 ( i ) ⊗ · · · ⊗ P m ( i ) where P j ( i ) ∈ P o s  X j  , for every j ∈ [ m ] and i ∈ [ k ] . The se t o f fully separable o perators is denote d Sep ( X 1 , X 2 , . . . , X m ) . This notation is he lpful in the context o f cone programming. In the setting o f quantum information, on e typically also has T r ( A ) = 1. T h e set of fully sep arable density ope rators is convex, compact, and has non-empty inte rior since it contains a ball around the normalized identity o perator [GB02, GB03, GB05]. W e use t he fact that any pure quantum state | ψ i ∈ C N can be d escribed approximately classi- cally using N · f ( N ) bits, for some function f : N → N . T he r esulting appr oximate d escription | ψ ′ i s atisfies k | ψ i − | ψ ′ i k ≤ N 2 − ( f ( N ) + 1 ) . W e also speak in te r ms o f quantum regi sters rather than quantum s t ates in the ne xt t wo sections . T o make the asso ciation precise, an n -qubit quantum register X is ass ociated with a vector s pace X = C 2 n and contains any eleme nt of D ( X ) . 6 Finally , moving to quantum o perations, t he notion of measurement use d in this paper is that of a Po s itive Operato r V alued Measure (POVM), g iven by a fin ite set of p o sitive semidefinite op- erators { Π 1 , . . . , Π r } ⊂ Pos ( X ) obeying r ∑ i = 1 Π i = 1 X . Regarding unitary operators, we use the fac t that any unitary operator acting on k qubits can be approxima ted within high p reci sion by a finite se t of one-qubit, tw o -qubit, and/or three-qubit unitary operators. Such a finite set is often referred to as an appr oxi mately universal set of quan- tum gates, and one such se t is comprise d of t h e T off oli, Hadamar d, and phase -s hift gate s. The Solovay-Kitaev th e orem implies that the action of an arbitrary unitary operator U on k qubits can be simulated by a composition e U of O ( 4 k poly ( log ( 1 / ǫ ) ) ) many univers al gates, such that    U − e U    ∞ ≤ ǫ [NC00]. 2.1 Relevant quantum complexity classes A p romise problem A = ( A yes , A no ) is a partition o f the se t { 0, 1 } ∗ into t hree disjoint subsets : the set A yes denote s the set of YES-instances of the problem, the set A no denote s the set of NO- instances o f t he problem, and the set { 0, 1 } ∗ \ ( A yes ∪ A no ) is the set of disallowed string s (we ar e pr omised the input does not fall into this last set). W e now d efine QMA ( m ) , o r QMA with m unentangled provers. Definition 2.2 (QMA ( m ) ) . Let p : N → N be a polynomially bounded function, and m : N → N a function. A pr omise problem A = ( A yes , A no ) is in the clas s QMA ( m ) if ther e exists a polynomia l-time generate d family of verification cir cuits Q = { Q n | n ∈ N } with the following prope rties: 1. Each Q n acts on n + p ( n ) input qu bits, and outputs one qu bit. 2. (Completeness) For every x ∈ A ye s , ther e exist m ( | x | ) quantum pro ofs | ψ 1 i , | ψ 2 i , . . . , | ψ m ( | x | ) i ∈ C 2 p ( | x | ) such that Pr [ Q | x | accepts ( x , | ψ 1 i ⊗ . . . ⊗ | ψ m ( | x | ) i ) ] ≥ 2/ 3. 3. (Soundness) For any x ∈ A no and any m ( | x | ) quantum pr oofs | ψ 1 i , | ψ 2 i , . . . , | ψ m ( | x | ) i ∈ C 2 p ( | x | ) , we have Pr [ Q | x | accepts ( x , | ψ 1 i ⊗ . . . ⊗ | ψ m ( | x | ) i ) ] ≤ 1/ 3. Furthermor e, the class QMA ( poly ) is defined as QMA ( poly ) = S m ∈ poly QMA ( m ) . W e remark that the cons tants 2/ 3 and 1 /3 can be replaced by any a , b ≥ 0, respectively , such that a − b ≥ 1 /poly ( n ) . T h is d oes not change the exp ressive power of the p roof syst em. All complexity classes considered in this paper are variants o f QMA ( m ) and satisfy the p rop- erties mentioned above in Definition 2.2. W e define t he following variants, which are relevant to this paper . 7 1. [QMA and MQA] Th e class QMA is simply QMA(1). If we r eplace t h e qu antum proofs in the d efinition o f QMA w ith a polyn o mial-siz e classical proof s tring, the correspond ing class is denote d MQA. 2. [SepQMA(poly)] T he class Se p QMA ( poly ) is a subclass of QMA ( poly ) , wherein Arthur ’s measurement operator corresponding to o utcome accept is a fully s eparable op e rator across the proofs. 3. [QMA log ( po ly ) ] The class QMA log ( poly ) is a s ubclass o f QMA ( poly ) , wherein each Merlin’s message to Arthur is O ( log ( | x | ) ) qubits in leng th. For clarity , w e give a formal d efinition of the variant of Be llQMA w e introduce, Be llQMA [ r , m ] . Definition 2.3 (BellQMA [ r , m ] ) . Let r , m : N → N be two functions. We say that a pro mise pro blem A = ( A yes , A no ) is in Be llQMA [ r , m ] if th er e exis ts a QMA ( m ) verification pr otocol in which Arthur is r estricte d to act as follows. 1. Arthur perfo rms a polynomia l-time qu antum computation on the input x and genera tes a descr iption of quantum circuits V 1 ( x ) , . . . , V m ( x ) , one for each of the m pr overs . 2. (Stage 1) Arthur simultaneously measures all m quantum pr oofs by applying V i ( x ) to the i -th quantum pr oof, wher e the action of V i ( x ) can be described by a unitary operato r fol lowed by measur ement in the standard basis. The label of the i -th measure ment outcome is stor ed as a classical string y i also identified as an element of [ r ( | x | ) ] . 3. (Stage 2) Arthur runs an efficient quantum verificatio n cir cuit on input x and measure ment outcomes ( y 1 , . . . , y m ) to decide w hether to accept or r eject. Note that the key distinction between BellQMA [ r , m ] and B ellQMA ( po ly ) is t hat the former has the number of me asurement out comes in Stage 1 of the protocol bounde d by r ( | x | ) , whe reas the latter may allow exponentially many possible outcomes. Throughout this paper , we us e the notation BellQMA [ p oly, poly ] to d enote BellQMA [ poly, poly ] : = [ r ∈ poly [ m ∈ poly BellQMA [ r , m ] . W e remark that, as in [CD10], our B e llQMA protocols ar e allowed to use a quan tu m verification cir cuit in Stage 2, whereas originally in references [Bra08, A BD + 09] only classical p rocessing of measurement ou tcomes { y i } was allowed in order to emulate the n o tion of a Bell experime nt p er- formed by Art h u r . W e again remark that Theorem 1.2 ho lds even if Arthu r is restricted to do classical processing on t he measurement o utcomes. 2.2 Cone programming W e now briefly revi ew basic notions in conic optimization (or cone p rogramming), which is a generalization of semidefinite optimization. W e say that a set K in an underlying Euclidean s pace is a cone if x ∈ K implies that λ x ∈ K for all λ > 0. A cone K is convex if x , y ∈ K implies that x + y ∈ K . Cone programs are concerned with optimizi ng a linear function over the inters ection of a convex cone and an affine space. It ge neralizes se veral well-studied models of optimization including semidefinite programming ( K = Pos ( X ) ) and linear programming ( K = R n + ). In this paper , we are primarily concerned with the con e of fully se parable operators Sep ( X 1 , X 2 , . . . , X m ) which recal l is a closed, convex cone with non-empty interior . 8 Associated with a cone K is its dual cone K ∗ defined as K ∗ = { S : h X , S i ≥ 0 for all X ∈ K } . A cone pr ogram associates the following 4-tuple ( C , b , A , K ) t o an opt imization p roblem described as: supremum: h C , X i subject to: A ( X ) = b , X ∈ K . Here A : Span ( K ) → R m is a linear transformation. No te that the inner product is defined as in the Euclidean space. Fo r instance, if the cone under cons ideration is the set of po sitive semidefinite or se parable operator s , then th e inner product is the standard Hilbert-Schmidt inner product over the space of Hermitian ope rators. W e say that the cone program is feasible if { X : A ( X ) = b } ∩ K is non-empty and strictly feasi ble if { X : A ( X ) = b } ∩ int ( K ) is non-empty , where int ( · ) d enotes the interior of a set. Cone programs come in primal-dual pairs: Primal problem (P) supremum: h C , X i subject to: A ( X ) = b , X ∈ K . Dual problem (D) infimum: h b , y i subject to: A ∗ ( y ) = C + S , S ∈ K ∗ . Here A ∗ is the adjoint of A . A convex cone K is closed if and only if K = K ∗ ∗ . In other words, the dual of the cone K ∗ is the o riginal cone K . Thus, if K is not closed w e need t o “order ” the primal-dual pairs since K 6 = K ∗ ∗ implying the dual of the dual p robl em is not equal to th e primal problem. Since the convex cone o f fully separable op erators is closed, ordering the primal-dual pairs is not an issu e in our case. Similar to linear pr ogramming and s emidefinite programming, cone pr ogramming has a r ich duality theo r y . Lemma 2.4 (W eak Duality) . If X is primal feasible and ( y , S ) is dual feasible then h b , y i − h C , X i = h X , S i ≥ 0. This result can be use d to s how up p er bound s on the value of the primal problem or lower bounds on the value of the dual problem. The re is also a no tion of str ong duality . W e say t hat str ong duality holds for a problem (P) if the optimal value of (P) e quals the o ptimal value o f (D) and (D) attains an o ptimal solution. Below we give a condition that guarantees strong du ality for (P). Theorem 2.5 (Strong Duality , V ersion 1) . If (P) is strictly feasible and the optimal value is bounded from above, then stro ng duality holds for (P) , i.e., (D) attains an optimal solution and the optimal values for (P) and (D) coincide. In this pape r , we ar e concerned with close d, convex cones with non-empty interior . Since the dual of t he dual p robl em is t h e primal problem when K is closed , we can use the following stronger vers ion o f st rong duality . Theorem 2.6 (Str ong Dua lity , V ersion 2) . Suppose K is a closed, convex cone. If (P) and (D) ar e both strictly fe asible then str ong duality holds for both problems, i.e., both pr oblems attain an optimal solution and the optimal values coincide. W e refer the r eader to the w o rk of T unc ¸ e l and W olkowicz [TW08] and the references therein for more d e tails o n cone programming du ality . 9 3 Equivalence of MQ A and QMA log ( poly ) W e now prove Theorem 1.1 which states that MQA = QMA log ( poly ) . W e first s h o w the direction MQA ⊆ QMA log ( poly ) . Le t A = ( A yes , A no ) be a pr omise pr oblem in MQA and let x ∈ { 0, 1 } n be the input string. Suppo s e the MQA prover s e nds an m -bit classical p roof to the verifier , for polynomially bounded m . Then the following simple QMA log ( m ) p rotocol ac hieves the des ir ed containment: QMA log ( m ) Protocol 1. Embed classical bits into qubits. E ach (unentangled) prover i ∈ [ m ] se nds a single qubit | ψ i i ∈ C 2 to Art hur . If the i -th prover is hone st, his/he r qubit is the computational basis state corresponding to t he i -t h bit of the classical MQA proof. 2. Make things classical again. Arthur measures all proofs in the computational basis, obtaining a classical string y ∈ { 0, 1 } m . 3. Run MQA verification. Arthur runs the MQA ver ification circuit on x and y and accepts if and only if acceptance occurs in the MQA ve r ification. The comp let eness property follows straightfo r w ar dly . The s oundness property is also easy to observe. Note that Arthur runs the MQA verification on a classical string y and hence he accepts the string with probabili ty at most 1 /3. T o sho w the reverse containment, let A = ( A yes , A no ) be a p romise problem in QMA log ( poly ) and let x ∈ { 0, 1 } n be the input string. Supp ose we have a QMA log ( m ) protocol for polynomially bounded m , where p rover i s ends a ⌈ c log n ⌉ -qubit s tate | ψ i i fo r s ome constant c > 0. Let r ( n ) = 2 ⌈ c log n ⌉ = O ( n c ) . The MQA protocol proceeds as follows: MQA Protocol 1. Describe proofs classic ally . The prover sends m classical registers repr esented by the t uple ( C 1 , C 2 , . . . , C m ) , e ach of length 2 n · r ( n ) to Ar t hur . If the prover is honest , register C i contains a classical des cription of t he i -t h quantum proof of the QMA log ( m ) p rotocol. 2. State preparation. Using the contents of register C i , fo r e very choice of i ∈ [ m ] , Art h u r pr e- pares the st ate | ψ i i by firs t dete rmining a unitary U i such that U i | 0 . . . 0 i = | ψ i i , and then implementing U i with high precision using a finite set of approxima tely univers al gates , ob- taining states | ψ ′ i i . 3. Run QMA log ( m ) verification. Arthur runs the QMA log ( m ) verification cir cuit on the state | ψ ′ 1 i ⊗ · · · ⊗ | ψ ′ m i and accepts if and only if acceptance occurs in th e QMA log ( m ) ver ification. Observe that each classical register C i is of s ize polynomial in n , implying the overall proof length is of polynomial size. In Step 1, the prover us e s n bits to repr esent the r eal and imaginary p arts of each of the polynomially many entities ( r ( n ) entries ) requir ed to describe each | ψ i . Let the unit vector de scribed by register C i be deno ted | ψ i i . In S tep 2, U i is e asily found as the u nitary that maps | 0 . . . 0 i to | ψ i i as the inverse of t he unitary that maps | ψ i i to | 0 . . . 0 i . Such a unitary can be easily decomposed into a pr oduct of polynomially many 2 × 2 rotations on an r ( n ) -dimensional real space and a d iagonal unitary as follows. Th e firs t step is to convert the vector | ψ i i into a 10 real vector by applying an appropriate diagonal unitary operator . The second step is to convert the r esulting r eal unit vector into | 0 . . . 0 i by shifting t he amplitudes of any st and ard basis othe r than | 0 . . . 0 i to | 0 . . . 0 i . Each of these unitary ope r ato rs can be implemented by a finite set of approxima tely universal gates (see Be rnstein and V azirani [BV97] for de tails). This step also incurs some error , which can be made exponent ially small. Since Steps 1 and 2 can be performed to within inverse e xponential error , we thus can ens u re k | ψ i i − | ψ ′ i i k ≤ ǫ for all i ∈ [ m ] and for inverse e xponential ǫ > 0. By Lemma 2.1, it follows that the overall precision error is at most m ǫ for polyno mial m , and thus the completene ss and soundne s s of the protocol ar e bound e d from below and above by (respectively) 2 3 − m ǫ and 1 3 + m ǫ . Alternatively , the containment QMA log ( poly ) ⊆ MQA can be s hown us ing a slightly differ - ent protocol 1 , w here Merlin sends classical des criptions of the quantum cir cuits that generate t he quantum proofs from | 0 . . . 0 i instead of classical d escriptions of the proofs. 4 Equivalence of Bel lQMA [ poly , poly ] and QM A W e no w s how T h e orem 1.2, i.e., that BellQMA [ r , m ] = QMA for polyno mially-bounded functions r and m . For notational convenience, let Π j ( i ) d enote Arthur ’s i -th POVM element in Stage 1 of the BellQMA verification protocol for the j -th p rover (i.e. ∑ r i = 1 Π j ( i ) = 1 ), whe re we assu me without loss of generality that the number of pos sible out comes is e xactly r for e ach p rover , and w h e re j ∈ [ m ] for m t he number of provers. W e proceed as follows. Let A = ( A yes , A no ) be a p romi se p robl em, and x be an input st ring of length n : = | x | . As mentioned in Section 1, the containment QMA ⊆ BellQMA [ poly, poly ] follows straightforwardly since QMA ⊆ BellQMA [ 2, 1 ] . For t he reverse containment, s uppose we have a B ellQMA [ r , m ] protocol for polynomially bound ed functions r , m : N → N w ith completeness 2/3 and so undness 1/3. W e show that th is protocol can be simulated by a QMA p rotocol whe re Merlin sends the following proof to Art hur . Merlin’s proof consists of two registers ( X , Y ) , which should be t h o ught o f as t he classical and quantum registers, respectively . Suppose o ptimal proofs for the BellQMA [ r , m ] protocol for input x are given by ρ j for j ∈ [ m ] . Then, in the qu antu m register Y , an hones t Merlin should sen d many copies o f the state ρ j . Specifically , Y is partitioned into m registers Y j , one fo r e ach original prover , and each Y j should contain k copies o f ρ j , for k a car efully chosen polynomial. In other wo rds, Y should contain the state [ ρ ⊗ k 1 ] Y 1 ⊗ · · · ⊗ [ ρ ⊗ k m ] Y m . W e further view each Y j as a block of registers ( Y 1 j , . . . , Y k j ) w here Y l j should contain the l -th copy of ρ j . In t he classical register X , an ho nest Merlin prepar es a quantum st ate in the computational basis, which intuitively corresponds to a bit string d escribing t h e m cla ssical probabil ity distribu- tions Arthur induces upon applying the measurement ope r ation corr esponding t o Stage 1 o f the BellQMA verification to each of the op timal p roofs ρ j , respectively . More formally , we partition X into m r registers X i j corresponding to each of the j ∈ [ m ] provers and i ∈ [ r ] POVM ou tcomes pe r prover . The content o f X i j should be p j ( i ) : =  Π j ( i ) , ρ j  , truncated to α bits of precision ( α polyno- miall y bounded ), such that ∑ r i = 1 p j ( i ) = 1. For example, if the j -th prover ’s p roof was t h e single qubit s t ate ρ j = | 0 i h 0 | , with Π j ( 1 ) = | 0 i h 0 | and Π j ( 2 ) = | 1 i h 1 | , then X j = ( 1, 0 ) . W e remark that X plays the role of the classical “consiste ncy check” string de scribed in Se ction 1. 1 This protocol was mentioned to us by Richard Cl eve. 11 Of cou rse, Merlin may elect to be dishon e st and choos e not to s end a p roof of t he above form to A r t hur by , e.g. , sending a quantum st ate which is e ntangled acr oss the registers ( X , Y ) . T o catch this, our QMA protocol is define d as follows: QMA Protocol 1. Merlin send s Art hur a quantum state in registers ( X , Y ) , for X and Y defined as above. 2. Force X to be class ical. Arthur measures register X in the computational basis and reads the measurement out come. This forces X t o essent ially be a classical register of bits, and d estroys any entanglemen t or correlations betwee n X and Y . 3. X sho uld contain probability distributions. Arth u r checks wheth e r the content of registers X j form a probabil ity distribution p j , i.e., that ∑ r i = 1 p j ( i ) = 1. Arthur rejects if this is not t he case. 4. Consistency check: C an the quantum st ates in Y reproduce the distr ibutions in X ? Art hur picks independe ntly and uniformly at random, an index j ∈ [ m ] and anothe r inde x i ∈ [ r ] . He applies the measurement { Π j ( i ) } r i = 1 separately t o each register Y 1 j , . . . , Y k j , and counts the number of times outcome i appears, which w e denote hencefort h as n j ( i ) . Art hur rejects if     n j ( i ) k − p j ( i )     ≥ 1 p for p a car efully chos e n polynomial. 5. Run Stage 2 of the BellQMA verification and repeat for error amplification. For each prover j , Ar t hur s amples an out come from [ r ] according to the distribution in ( X 1 j , . . . , X r j ) , and runs Stage 2 o f the BellQMA verification on the resulting set of samples. He repeats this p rocess independe ntly a polyno mial number of times q , and accepts if and only if the BellQMA proce- dure accepts on the majority of the runs. Let us d iscuss the intuition behind the verification procedure above. The key ste p above is Step 4, where Arthur cr o ss-checks that t h e classica l distributions sent in X real ly can be obtained by measuring m quantum proofs, which fo r an honest Me rlin should be une ntangled. In this sense, o ur protocol can alternatively be viewed as using qu antum proofs ( Y ) to check validity of a classical p roof ( X ). Intuitively , the reason why entanglement in Y d oes not he lp a disho nest Merlin in S t ep 3 is due to the local nature of Art hur ’s checks/measu rements. Finall y , once Art hur is satisfied that X cont ains valid d istributions, he runs Step 5. W e remark that repetition is used here in order t o boost the p roba bility o f acceptance in the x ∈ A yes case to e xponentially close to 1, which is requir e d to sep arate it from the x ∈ A no case, whe re the probabili ty of catching a dishones t Me rlin is only inverse polynomially bounded away fr om 1. Once such a gap exists, standard amplification te chniques [KW00, MW05] can be us ed to further impr ove completeness and sound ness parameters. T o formally analyze complete ness and sound ness of the protocol, we assign t he following values to the p aramete rs ment ione d above, all of which are polynomial in n in our set t ing: q = 50 n and p = 20 m r and k = 5 p 3 and α = 20 n m r . Completeness. Intuitively , when x ∈ A yes , Merlin passes Step 4 with p roba bility e xponentially close t o 1 since he has no incentive to cheat — he can se nd an u nentangled p roof in Step 1 to Arthur corresponding to the optimal p roofs ρ j in the B ellQMA protocol, s uch that the exp e cted value of 12 n j ( i ) / k is inde ed p j ( i ) . Arthur ’s checks in Ste p 4 are then independe nt local trials, allowing a Chernoff bound to be applied. W e t hen show that Merlin p asses e ach run in Ste p 5 with con- stant probabil ity , and applying the Cher n o f f bound a se con d time yields the desired completeness exponent ially close to 1 for the protocol. T o state this formally , suppos e Merlin is hones t and s ends registers ( X , Y ) in the de sired form, i.e., X i j contains p j ( i ) =  Π j ( i ) , ρ j  up to α bi ts of precision, and Y l j contains ρ j . Then, the e x p ected value of t h e random varia ble n j ( i ) is E [ n j ( i ) ] = k  Π j ( i ) , ρ j  , which i s equal t o k · p j ( i ) up to t he error incurred by repr esenting p j ( i ) us ing α bits of precision. In ot her words,     E [ n j ( i ) ] k − p j ( i )     < 1 2 α < 1 2 p . (1) W e can hence uppe r bound th e probab ility of rejecting in Ste p 3 by Pr      n j ( i ) k − p j ( i )     ≥ 1 p  < Pr      n j ( i ) k − E [ n j ( i ) ] k     ≥ 1 2 p  ≤ 2 e xp  − 5 p 4  where the first inequality follows from Eq. (1 ) and the second from the Chernoff bou n d . Thus, Merlin passes Step 4 with probabi lity ex p onentially close to 1. W e now turn to th e final st ep. Since x ∈ A yes , we know that the optimal dist ributions, deno ted q j : =   Π j ( 1 ) , ρ j  , . . . ,  Π j ( r ) , ρ j   for j ∈ [ m ] , obtained in Stage 1 of the o riginal BellQMA pr o- tocol are no w accepted in Stage 2 with probabili ty at least 2/3. However , in our case, Me rlin was only able to specify each q j up to α bi ts of pr ecision p er entry as the dist ributions p j . T o analyze how this affects the probab ility of acceptance, let P j and Q j be d iagonal ope rators with entries P j ( i , i ) = p j ( i ) and Q j ( i , i ) =  Π j ( i ) , ρ j  , respectively . L etting Λ accept denote the POVM element corresponding to outcome accept in Stage 2 of the BellQMA protocol, we thus boun d the change in acceptance probabil ity by:       T r   Λ accept   m O j = 1 P j − m O j = 1 Q j           ≤     m O j = 1 P j − m O j = 1 Q j     tr ≤ m ∑ j = 1   P j − Q j   tr = m ∑ j = 1 r ∑ i = 1 | p j ( i ) −  Π j ( i ) , ρ j  | ≤ mr 2 20 n mr where t h e first inequality follows fr om the fact that | T r ( A B ) | ≤ k A k ∞ · k B k tr and the se cond inequality follows from L emma 2.1. Therefore, the p roba bility o f su cces s for each of the q runs of the BellQMA protocol in Step 5 is at least  2 3 − mr 2 20 n mr  > 0.6. Since each run is indepen d ent, app lying the Chernoff boun d yields th at Arthur accepts Mer lin’s proof in Step 5 with pr obability at least 1 − 2 exp ( − 0.02 q ) , as des ir ed. There ma y be some err or incurr ed in sampling, w hich can be assumed to be e xponentially small so t hat t h e s uccess p roba- bility of each run is still at least 0.6. 13 Sound ness. W e no w p rove that when x ∈ A no , a dish o nest Me rlin can win with probab ility at most inverse polynomially bound ed away from 1. T o show this, we bound the probability of passing St e p 4 by relating the quantity p j ( i ) to the exp ected value of n j ( i ) / k , and then apply the Markov bound. The des ir ed relationship follows by observing first that the expe cted value of n j ( i ) / k is precisely the probab ility o f obtaining outcome i when measuring proof j of some (honest) unentang led strate gy , followed by ar g uing that t he distribution p j must hence be far fr om t his latte r (honest ) distribution if Merlin is t o pass Step 5 with probabili ty at least 1 /2 (since x ∈ A no ). C ombining th e se facts, we find that A r t hur dete cts a cheating Merlin with inverse polynomial probabi lity in S tep 4. More formally , let the quantum register Y j contain an arbitrary quantum state σ j whose re- duced state s in registers Y l j for l ∈ [ k ] ar e given by σ j ( l ) , and defin e ξ j : = 1 k k ∑ l = 1 σ j ( l ) . By the linearity o f exp ectation, the expected value of the random variable n j ( i ) / k is E  n j ( i ) k  = 1 k k ∑ l = 1  Π j ( i ) , σ j ( l )  =  Π j ( i ) , ξ j  . Our goal is to lower bound the expression Pr      n j ( i ) k − p j ( i )     ≥ 1 p  . (2) T o achieve this, we first su bs titute p j ( i ) above with a quantity involving E [ n j ( i ) / k ] , and t hen apply the Markov bound. T o r elate E [ n j ( i ) / k ] to p j ( i ) , we first r emark that in order for Merlin to pass e ach run of Step 5 with probabili ty expone ntially close to 1, h e must se nd pr obability distributions p j , w hich are accepted by Stage 2 o f the BellQMA verification with probabi lity at least 1/2. Let q j ( i ) : =  Π j ( i ) , ξ j  . Let us imagine a B ellQMA protocol wher e the j -th Merlin sends ξ j as his quantum proof. Since x ∈ A no , by t he sound ness property of the BellQMA ( m ) proof s ystem, t he success probability of the Merlins is at most 1/3. In other wor ds, sampling outcomes fr o m the p roba bility distribu- tions ( q j ( 1 ) , . . . , q j ( r ) ) and then running the second stage o f the BellQMA verification will yield outcome accept with probabili ty at most 1 / 3. Also, o bserve t hat E  n j ( i ) k  = q j ( i ) . It follows that by letting P j and Q j be diagonal ope rators with the probabili ty vectors p j and q j on their diagonals, respectively , and Λ accept the POVM element corresponding to outcome accept in Stage 2 of the B e llQMA protocol, we have 1 10 <       T r   Λ accept   m O j = 1 P j − m O j = 1 Q j           ≤     m O j = 1 P j − m O j = 1 Q j     tr ≤ m ∑ j = 1   P j − Q j   tr . 14 Here, the (loose) lower bound of 1/10 comes from t he following two obse r vations. First , the distributions repr esented by the diagonal operators Q j ’s are derived from a BellQMA protocol and the refor e achieve a success proba bility at most 1/ 3 by the s o undness property of the Be l- lQMA verification. Second, the distributions repr esente d by the diagonal oper ato rs P j ’s have to achieve a success p roba bility strictly greater than 1/2 per run to guarantee that Merlin wins S tep 5 with probabi lity expone ntially close t o 1. Combining these two, we g e t that the dif fer ence be- tween t he success probabi lities obtained by distributions described by operators { P j : j ∈ [ m ] } and { Q j : j ∈ [ m ] } should be at least 1/ 6 modulo the error incurred d ue to finite precision when encoding the d istributions p j . The use of t he constant 1/ 10 overcompensates for t his precision error . Hence, there e xists a j such that   P j − Q j   tr = r ∑ i = 1 | p j ( i ) − q j ( i ) | ≥ 1 10 m implying the existe nce of an i such that | p j ( i ) − q j ( i ) | ≥ 1 10 mr . (3) This is our des ired relati onship betwee n p j ( i ) and E [ n j ( i ) / k ] = q j ( i ) . Note that the p roba bility of picking pair ( i , j ) in Ste p 4 is 1/ mr . W e n o w s ubstitute this relationship into Eq. (2) and apply the Markov bound. Specifically , choose i and j as in Eq. (3), and ass ume th at p j ( i ) >  Π j ( i ) , ξ j  . Then, we have Pr      n j ( i ) k − p j ( i )     < 1 p  < Pr  n j ( i ) k − E  n j ( i ) k  > 1 10 mr − 1 p  ≤ 1 − 1 2 p . The case of p j ( i ) <  Π j ( 1 ) , ξ j  is s imila r . W e conclude that a dish o nest Merlin is caught in Step 4 with probability at least 1/2 p . Therefor e, the p roba bility t hat Art hur p roceeds t o Step 5 is upper bounded by  1 mr   1 − 1 40 mr  +  1 − 1 mr  ( 1 ) = 1 − 1 40 m 2 r 2 where the first term r epresents the case wher e Arthur selects the correct pair ( i , j ) to check, and the se cond term the complementary case, in which we assume the cheating prover can w in with probab ility 1. Hence the overall s uccess p roba bility of a dishones t Merlin is at most 1 − 1/40 m 2 r 2 , which is bounde d away from 1 by an inverse polynomial. Finally , as me ntioned before, since m and r are polynomially bounded functions, we have that the complete ness is ex p onentially close to 1, while the s o undness is bounded away fr om 1 by an inverse polynomial. By known amplification techniques for QMA protocols [KW00, MW05], o ne can amplify the completenes s and s oundnes s errors to be exponentially close to 0. This proves ou r desired containment. 5 Perfect parallel re pe tition for SepQMA ( poly ) W e now sho w Theorem 1.3, i.e., that t he class Se p QMA ( m ) admits perfect parallel repetition. B e - fore we proceed, recall that the closed convex con e Sep ( X 1 , . . . , X m ) is de fined to contain operators of the form k ∑ i = 1 P 1 ( i ) ⊗ · · · ⊗ P m ( i ) 15 where P j ( i ) ∈ Po s  X j  , for every j ∈ [ m ] and i ∈ [ k ] . T his is the con e of interest and it is know n to be closed and convex with non-empty inter ior . Given C to be the me asurement op e rator corre- sponding to o utcome accept , the maximum s u ccess p roba bility of the Me rlins in any QMA ( m ) pro- tocol can be written as t he maximum of h ρ , C i , whe re ρ is a density o p erator in Sep ( X 1 , . . . , X m ) . By standard convexity ar gument, one can always assume that the maximum is achieved by a pure product st ate. For th e remainder of t he section, it will be convenien t for us to distinguish two instances o f SepQMA ( m ) protocols as the first and second protocol. For t h e first SepQMA ( m ) pr otocol we can w rite the maximum acceptance p roba bility as the o ptimal value o f the primal problem in the following primal-dual pair (where the operator C 1 is Arthur ’s POVM element corr esponding to outcome accept ): Primal problem (P 1 ) maximiz e: h ρ 1 , C 1 i subject to: T r ( ρ 1 ) = 1, ρ 1 ∈ Sep ( X 1 , . . . , X m ) , Dual problem (D 1 ) minimiz e: t 1 subject to: t 1 1 X = C 1 + W 1 , W 1 ∈ Se p ( X 1 , . . . , X m ) ∗ , where X denotes X 1 ⊗ · · · ⊗ X m . The use of “maximum” and “minimum” is justified in the above programs since ρ 1 = 1 X dim ( X ) and ( t 1 , W 1 ) = ( 2, 2 1 X − C 1 ) ar e strictly feasible solutions for ( P 1 ) and ( D 1 ) , respectively [GB02, GB03, GB05]. Hen ce , by Theo- rem 2.6, strong duality holds for both problems, i.e., both problems attain an op timal solution and the optimal val ues are the same. W e note t hat the the d ual cone contains the s et of enta nglement witnesses in the t h e ory of entanglement, s ee [HHH H 09]. W e can similarly formulate the acceptance probab ility of the s econd protocol as Primal problem (P 2 ) maximiz e: h ρ 2 , C 2 i subject to: T r ( ρ 2 ) = 1, ρ 2 ∈ Se p ( Y 1 , . . . , Y m ) , Dual problem (D 2 ) minimiz e: t 2 subject to: t 2 1 Y = C 2 + W 2 , W 2 ∈ Se p ( Y 1 , . . . , Y m ) ∗ , where Y de n o tes Y 1 ⊗ · · · ⊗ Y m . Since we are considering SepQMA protocols it holds that C 1 ∈ Se p ( X 1 , . . . , X m ) and C 2 ∈ Se p ( Y 1 , . . . , Y m ) . Given the t wo cone programs above, the maximum acceptance probabil ity of the two-fold repetition of the protocol can hence be expressed as Primal problem (P ) Dual problem (D ) maximiz e: h ρ , C 1 ⊗ C 2 i minimiz e: t subject to: T r ( ρ ) = 1, subject to: t 1 X ⊗ Y = C 1 ⊗ C 2 + W , ρ ∈ Se p ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) , W ∈ Sep ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ . Note that the ope rators ρ and W are elements of He r m ( X 1 ⊗ · · · ⊗ X m ⊗ Y 1 ⊗ · · · ⊗ Y m ) . T o show Th e orem 1.3, observe that if ρ 1 and ρ 2 ar e any respective op t imal solutions o f (P 1 ) and (P 2 ), t h e n ρ 1 ⊗ ρ 2 is a feasible s olution of (P). Therefor e the optimal value of (P) is at least the product of the o ptimal values of (P 1 ) and (P 2 ). It remai ns to s how that in fact n o o t her strateg y for 16 the prover can perform better than this hones t strate gy . T o do so , we demonstr ate a dual feasible solution for (D) attaining t h is s ame objective value. More formally , let ( t 1 , W 1 ) and ( t 2 , W 2 ) be respective dual optimal solutions of ( D 1 ) and ( D 2 ) . By strong duality , t 1 is the op t imal value of ( P 1 ) and t 2 is the op t imal value of ( P 2 ) . W e show that t 1 · t 2 is an upper bound on the o ptimal value of ( P ) by exhibi ting a s o lution ( t 1 · t 2 , W ) which is feasible in ( D ) , for some W ∈ Se p ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ . W e first prove the foll owing use ful lemma. Lemma 5.1. For complex Euclidean spaces X 1 , . . . , X m and Y 1 , . . . , Y m , the following two containments hold: • Sep ( X 1 , . . . , X m ) ∗ ⊗ Sep ( Y 1 , . . . , Y m ) ⊆ S ep ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ , and • Sep ( X 1 , . . . , X m ) ⊗ Se p ( Y 1 , . . . , Y m ) ∗ ⊆ S ep ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ . Pro of. W e p rove the first condition as the second is ne arly identical. Fix W ∈ Sep ( X 1 , . . . , X m ) ∗ and C ∈ Sep ( Y 1 , . . . , Y m ) . The n for S ∈ Sep ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) , we have h W ⊗ C , S i = h W , T r Y [ S ( 1 X ⊗ C ) ] i ≥ 0 if T r Y [ S ( 1 X ⊗ C ) ] ∈ Se p ( X 1 , . . . , X m ) . The refore, it suffices to prove that T r Y [ S ( 1 X ⊗ C ) ] ∈ Se p ( X 1 , . . . , X m ) . T o this end, let S = k ∑ i = 1 m O l = 1 ρ i ( l ) and C = k ′ ∑ j = 1 m O l = 1 σ j ( l ) where ρ i ( l ) ∈ Pos ( X l ⊗ Y l ) and σ j ( l ) ∈ Pos ( Y l ) for all i ∈ [ k ] , j ∈ [ k ′ ] , and l ∈ [ m ] . No w w e can write T r Y [ S ( 1 X ⊗ C ) ] as T r Y " k ∑ i = 1 m O l = 1 ρ i ( l ) ! 1 X ⊗ k ′ ∑ j = 1 m O l = 1 σ j ( l ) ! # = k ∑ i = 1 k ′ ∑ j = 1 m O k = 1 T r Y k  ρ i ( k )  1 X k ⊗ σ j ( k )   . Hence, T r Y [ S ( 1 X ⊗ C )] ∈ Se p ( X 1 , . . . , X m ) since T r Y k  ρ i ( k )  1 X k ⊗ σ j ( k )   is positive semidefinite for all i , j , k . The latter follows since for pos itive s e midefinite A X ⊗ Y and B Y , T r Y [ A X ⊗ Y ( I X ⊗ B Y ) ] = T r Y [( I X ⊗ B 1 2 Y ) A X ⊗ Y ( I X ⊗ B 1 2 Y ) ]  0, because the partial tr ace preserves positive semide fin ite ness. This concludes th e proof. W e n o w us e Lemma 5.1 to construct two op erators in Se p ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ , the ap- propriate convex combi nation of which is the d ual fe asible solution we are se eking. Specifically , observe first that s ince for the two instances of t he SepQMA ( m ) protocol, we have C 1 ∈ Se p ( X 1 , . . . , X m ) and C 2 ∈ Se p ( Y 1 , . . . , Y m ) , and since 1 X and 1 Y ar e fully se parable op e rators, it follows that t 1 1 X + C 1 ∈ Sep ( X 1 , . . . , X m ) and t 2 1 Y + C 2 ∈ Sep ( Y 1 , . . . , Y m ) 17 for all t 1 , t 2 ≥ 0. Us ing Lemma 5.1, we thus obtain operators ( t 1 1 X − C 1 ) ⊗ ( t 2 1 Y + C 2 ) ∈ Se p ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ (4) and ( t 1 1 X + C 1 ) ⊗ ( t 2 1 Y − C 2 ) ∈ Se p ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ (5) where t 1 1 X − C 1 ∈ Sep ( X 1 , . . . , X m ) ∗ by the constraints of ( D 1 ) , and similarly for t 2 1 Y − C 2 . Since Sep ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ is a convex cone, it follows that the average of Eqs . (4) and (5) yields the desired op erator W : = t 1 · t 2 1 X ⊗ Y − C 1 ⊗ C 2 ∈ Sep ( X 1 ⊗ Y 1 , . . . , X m ⊗ Y m ) ∗ . W e conclude that ( t 1 · t 2 , W ) is a feasible solution of the d ual problem (D) with objective value t 1 · t 2 as des ired. This con clude s the proof o f T h e orem 1.3. W e note t hat the re are instances of QMA ( poly ) protocols, which ar e not Se p QMA ( poly ) pro- tocols, that admit perfect parallel repetition. Altho ugh this fact is known in the literature (see Harrow and Mont anar o [HM10] for de tails), we provide a concr ete example below . First, not e that th e maximum acceptance probability of Arthur in a QMA ( m ) p rotocol is uppe r bounded by k C k ∞ , where C is the accepting measurement operato r . Now , consider the two -qubit POVM operato r C : = 1 2 | 00 i h 00 | + 1 2 | Ψ + i h Ψ + | where | Ψ + i : = 1 √ 2 | 01 i + 1 √ 2 | 10 i . W e can easily check that C has two eigenvalues, 0 and 1/2, and two principal eigenvectors | 00 i and | Ψ + i , o ne of which is a product state. It follows that t h e maximum acceptance p roba bility is 1/2. By the multiplicative property o f the infinity-norm under tensor pr oducts, it holds that the maximum acceptance probabil ity of the k -fold repetition is ex actly 1/ 2 k . W e now argue t hat C is no t a sep arable operato r . S u ppose for the s ake of con t radiction that C can be written as n ∑ i = 1 ρ i ⊗ σ i for some ρ i , σ i ∈ Pos ( C 2 ) . The n we have 0 = h C , | 11 i h 11 | i = n ∑ i = 1 h 1 | ρ i | 1 i h 1 | σ i | 1 i which implies ρ i | 1 i = 0 or σ i | 1 i = 0 for all i ∈ [ n ] . T his leads to the contradiction 1 4 = h C , | 01 i h 10 | i = n ∑ i = 1 h 1 | ρ i | 0 i h 0 | σ i | 1 i = 0. Alternatively , one can show that C is no t sep arable by o bs erving that C has a non -p o sitive partial transpose [Per96, HH H 96]. 18 6 Conclusions and open problems In t his paper , we have studied t hree variants of multi-prover qu antu m Merlin- Arthur pr oof s ys- tems. W e firs t showe d th at a system with polynomially many provers is indeed st rictly more powerful t han a single prover syst em if messages are restricted to be logarithmic in length, unless BQP = MQA. W e next showed that polyno mially many provers do n o t provide additional ex- pressive power over a single prover in the s etting where the verifier is restricted to firs t applying unentangled and non-adaptive measurements w ith at most a polyno mial nu mber of outcomes per proof. Both of these quest ions make steps tow ards underst anding the major ope n question of whether QMA w ith polynomially many provers is more powe rful than QMA. Finally , w e use d cone p rogrammi ng duality to give an alternate proof of the fact that perfect parallel repetition holds whe never a QMA verifier ’s POVM element correspond ing t o accept is a fully separable op- erator . A con s equence of o ur fir s t result is that the t wo variants of t he class QMA ( poly ) , whe re Mer- lins send logarithmic-size proofs and Merlins se nd constant-size proofs are equal. A natural qu e s- tion concerning our first r esult is to underst and the expressive powe r of t he variant of QMA ( poly ) , where Merlins are restricted to se nd p oly log ( | x | ) qubits to Arthur . A nother open q u estion con- cerning the results presente d in this paper is the relationship between BellQMA ( poly ) and QMA. W e believe t hat understanding the complexity of Be llQMA protocols, or more gene rally LOCC- QMA protocols, will s hed new light on the bigger quest ion pertaining to QMA(2) and QMA. An- other avenue of interest is to find further applic ations of the cone programming characterization of multi-prover quantum Merlin-Arthur proof systems . A st raightforward question concerning the parallel repetition result presented in this paper is to investigate whether cone p rogramming duality can be used to analyze the product state test in the Ref. [HM10]. Anothe r question one can ask is to find o t her classes of QMA ( m ) protocols that admit a pe r fe ct p arallel repetition t heorem. Acknowledgements W e thank Richar d Cleve, T s uyoshi Ito, I ordanis Kerenidis, A s hwin N ayak, Oded Re gev , and Le v- ent T unc ¸ el for insightful discus s ions. Most of this work was completed at IQC at the Un ivers ity of W aterloo during the autho rs’ graduate s tudies. W e also th ank the EU-Canada Exchange Program and L IAF A, Paris for their hosp itality , where part of this work was complete d . S G acknowled g es support from NSERC, NS E RC MSF SS, the David R. Cherito n Graduate Scholarship program, t he President’s Graduate Scholarship, and CIF AR. JS acknowledges support from NSERC, MIT ACS, ERA (Ontario), and the President’s Graduate Scholarship. S U acknowledges suppo rt in p arts fr om CIF AR, MIT ACS, NS ERC, Ontario’s Ministry of Rese arch and Innovation, QuantumW o r k s, the U.S. A.R. O., Davi d R. Cheriton Graduate Scholarship, t he Mike and Ophelia Graduate Fellow- ship, and internal g rants of Centre for Quantum T echnologies . References [Aar06] S. Aaronson. QMA/qpoly is contained in PSP ACE/poly: De-Merlinizing quantum protocols. In Pro ceedings of the 21st An nual IEEE C onfer ence on Computational C omplex- ity , pages 273–286 , 2006. [AB09] S. Arora and B. Barak. Computati onal Complexity: A Modern A ppr oach . Cambri dge University Press, 2009. 19 [ABD + 09] S. Aaronson, S. Beigi, A. Drucker , B. Feff erman, and P . Sho r . The power of unentan- glement. T heory of Computing , 5:1–42, 2009. [ABOBS08] D. A har onov , M. Be n-Or , F . Brand ˜ ao, and O. Sattath. The pursu it for uniquenes s : Ex- tending Valiant-Vazirani t heorem to the probabili stic and quantum set tings. A vail- able at arXiv .org e-Print quant-ph/0810.4840v 1, 2008. [AGIK09] D. Aharonov , D. Gott esman, S. I r ani, and J. Kempe . The powe r of quantum syste ms on a line. Communications in Mathematical Physics , 287(1):41 –65, 2009. [AK07] S. Aaronson and G. Kupe r be rg. Quantum vers u s classical proofs and advice. Theory of Computing , 3:129–157 , 2007. [ALM + 98] S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szege dy . Proof verification and the har dness of approximation problems. Journal of the ACM , 45(3):50 1–555 , 1998. [AN02] D. Ahar onov and T . Naveh. Quantum NP - a survey . A vailable at arXiv .org e-Print quant-ph/021007 7v1, 2002. [AS98] S. Arora and S. Safra. Probabi listic checking of proofs: A ne w characterization of NP. Journal of the A CM , 45(1):70–12 2, 19 98. [BCY11] F . Brand ˜ ao, M. Christandl, and J. Y ar d. A qu asipo lynomial-time algorithm for the quantum sep arabili ty problem. A vailable at arXiv .org e-Print quant-ph/1011.2751 v2, 2011. [Bei08] S. Beigi. NP vs QMA log ( 2 ) . A vail able at arXiv .org e-Print quant-ph/0810.5109v 1, 2008. [Bra06] S. Bravyi. Efficient algorithm for a quantum analogue o f 2-SA T. A vai lable at arXiv .org e-Print quant-ph/060210 8, 2006. [Bra08] F . B rand ˜ ao. Entanglement Theory and the Quantum Simu lation of Many-Body P hysics . PhD thesis, Imper ial College Lond o n, Lond on, 2008. A vailable at arXiv . org e -Print quant-ph/1011.27 51v2. [BS07] S. Beigi and P . Shor . On the comp lex ity of computing zero-error and Holevo capacity of quantum channels. A vailable at arXiv .o rg e -Print quant-p h /0709.2090, 2007. [BT09] H. Blier and A. T app. All languages in NP have very short quantum pr oofs. I n Pro ceedings of the 3rd International Confer ence on Quantum, Nano and Micr o T echnologie s , pages 34–37, 2009. A vaila ble at arXiv .or g e-Print quant-ph/0709.073 8v2, firs t pos ted in 2007. [BV97] E. Bernst e in and U. V azirani. Quantum comp lex ity t heory . SIAM Journal on Comput- ing , 26(5):141 1–147 3, 1997. [CD10] J. Chen and A . Drucker . Sho rt multi-prover quantum proofs for S A T without entan- gled measurements. A vailable at arXiv . org e -Print quant-ph/1011.07 16v2, 2010. [CF11] A. Chi esa and M. Forbes. Improved soundne s s for QMA with multiple provers. A vailab le at arXiv .org e-Print quantum-ph/1108.2098 v1, 2011. 20 [Coo71] S. Cook. The complexity of theo rem p rovi ng procedures. In Pro ceedings of the 3rd Annual ACM Symposium on T heory of Computing , page s 151–158, 1971. [CSUU08] R. Cleve, W . Slofstra, F . Unger , and S. Upadhyay . Perfe ct parallel r epetition theorem for quantum XOR p roof syst ems. Computational Complexity , 17(2):282–2 99, 2008. [GB02] L. Gurvits and H. Barnum. L ar gest separable balls around the maxima lly mixed bi- partite quantum state. Physical Review A , 66(6):06 2311, 2002. [GB03] L. Gurvits and H. B arnum. Separable balls around the maximally mixed multipartite quantum states. Physical Review A , 68(4):04231 2, 20 03. [GB05] L. Gurvits and H. B arnum. B etter bound on the exponen t of the radius of th e multi- partite separable ball. Physical Review A , 72(3):03 2322, 2005. [GNN12] F . Le Gall, S. N akagawa, and H . Nishimura. On QMA protocols with t wo short quan- tum proofs. Quantu m Information and C omputation , 12(7&8):058 9–060 0, 2012 . [Gut09] G . Guto ski. Quantum strategies and local operations . PhD Thesis, University of W aterloo, 2009. A vaila ble at arXiv .org e-Print quant-ph/1003.0038. [HHH96] M. Horodecki, P . Horodecki, and R. Horodecki. Se parability of mixed states: neces- sary and sufficient cond itions. Physics Letters A , 223(1):1– 8, 1996. [HHHH09] R. H orodecki, P . Horodecki, M. Horodecki, and K. Horodecki. Quantum entangle- ment. Review of Modern Physics , 81(2):865–9 42, 2009. [HM10] A. Harrow and A. Montanaro. An ef ficient t e st for pr oduct states , with applic ations to quantum Merlin- Arthur games. In Pr oceedings of the 51st IEEE Annual Symposium on Foundations of Computer Science , pages 633–642 , 2010. [JGL10] S. Jordan, D. Gosse t, and P . Love. QMA-complete problems for sto quastic Hamiltoni- ans and Markov matrices. Physical R eview A , 81(3):03 2331, 2010. [JW06] D. Janzing and P . W ocjan. BQP-complete problems concerning mixing properties of classical rando m walks on sparse graphs. A vailable at arXiv .org e -Print quant- ph/061023 5v2, 2006. [KKR06] J. Kempe, A. Kitaev , and O. Rege v . The complexity o f the local Hamiltonian p roblem. SIAM Journal on Computing , 35(5):1 070–1 097, 200 6. [KMY03] H. Kobayashi, K. Matsumo t o, and T . Y amakami. Quantum Merlin Arthur proof s ys- tems: Are multiple Merlins more helpful to Art hur? In Proce edings of the 14th In terna- tional S ymposium on Algorithms and Computation , page s 189–19 8, 2003. V olume 2906 of Lectur e Notes in Computer Science , Springe r . [KR03] J. Kempe and O. Regev . 3-local Hamiltonian is QMA-complete. Quantum Information and Computation , 3(3):258– 264, 2003. [KR T10] J. Kempe, O. Rege v , and B . T oner . Unique games with entangled provers are easy . SIAM Journal on Computing , 39(7):3 207–3 229, 201 0. [KSV02] A. Kitaev , A. Shen, and M. V yalyi. Classical and Quantum Computation . American Mathematical Society , 2002. 21 [KW00] A. Kitaev and J. W atrous. Parallelization, amplification, and exp onential time simu- lation of quantum interactive pr oof systems. In Proceed ings of the 32nd Annual ACM Symposium on Theory of Computing , pages 608–617 , 2000. [LCV07] Y .-K. Liu, M. Christandl, and F . V erstraete. Quantum comput ational complexity of the N-repr esentability problem: QMA complete. Physical Review Letters , 98(11):11050 3, 2007. [Lev73] L . L evin. U n ivers al search probl ems. Proble ms of Information T ransmission , 9(3):265– 266, 1973. In Russian. [Liu06] Y .-K. Liu. Consistency of local density matrices is QMA-complete. In Proceed ings of the 10th Internatio nal Workshop on Randomizatio n and Computation , pages 438–4 49, 2006. V olume 4110 of Lecture Notes in Computer Science , Springe r . [MW05] C. Marriott and J. W atrous. Quantum A rthur-Merlin games. C omputational Complexity , 14(2):1 22–152 , 200 5. [NC00] M. Nielsen and I . Chuang. Quantum C omputation and Quan tu m Information . Cam- bridge University Press, 2000. [OT08] R. Oliveira and B . T erhal. The complexity of quantum spin sys tems on a two- dimensional square lattice. Quantum Information and Computation , 8(10 ):0900– 0924, 2008. [Per96] A. P e res. Separability criterion for density matrices. Physical Revie w Letters , 77:1413, 1996. [Ros11] B. Rosge n. T esting non-isometry is QMA-complete. In P r oceedings of the 5th C onfer- ence on Theory of Quantum C omputation, Communication, and Cryptogr aphy , page s 63–76, 2011. V olume 6519 of Lecture Notes in Computer S cience , S pringer . [SV09] N . Schuch and F . V erstraet e . Comput ational complexity of inte r acting electrons and fundamental limitations of de n s ity functional theory . Natur e Physics , 5:732–735, 2009. [TW08] L. T unc ¸ el and H. W olko wicz. S trong duality and minimal repr esentations for cone optimization. T echnical Re port CORR 2008-07, Department of Combinatorics and Optimization, University of W aterloo, W aterloo, Ontario, Canada, A ugust 2008 (re- vised: December 2008). [W at02] J. W atrous. Limits on the p ower of quantum statistical zero-knowledge. I n Pr oceedi ngs of the 50th An nual IEEE Symposium on Foun dations of Computer Science , pages 459–46 8, 2002. [W at09] J. W atrous. Encyclopedia of Complexity and System Science , chapter Quantum Comput a- tional Complexity . Springe r , 2009. [WMN10] T .-C. W ei, M. Mo sca, and A. N ayak. Interacting boson pr oblems can be QMA-hard. Physical Review Letters , 104(4):0405 01, 2010. [WY08] P . W ocjan and J. Y ard. The Jones polynomial: Quantum algorithms and applic ations in quantum complexity theory . Q u antum Information and C omputation , 8(1&2):0147 – 0180, 2008. 22