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49
Succinct Quantum Proofs for Properties of Finite Groups
 In Proc. IEEE FOCS
, 2000
"... In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite g ..."
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Cited by 64 (3 self)
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In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomiallength) quantum proofs for the Group NonMembership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossibleit is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group NonMembership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for nonmembership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.
ZeroKnowledge Against Quantum Attacks
 STOC'06
, 2006
"... This paper proves that several interactive proof systems are zeroknowledge against general quantum attacks. This includes the wellknown GoldreichMicaliWigderson classical zeroknowledge protocols for Graph Isomorphism and Graph 3Coloring (assuming the existence of quantum computationally conceal ..."
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Cited by 34 (0 self)
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This paper proves that several interactive proof systems are zeroknowledge against general quantum attacks. This includes the wellknown GoldreichMicaliWigderson classical zeroknowledge protocols for Graph Isomorphism and Graph 3Coloring (assuming the existence of quantum computationally concealing commitment schemes in the second case). Also included is a quantum interactive protocol for a complete problem for the complexity class of problems having “honest verifier” quantum statistical zeroknowledge proofs, which therefore establishes that honest verifier and general quantum statistical zeroknowledge are equal: QSZK = QSZK HV. Previously no nontrivial proof systems were known to be zeroknowledge against quantum attacks, except in restricted settings such as the honestverifier and common reference string models. This paper therefore establishes for the first time that true zeroknowledge is indeed possible in the presence of quantum information and computation.
Interaction in Quantum Communication and the Complexity of Set Disjointness
, 2001
"... One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible ..."
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Cited by 33 (7 self)
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One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible classically [1], [2], [3]. Moreover, some of these methods have a very simple structurethey involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a \simpler" quantum protocolone that has similar eciency, but uses fewer message exchanges.
Quantum ArthurMerlin games
 Computational Complexity
"... Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the complex ..."
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Cited by 33 (2 self)
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Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in powerover BQP and a simple proof that QMA ` PP. ffl In the case of three or more messages, quantum ArthurMerlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any languagehaving a quantum interactive proof system there exists a threemessage quantum ArthurMerlin game in whichArthur's only message consists of just a single coinflip that achieves perfect completeness and soundness errorexponentially close to 1/2. ffl Any language having a twomessage quantum ArthurMerlin game is contained in BP \Delta PP. This gives somesuggestion that three messages are stronger than two in
Quantum multiprover interactive proof systems with limited prior entanglement
 Journal of Computer and System Sciences
"... This paper gives the first formal treatment of a quantum analogue of multiprover interactive proof systems. In quantum multiprover interactive proof systems there can be two natural situations: one is with prior entanglement among provers, and the other does not allow prior entanglement among prov ..."
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Cited by 32 (5 self)
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This paper gives the first formal treatment of a quantum analogue of multiprover interactive proof systems. In quantum multiprover interactive proof systems there can be two natural situations: one is with prior entanglement among provers, and the other does not allow prior entanglement among provers. This paper focuses on the latter situation and proves that, if provers do not share any prior entanglement each other, the class of languages that have quantum multiprover interactive proof systems is equal to NEXP. It implies that the quantum multiprover interactive proof systems without prior entanglement have no gain to the classical ones. This result can be extended to the following statement of the cases with prior entanglement: if a language L has a quantum multiprover interactive proof system allowing at most polynomially many prior entangled qubits among provers, L is necessarily in NEXP. Another interesting result shown in this paper is that, in the case the prover does not have his private qubits, the class of languages that have singleprover quantum interactive proof systems is also equal to NEXP. Our results are also of importance in the sense of giving exact correspondances between quantum and classical complexity classes, because there have been known only a few results giving such correspondances.
Limits on the Power of Quantum Statistical ZeroKnowledge
, 2003
"... In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK ..."
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Cited by 30 (4 self)
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In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK
Quantum Merlin Arthur proof systems, manuscript
, 2001
"... Quantum MerlinArthur proof systems are a weak form of quantum interactive proof systems, where mighty Merlin as a prover presents a proof in a pure quantum state and Arthur as a verifier performs polynomialtime quantum computation to verify its correctness with high success probability. For a more ..."
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Cited by 26 (7 self)
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Quantum MerlinArthur proof systems are a weak form of quantum interactive proof systems, where mighty Merlin as a prover presents a proof in a pure quantum state and Arthur as a verifier performs polynomialtime quantum computation to verify its correctness with high success probability. For a more general treatment, this paper considers quantum “multipleMerlin”Arthur proof systems in which Arthur uses multiple quantum proofs unentangled each other for his verification. Although classical multiproof systems are easily shown to be essentially equivalent to classical singleproof systems, it is unclear whether quantum multiproof systems collapse to quantum singleproof systems. This paper investigates the possibility that quantum multiproof systems collapse to quantum singleproof systems, and shows that (i) a necessary and sufficient condition under which the number of quantum proofs is reducible to two and (ii) using multiple quantum proofs does not increase the power of quantum MerlinArthur proof systems in the case of perfect soundness. Our proof for the latter result also gives a new characterization of the class NQP, which bridges two existing concepts of “quantum nondeterminism”. It is also shown that (iii) there is a relativized world in which coNP (actually coUP) does not have quantum MerlinArthur proof systems even with multiple quantum proofs. 1 1
Quantum Entanglement and Communication Complexity
 SIAM J. COMPUT
, 1998
"... We consider a variation of the communication complexity scenario, where the parties are supplied with an extra resource: particles in an entangled quantum state. We note that "quantum nonlocality" can be naturally expressed in the language of communication complexity. These are communication complex ..."
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Cited by 26 (6 self)
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We consider a variation of the communication complexity scenario, where the parties are supplied with an extra resource: particles in an entangled quantum state. We note that "quantum nonlocality" can be naturally expressed in the language of communication complexity. These are communication complexity problems where the "output" is embodied in the correlations between the outputs of the individual parties. Without entanglement, the parties must communicate to produce the required correlations; whereas, with entanglement, no communication is necessary to produce the correlations. In this sense, nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication. We show how to transform examples of nonlocality into more traditional communication complexity problems, where the output is explicitly determined by each individual party. The resulting problems require communication with or without entanglement, but the required communication is less when entanglement is available. All these results are a noteworthy contrast to the wellknown fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties.
Toward a general theory of quantum games
 In Proceedings of 39th ACM STOC
, 2006
"... Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multipleround interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that genera ..."
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Cited by 20 (10 self)
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Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multipleround interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that generalizes the ChoiJamiol/kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting onlyon the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simpleproof of Kitaev's lower bound for strong coinflipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games. 1 Introduction The theory of games provides a general structure within which both cooperation and competitionamong independent entities may be modeled, and provides powerful tools for analyzing these models. Applications of this theory have fundamental importance in many areas of science.This paper considers games in which the players may exchange and process quantum information. We focus on competitive games, and within this context the types of games we consider arevery general. For instance, they allow multiple rounds of interaction among the players involved, and place no restrictions on players ' strategies beyond those imposed by the theory of quantuminformation. While classical games can be viewed as a special case of quantum games, it is important tostress that there are fundamental differences between general quantum games and classical games. For example, the two most standard representations of classical games, namely the normal formand extensive form representations, are not directly applicable to general quantum games. This is due to the nature of quantum information, which admits a continuum of pure (meaning extremal)