Quantum Information and Variants of Interactive Proof Systems

For nearly three decades, the model of interactive proof systems and its variants have been central to many important and exciting developments in computational complexity theory such as exact characterization of some well known complexity classes, development of probabilistically checkable proof systems and theory of hardness of approximation, and formalization of fundamental cryptographic primitives. On the other hand, the theory of quantum information, which is primarily concerned with harnessing quantum mechanical features for algorithmic, cryptographic, and information processing tasks has found many applications. In the past three decades, quantum information has been used to develop unconditionally secure quantum cryptography protocols, efficient quantum algorithms for certain problems that are believed to be intractable in classical world, and communication efficient protocols. In this thesis, we study the impact of quantum information on the models of interactive proof systems and their multi-prover variants. We study various quantum models and explore two questions. The first question we address pertains to the expressive power of such models with or without resource constraints. The second question is related to error reduction technique of such proof systems via parallel repetition. The question related to the expressive power of models of quantum interactive proof systems and their variants lead us to the following results. (1) We show that the expressive power of quantum interactive proof systems is exactly PSPACE, the class of problems that can be solved by a polynomial-space deterministic Turing machines and that also admit a classical interactive proof systems. This result shows that in terms of complexity-theoretic characterization, both the models are equivalent. The result is obtained using an algorithmic technique known as the matrix multiplicative weights update method to solve a semidefinite program that characterizes the success probability of the quantum prover. (2) We show that polynomially many logarithmic-size unentangled quantum proofs are no more powerful than a classical proof if the verifier has the ability to process quantum information. This result follows from an observation that logarithmic-size quantum states can be efficiently represented classically and such classical representation can be used to efficiently generate the quantum state. (3) We also establish that the model of multi-prover quantum Merlin Arthur proof system, where the verifier is only allowed to apply nonadaptive unentangled measurement on each proof and then a quantum circuit on the classical outcomes, is no more powerful than QMA under the restriction that there are only polynomial number of outcomes per proof. This result follows from showing that such proof systems also admit a QMA verification procedure. The question related to error reduction via parallel repetition lead us to following results on a class of two-prover one-round games with quantum provers and a class of multi-prover QMA proof systems. (1) We establish that for a certain class of two-prover one-round games known as XOR games, admit a perfect parallel repetition theorem in the following sense. When the provers play a collection of XOR games, an optimal strategy of the provers is to play each instance of the collection independently and optimally. In particular, the success probability of the quantum provers in the n-fold repetition of an XOR game G with quantum value w(G) is exactly (w(G))^n. (2) We show a parallel repetition theorem for two-prover one-round unique games. More specifically, we prove that if the quantum value of a unique game is 1-e, then the quantum value of n-fold repetition of the game is at most (1-e^2/49)^n. We also establish that for certain class of unique games, the quantum value of the n-fold repetition of the game is at most (1-e/4)^n. For the special case of XOR games, our proof technique gives an alternate proof of result mentioned above. 3. Our final result on parallel repetition is concerned with SepQMA(m) proof systems, where the verifier receives m unentangled quantum proofs and the measurement operator corresponding to outcome "accept" is a fully separable operator. We give an alternate proof of a result of Harrow and Montanaro [HM10] that states that perfect parallel repetition theorem holds for such proof systems. The first two results follow from the duality of semidefinite programs and the final result follows from cone programming duality

Quantum Information and Variants of Interactive Proof Systems

For nearly three decades, the model of interactive proof systems and its variants have been central to many important and exciting developments in computational complexity theory such as exact characterization of some well known complexity classes, development of probabilistically checkable proof systems and theory of hardness of approximation, and formalization of fundamental cryptographic primitives. On the other hand, the theory of quantum information, which is primarily concerned with harnessing quantum mechanical features for algorithmic, cryptographic, and information processing tasks has found many applications. In the past three decades, quantum information has been used to develop unconditionally secure quantum cryptography protocols, efficient quantum algorithms for certain problems that are believed to be intractable in classical world, and communication efficient protocols. In this thesis, we study the impact of quantum information on the models of interactive proof systems and their multi-prover variants. We study various quantum models and explore two questions. The first question we address pertains to the expressive power of such models with or without resource constraints. The second question is related to error reduction technique of such proof systems via parallel repetition. The question related to the expressive power of models of quantum interactive proof systems and their variants lead us to the following results. (1) We show that the expressive power of quantum interactive proof systems is exactly PSPACE, the class of problems that can be solved by a polynomial-space deterministic Turing machines and that also admit a classical interactive proof systems. This result shows that in terms of complexity-theoretic characterization, both the models are equivalent. The result is obtained using an algorithmic technique known as the matrix multiplicative weights update method to solve a semidefinite program that characterizes the success probability of the quantum prover. (2) We show that polynomially many logarithmic-size unentangled quantum proofs are no more powerful than a classical proof if the verifier has the ability to process quantum information. This result follows from an observation that logarithmic-size quantum states can be efficiently represented classically and such classical representation can be used to efficiently generate the quantum state. (3) We also establish that the model of multi-prover quantum Merlin Arthur proof system, where the verifier is only allowed to apply nonadaptive unentangled measurement on each proof and then a quantum circuit on the classical outcomes, is no more powerful than QMA under the restriction that there are only polynomial number of outcomes per proof. This result follows from showing that such proof systems also admit a QMA verification procedure. The question related to error reduction via parallel repetition lead us to following results on a class of two-prover one-round games with quantum provers and a class of multi-prover QMA proof systems. (1) We establish that for a certain class of two-prover one-round games known as XOR games, admit a perfect parallel repetition theorem in the following sense. When the provers play a collection of XOR games, an optimal strategy of the provers is to play each instance of the collection independently and optimally. In particular, the success probability of the quantum provers in the n-fold repetition of an XOR game G with quantum value w(G) is exactly (w(G))^n. (2) We show a parallel repetition theorem for two-prover one-round unique games. More specifically, we prove that if the quantum value of a unique game is 1-e, then the quantum value of n-fold repetition of the game is at most (1-e^2/49)^n. We also establish that for certain class of unique games, the quantum value of the n-fold repetition of the game is at most (1-e/4)^n. For the special case of XOR games, our proof technique gives an alternate proof of result mentioned above. 3. Our final result on parallel repetition is concerned with SepQMA(m) proof systems, where the verifier receives m unentangled quantum proofs and the measurement operator corresponding to outcome "accept" is a fully separable operator. We give an alternate proof of a result of Harrow and Montanaro [HM10] that states that perfect parallel repetition theorem holds for such proof systems. The first two results follow from the duality of semidefinite programs and the final result follows from cone programming duality

Entangled Games Are Hard to Approximate

We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1-1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games

Non-Cooperative Rational Interactive Proofs

Interactive-proof games model the scenario where an honest party interacts with powerful but strategic provers, to elicit from them the correct answer to a computational question. Interactive proofs are increasingly used as a framework to design protocols for computation outsourcing. Existing interactive-proof games largely fall into two categories: either as games of cooperation such as multi-prover interactive proofs and cooperative rational proofs, where the provers work together as a team; or as games of conflict such as refereed games, where the provers directly compete with each other in a zero-sum game. Neither of these extremes truly capture the strategic nature of service providers in outsourcing applications. How to design and analyze non-cooperative interactive proofs is an important open problem. In this paper, we introduce a mechanism-design approach to define a multi-prover interactive-proof model in which the provers are rational and non-cooperative - they act to maximize their expected utility given others\u27 strategies. We define a strong notion of backwards induction as our solution concept to analyze the resulting extensive-form game with imperfect information. We fully characterize the complexity of our proof system under different utility gap guarantees. (At a high level, a utility gap of u means that the protocol is robust against provers that may not care about a utility loss of 1/u.) We show, for example, that the power of non-cooperative rational interactive proofs with a polynomial utility gap is exactly equal to the complexity class P^{NEXP}

Quantum de Finetti Theorems under Local Measurements with Applications

Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results: We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the exponential time hypothesis. We show that the maximum winning probability of free games can be estimated in polynomial time by linear programming. We also show that 3-SAT with m variables can be reduced to obtaining a constant error approximation of the maximum winning probability under entangled strategies of O(m^{1/2})-player one-round non-local games, in which the players communicate O(m^{1/2}) bits all together. We show that the optimization of certain polynomials over the hypersphere can be performed in quasipolynomial time in the number of variables n by considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy of semidefinite programs. As an application to entanglement theory, we find a quasipolynomial-time algorithm for deciding multipartite separability. We consider a result due to Aaronson -- showing that given an unknown n qubit state one can perform tomography that works well for most observables by measuring only O(n) independent and identically distributed (i.i.d.) copies of the state -- and relax the assumption of having i.i.d copies of the state to merely the ability to select subsystems at random from a quantum multipartite state. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor improvements. v3: added some explanations, mostly about Theorem 1 and Conjecture 5. STOC version. v4, v5. small improvements and fixe

Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher