Consistency of Local Density Matrices Is QMA-Complete

Liu, Yi-Kai

doi:10.1007/11830924_40

Cited by 78 publications

(80 citation statements)

References 16 publications

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“…The exponential complexity in n merely reflects the fact that we must supply an exponentially increasing amount of initial data in the form of the reduced states. Note that this does not contradict the results of [29] discussed earlier, as the compatibility problem considered there involves having only a number of reduced states that is polynomial in the number of quantum systems n, and each reduced state describes the state of a number of systems below some fixed constant. Hence the problem size here is only a polynomial in n, leading to very different conclusions regarding the complexity of the problem.…”

Section: Complexitysupporting

confidence: 61%

“…the number of known reduced states, and the number of systems described by these states. Let us consider the case discussed in [29] (to reiterate: we have a number of reduced states p(n), a polynomial in n, each describing a number of systems less than some constant k). In this case, the problem size is O(p(n)), whereas the dimension of the matrices involved is D, and there are O(D 2 ) free variables (since fixing polynomially many reduced states describing at most k systems can only fix polynomially many coefficients in B 0 ).…”

Section: A Partial Knowledge Of Reduced Statesmentioning

confidence: 99%

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Compatibility of subsystem states and convex geometry

Hall

2007

Phys. Rev. A

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The subsystem compatibility problem, which concerns the question of whether a set of subsystem states are compatible with a state of the entire system, has received much study. Here we attack the problem from a new angle, utilising the ideas of convexity that have been successfully employed against the separability problem. Analogously to an entanglement witness, we introduce the idea of a compatibility witness, and prove a number of properties about these objects. We show that the subsystem compatibility problem can be solved numerically and efficiently using semidefinite programming, and that the numerical results from this solution can be used to extract exact analytic results, an idea which we use to disprove a conjecture about the subsystem problem made by Butterley et al. [Found. Phys. 36 83 (2006)]. Finally, we consider how the ideas can be used to tackle some important variants of the compatibility problem; in particular, the case of identical particles (known as N -representability in the case of fermions) is considered.

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Section: Complexitysupporting

confidence: 61%

Section: A Partial Knowledge Of Reduced Statesmentioning

confidence: 99%

Compatibility of subsystem states and convex geometry

Hall

2007

Phys. Rev. A

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“…The Consistency of Density Operators problem was shown to be QMA-complete with respect to Cook reductions in [123]. We refer to the survey [34] for a more extensive list of QMA-complete problems.…”

Section: Chapter Notesmentioning

confidence: 99%

Quantum Proofs

Vidick

Watrous

2016

FNT in Theoretical Computer Science

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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 \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

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“…Watrous showed that Group non-Membership was a problem in QMA and based on this result he constructed an oracle under which MA is strictly included in QMA. Since then, various problems have been proven to be complete for QMA [16,18,23,17,24]. A potentially weaker quantum extension of MA, namely QCMA, was defined by Aharonov and Naveh [3]: in the case of QCMA, the verifier is still a quantum polynomial-time algorithm, but the message of the prover can only be classical.…”

Section: Introductionmentioning

confidence: 99%