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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 co ..."
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Cited by 71 (4 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
NPcomplete problems and physical reality
 ACM SIGACT News Complexity Theory Column, March. ECCC
, 2005
"... Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Mal ..."
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Cited by 55 (6 self)
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Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, MalamentHogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NPcomplete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1
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 g ..."
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Cited by 43 (13 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)
Parallel approximation of noninteractive zerosum quantum games
, 2008
"... This paper studies a simple class of zerosum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players ’ payoffs. We prove that an equilibrium point of any such game can be approxi ..."
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Cited by 13 (3 self)
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This paper studies a simple class of zerosum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players ’ payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that oneturn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space. 1
Upper bounds for quantum interactive proofs with competing provers
 In Proceedings of the 20th Annual IEEE Conference on Computational Complexity
, 2005
"... Refereed games are interactive proof systems with two competing provers: one that tries to convince the verier to accept and another that tries to convince the verier to reject. In quantum refereed games, the provers and verier may perform quantum computations and exchange quantum messages. One may ..."
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Cited by 10 (5 self)
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Refereed games are interactive proof systems with two competing provers: one that tries to convince the verier to accept and another that tries to convince the verier to reject. In quantum refereed games, the provers and verier may perform quantum computations and exchange quantum messages. One may consider games with a bounded or unbounded number of rounds of messages between the verier and provers. In this paper, we prove classical upper bounds on the power of both oneround and manyround quantum refereed games. In particular, we use semidenite programming to show that manyround quantum refereed games are contained in NEXP. It then follows from the symmetric nature of these games that they are also contained in coNEXP. We also show that oneround quantum refereed games are contained in EXP by supplying a separation oracle for use with the ellipsoid method for convex feasibility. 1.
Equilibrium value method for the proof of QIP=PSPACE
, 2009
"... We provide an alternative proof of QIP=PSPACE to the recent breakthrough result [JJUW09]. Unlike solving some semidefinite programs that captures the computational power of quantum interactive proofs, our method starts with one QIPComplete problem which computes the diamond norm between two admissi ..."
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Cited by 8 (4 self)
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We provide an alternative proof of QIP=PSPACE to the recent breakthrough result [JJUW09]. Unlike solving some semidefinite programs that captures the computational power of quantum interactive proofs, our method starts with one QIPComplete problem which computes the diamond norm between two admissible quantum channels. The key observation is that we can convert the computation of the diamond norm into the computation of some equilibrium value. The later problem, different from semidefinite programs, is of better form, easier to solve and could be interesting for its own sake. The multiplicative weight update method is also applied to solve the equilibrium value problem, however, in a relatively simpler way than the one in the original proof [JJUW09]. Furthermore, we provide a generalized form of equilibrium value problems which can be solved in the same way as well as comparisons to semidefinite programs.
L.: Secure twoparty quantum evaluation of unitaries against specious adversaries
 Advances in Cryptology, Proceedings of Crypto 2010
, 2010
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Parallel approximation of minmax problems with applications to classical and quantum zerosum games. Computational Complexity, 22(2):385428, 2013, the special issue of CCC 2012
 In Proceedings of the 27rd Annual IEEE Conference on Computational Complexity (CCC 2012
, 2012
"... Abstract This paper presents an efficient parallel algorithm for a new class of minmax problems based on the matrix multiplicative weight (MMW) update method. Our algorithm can be used to find nearoptimal strategies for competitive twoplayer classical or quantum games in which a referee exchange ..."
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Cited by 6 (3 self)
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Abstract This paper presents an efficient parallel algorithm for a new class of minmax problems based on the matrix multiplicative weight (MMW) update method. Our algorithm can be used to find nearoptimal strategies for competitive twoplayer classical or quantum games in which a referee exchanges any number of messages with one player followed by any number of additional messages with the other. This algorithm considerably extends the class of games which admit parallel solutions and demonstrates for the first time the existence of a parallel algorithm for any game (classical or quantum) in which one player reacts adaptively to the other. As a direct consequence, we prove that several competingprovers complexity classes collapse to PSPACE such as QRG Competitive multiturn twoplayer (say, Alice and Bob) games are often studied in the classical game theory either from the aspect of computing the game values or from the aspect of the complexity classes induced by those game models. For succinct games, exponentialtime algorithm exists for finding the exact value [KM92, KMvS94] and it is also EXPhard to approximate the game value [FIKU08, FKS95]. The situation is much different for shorter games, where succinct twoturn games admit polynomialspace approximation scheme and are also PSPACEhard to approximate Those game settings naturally extend to quantum case where provers and referees are allowed to exchange and process quantum information. It is known that the class of problems that admit quantum refereed games, denoted by QRG, coincide with its classical counterpart RG and henceforth EXP [GW07]. Also there exists a polynomialspace approximation scheme for quantum oneturn refereed games [JW09]. However, much more remains unknown about quantum refereed games of small number of turns. In this paper, we consider the following class of competitive twoplayer refereed games, either classical or quantum, that subsumes all the quantum refereed games of small number of turns studied so far [Gut05, GW05, GW07]. (i) The referee exchanges several messages only with Alice. (ii) After processing this interaction with Alice, the referee exchanges several additional messages only with Bob. After further processing, the referee declares a winner. * Full version available at arXiv:1011.2787 [quantph]. Please refer to the most recent version as a new version will be posted very shortly.