Interactive proof systems form an important complexity model that has been central to many prominent results in computational complexity theory, such as those on probabilistically checkable proofs, hardness of approximation, and fundamental cryptographic primitives. In this thesis, we study the quantum-enhanced version of interactive proof systems, in which each party has access to quantum computing resources. We focus on its power and limitations, which lead us to the following results. (*) We provide an alternative and conceptually simpler proof of Jain et al.’s recent breakthrough result, which demonstrates that the expressive power of quantum interactive proof systems coincides with that of its classical counterpart, and therefore PSPACE. (*) We determine the complexity of several variants of quantum competing-prover interactive proof systems, which introduce zero-sum games into interactive proof systems. In particular, we prove that the complexity class of two-turn quantum refereed games coincides with PSPACE, answering an important open problem of Jain et al.’s. Our result suggests that a more general model called double quantum interactive proof systems coincides with PSPACE, which subsumes and unifies all previous results on short refereed games. (*) We contribute to the study of two-prover quantum Merlin-Arthur games (QMA(2)), which exhibit an intriguing tradeoff between an intrinsic quantum ingredient, entanglement, and computational power. We prove a PSPACE upper bound for a variant of QMA(2) that is to date the most general one known in PSPACE. We also provide a quasi-polynomial time algorithm for optimizing linear functions over separable states, giving an alternative to the one proposed by Brandao et al. Our main technical contribution behind the above results is the equilibrium value method for obtaining space-efficient algorithms for a class of optimization problems that arise naturally in quantum computation. We also contribute to QMA-complete problems, where we demonstrate that the “Non-Identity Check” problem remains QMA-complete on circuits of poly-logarithmic depths, improving upon polynomial depths from previous results.
Interactive proof systems form an important complexity model that has been central to many prominent results in computational complexity theory, such as those on probabilistically checkable proofs, hardness of approximation, and fundamental cryptographic primitives. In this thesis, we study the quantum-enhanced version of interactive proof systems, in which each party has access to quantum computing resources. We focus on its power and limitations, which lead us to the following results. () We provide an alternative and conceptually simpler proof of Jain et al.’s recent breakthrough result, which demonstrates that the expressive power of quantum interactive proof systems coincides with that of its classical counterpart, and therefore PSPACE. () We determine the complexity of several variants of quantum competing-prover interactive proof systems, which introduce zero-sum games into interactive proof systems. In particular, we prove that the complexity class of two-turn quantum refereed games coincides with PSPACE, answering an important open problem of Jain et al.’s. Our result suggests that a more general model called double quantum interactive proof systems coincides with PSPACE, which subsumes and unifies all previous results on short refereed games. (*) We contribute to the study of two-prover quantum Merlin-Arthur games (QMA(2)), which exhibit an intriguing tradeoff between an intrinsic quantum ingredient, entanglement, and computational power. We prove a PSPACE upper bound for a variant of QMA(2) that is to date the most general one known in PSPACE. We also provide a quasi-polynomial time algorithm for optimizing linear functions over separable states, giving an alternative to the one proposed by Brandao et al. Our main technical contribution behind the above results is the equilibrium value method for obtaining space-efficient algorithms for a class of optimization problems that arise naturally in quantum computation. We also contribute to QMA-complete problems, where we demonstrate that the “Non-Identity Check” problem remains QMA-complete on circuits of poly-logarithmic depths, improving upon polynomial depths from previous results.
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. \n \nOn 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. \n \nIn 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. \n \nThe question related to the expressive power of models of quantum interactive proof systems and their variants lead us to the following results. \n \n(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. \n\t \n(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. \n\t \n(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. \n \nThe 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. \n \n(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. \n\t \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. \n\t \n3. 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. \n \nThe first two results follow from the duality of semidefinite programs and the final result follows from cone programming duality.
This paper investigates the role of interaction and coins in quantum Arthur--Merlin games (also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur--Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs (to share with the prover). This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur--Merlin games where both of the two messages are quantum (denoted qq-QAM in this paper) does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq−QAM1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur--Merlin games: it is proved that, for any constant m≥2, the class of problems having m-turn quantum Arthur--Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur--Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.
This paper investigates the role of interaction and coins in quantum Arthur-Merlin games (also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM_1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m >= 2, the class of problems having m-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.
This thesis studies one-turn quantum refereed games, which are abstract zero-sum games with two competing computationally unbounded quantum provers and a computationally bounded quantum referee. The provers send quantum states to the referee, who plugs the two states into his quantum circuit, measures the output of the circuit in the standard basis, and declares one of the two players as the winner depending on the outcome of the measurement. The complexity class QRG(1) comprises of those promise problems for which there exists a one-turn quantum refereed game such that one of the players wins with high probability for the yes-instances, and the other player wins with high probability for the no-instances, irrespective of the opponent’s strategy. QRG(1) is a generalization of QMA (or co-QMA), and can informally be viewed as QMA with a no-prover (or co-QMA with a yes-prover). \n \nWe have given a full characterization of QRG(1), starting with appropriate definitions and known results, and building on to two new results about this class. Previously, the best known upper bound on QRG(1) was PSPACE. We have proved that if one of the provers is completely classical, sending a classical probability distribution instead of a quantum state, the new class, which we name CQRG(1), is contained in Ǝ · PP (non- deterministic polynomial-time operator applied to the class PP). We have also defined another restricted version of QRG(1) where both provers send quantum states, but the referee measures one of the quantum states first, and plugs the classical outcome into the measurement, along with the other prover’s quantum state, into a quantum circuit, before measuring the output of the quantum circuit in the standard basis. The new class, which we name MQRG(1), is contained in P · PP (the probabilistic polynomial time operator applied to PP). Ǝ · PP is contained in P · PP, which is, in turn, contained in PSPACE. Hence, our results give better containments than PSPACE for restricted versions of QRG(1).
This paper investigates the role of interaction and coins in quantum Arthur-Merlin games(also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m ≥ 2, the class of problems having m-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.
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 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 QSZK, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.