The class QMA plays a fundamental role in quantum complexity theory and it has found surprising connections to condensed matter physics and in particular in the study of the minimum energy of quantum systems. In this paper, we further investigate the class QMA and its related class QCMA by asking what makes quantum witnesses potentially more powerful than classical ones. We provide a definition of a new class, SQMA, where we restrict the possible quantum witnesses to the "simpler" subset states, i.e. a uniform superposition over the elements of a subset of n-bit strings. Surprisingly, we prove that this class is equal to QMA, hence providing a new characterisation of the class QMA. We also prove the analogous result for QMA(2) and describe a new complete problem for QMA and a stronger lower bound for the class QMA $_1$ .The class QMA plays a fundamental role in quantum complexity theory and it has found surprising connections to condensed matter physics and in particular in the study of the minimum energy of quantum systems. In this paper, we further investigate the class QMA and its related class QCMA by asking what makes quantum witnesses potentially more powerful than classical ones. We provide a definition of a new class, SQMA, where we restrict the possible quantum witnesses to the "simpler" subset states, i.e. a uniform superposition over the elements of a subset of n-bit strings. Surprisingly, we prove that this class is equal to QMA, hence providing a new characterisation of the class QMA. We also prove the analogous result for QMA(2) and describe a new complete problem for QMA and a stronger lower bound for the class QMA$_1$.

arXiv:1410.2882Oct 10, 20142 citations

Disturbed witnesses in quantum complexity theory

Friederike Anna Dziemba

QMA is the complexity class of computational problems that are efficiently verifiable by a quantum algorithm with the help of a witness in contrast to the smaller class BQP of problems efficiently solvable by a quantum algorithm without a witness. Like their classical counterparts NP and P, the class QMA is believed to be strictly larger than BQP, but the definitive answer remains one of the most fundamental open problems of complexity theory. An equality of QMA and BQP would imply that quantum computers can solve many physically and logically relevant problems efficiently, including the Local Hamiltonian problem and the Satisfiability problem for Boolean formulas. New approaches to gain more insight into the structure of BQP and QMA as well as which witness forms are sufficient for QMA are hence worth pursuing. This thesis comprises three research focuses: Firstly, we extend the uniform diagonalization theorem to complexity classes of promise problems in order to construct strictly intermediate problems between QMA and BQP under the assumption that these classes are unequal. The existence of strictly intermediate problems motivates our definition of noisy QMA classes, which form hierarchies of intermediate classes between QMA and BQP by restricting the witnesses to outputs of certain quantum channels. In our second research focus we apply the tool of concatenated coding to prove a bound on the witness noise up to which QMA stays robust. Besides a bound for general i.i.d. channels, we can prove that QMA stays robust if each witness qubit is disturbed by 18 % depolarizing or 27 % dephasing noise, while for complete depolarization or dephasing the noisy class obviously collapses to BQP and QCMA, respectively. In the third research focus we interpret the famous QPCP conjecture as robustness of the class QMA against high witness disturbance. Moreover, we consider a multiprover protocol by Fitzsimons and Vidick that constitutes a first step towards an important alternative formulation of the QPCP conjecture and achieve a reduction of the number of provers and an improvement of the acceptance probability for this protocol.QMA is the complexity class of computational problems that are efficiently verifiable by a quantum algorithm with the help of a witness in contrast to the smaller class BQP of problems efficiently solvable by a quantum algorithm without a witness. Like their classical counterparts NP and P, the class QMA is believed to be strictly larger than BQP, but the definitive answer remains one of the most fundamental open problems of complexity theory. An equality of QMA and BQP would imply that quantum computers can solve many physically and logically relevant problems efficiently, including the Local Hamiltonian problem and the Satisfiability problem for Boolean formulas. New approaches to gain more insight into the structure of BQP and QMA as well as which witness forms are sufficient for QMA are hence worth pursuing. This thesis comprises three research focuses: Firstly, we extend the uniform diagonalization theorem to complexity classes of promise problems in order to construct strictly intermediate problems between QMA and BQP under the assumption that these classes are unequal. The existence of strictly intermediate problems motivates our definition of noisy QMA classes, which form hierarchies of intermediate classes between QMA and BQP by restricting the witnesses to outputs of certain quantum channels. In our second research focus we apply the tool of concatenated coding to prove a bound on the witness noise up to which QMA stays robust. Besides a bound for general i.i.d. channels, we can prove that QMA stays robust if each witness qubit is disturbed by 18 % depolarizing or 27 % dephasing noise, while for complete depolarization or dephasing the noisy class obviously collapses to BQP and QCMA, respectively. In the third research focus we interpret the famous QPCP conjecture as robustness of the class QMA against high witness disturbance. Moreover, we consider a multiprover protocol by Fitzsimons and Vidick that constitutes a first step towards an important alternative formulation of the QPCP conjecture and achieve a reduction of the number of provers and an improvement of the acceptance probability for this protocol.

Jan 1, 20190 citations

Toward Separating QMA from QCMA with a Classical Oracle

Mark Zhandry

QMA is the class of languages that can be decided by an efficient quantum verifier given a quantum witness, whereas QCMA is the class of such languages where the efficient quantum verifier only is given a classical witness. A challenging fundamental goal in quantum query complexity is to find a classical oracle separation for these classes. In this work, we offer a new approach towards proving such a separation that is qualitatively different than prior work, and show that our approach is sound assuming a natural statistical conjecture which may have other applications to quantum query complexity lower bounds.QMA is the class of languages that can be decided by an efficient quantum verifier given a quantum witness, whereas QCMA is the class of such languages where the efficient quantum verifier only is given a classical witness. A challenging fundamental goal in quantum query complexity is to find a classical oracle separation for these classes. In this work, we offer a new approach towards proving such a separation that is qualitatively different than prior work, and show that our approach is sound assuming a natural statistical conjecture which may have other applications to quantum query complexity lower bounds.

Jan 1, 20250 citations

Toward Separating QMA from QCMA with a Classical Oracle

Mark Zhandry

arXiv:2411.01718Leibniz international proceedings in informaticsNov 4, 20240 citations

Quantum Proofs

Thomas Vidick, John Watrous

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 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*.

arXiv:1610.01664Foundations and Trends® in Theoretical Computer ScienceMar 31, 201644 citations

Zero-Knowledge Proof Systems for QMA

Anne Broadbent, Zhengfeng Ji, Fang Song +1 more

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.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.

SIAM Journal on ComputingJan 1, 202014 citations

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