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arXiv:1610.01664 · Foundations and Trends® in Theoretical Computer Science

Quantum Proofs

Thomas Vidick, John Watrous

Published Mar 31, 2016• 44 citations
arXivHTMLPDF

Abstract

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*.

DOI
10.1561/0400000068

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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*.
Quantum Proofs provides 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, it discusses 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*. Quantum Proofs is mainly intended for non-specialists having a basic background in complexity theory and quantum information. A typical reader may be a student or researcher in either area desiring to learn about the fundamentals of the (actively developing) theory of quantum interactive proofs.
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration.
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.
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.
Zero-knowledge and multi-prover systems are both central notions in classical and quantum complexity theory. There is, however, little research in quantum multi-prover zero-knowledge systems. This paper studies complexity-theoretical aspects of the quantum multi-prover zero-knowledge systems. This paper has two results: 1.QMIP* systems with honest zero-knowledge can be converted into general zero-knowledge systems without any assumptions. 2.QMIP* has computational quantum zero-knowledge systems if a natural computational conjecture holds. One of the main tools is a test (called the GHZ test) that uses GHZ states shared by the provers, which prevents the verifier's attack in the above two results. Another main tool is what we call the Local Hamiltonian based Interactive protocol (LHI protocol). The LHI protocol makes previous research for Local Hamiltonians applicable to check the history state of interactive proofs, and we then apply Broadbent et al.'s zero-knowledge protocol for QMA \cite{BJSW} to quantum multi-prover systems in order to obtain the second result.
In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs - notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP*, the set of languages decidable by multiprover interactive proofs with quantumly entangled provers, and the class PZK-MIP*, which is the set of languages decidable by MIP* protocols that furthermore possess the perfect zero knowledge property. Our main result is that the two classes are equal, i.e., MIP* = PZK-MIP*. This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP = PZK-MIP (in other words, all classical multiprover interactive protocols can be made zero knowledge). We prove our result by showing that every MIP* protocol can be efficiently transformed into an equivalent zero knowledge MIP* protocol in a manner that preserves the completeness-soundness gap. Combining our transformation with previous results, we obtain the corollaries that i) all languages that can be solved in non-deterministic double exponential time have zero knowledge MIP* protocols and ii) all co-recursively enumerable languages (which include undecidable problems as well as all decidable problems) have zero knowledge MIP* protocols with vanishing promise gap.