An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games

Harrow, Aram W.; Montanaro, Ashley

doi:10.1109/focs.2010.66

Cited by 40 publications

(121 citation statements)

References 37 publications

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“…Arguably, the biggest open question in the study of quantum Merlin-Arthur proof systems is whether QMA ¼ QMAð2Þ [note that Harrow and Montanaro have proved that QMAð2Þ ¼ QMAðkÞ for any polynomial k > 2 [37]]. On the one hand, there are natural problems from quantum physics that are in QMA(2) but not obviously in QMA [21,22,39].…”

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

“…This would, accordingly, give stronger applications, and possibly, solve the QMA vs QMA(2) puzzle due to the result of [37]. Another open question is, in Theorem 1, for a state supported on the symmetric subspace (also known as the Bose-symmetric state), whether its reduced states have pure-state approximations of the form R φ ⊗k dμðφÞ with φ pure.…”

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

“…In recent years, there have been significant advances on the structure of quantum Merlin-Arthur systems, where multiple unentangled proofs and possibly locally restricted measurements in the verification were considered [10,16,[35][36][37]. It has been proven that many natural problems in quantum physics are characterized by quantum Merlin-Arthur proof systems (see, e.g., [21,22,38,39]).…”

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

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Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Smith²

2015

Phys. Rev. Lett.

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We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multifold product states. The approximation is measured by distinguishability under measurements that are implementable by fully-one-way local operations and classical communication (LOCC). Our result strengthens Brandão and Harrow's de Finetti theorem where a kind of partially-one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.

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Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 99%

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Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Smith²

2015

Phys. Rev. Lett.

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“…(See Table 1 for a summary of such results. ) Harrow and Montanaro [13] recently answered several open problems regarding the class QMA(κ), by proving that amplification within QMA(κ) is possible and that QMA(poly(N)) = QMA(2); the "collapse" is achieved by giving an analysis of a product test, which allows a verifier to use the unentanglement promise of only two registers to ensure that states within a single register are close to a separable state.…”

Section: Introductionmentioning

confidence: 99%

“…Harrow and Montanaro [13], through their product test, reduce the number of provers of [1] to only two, thereby obtaining a two-prover QMA protocol for 3SAT, with perfect completeness and constant soundness gap, where each prover sends Θ( √ N) qubits.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Chiesa¹,

Forbes²

2013

Chicago J. of Theoretical Comp. Sci.

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We present three contributions to the understanding of QMA with multiple provers:• We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM '09], yielding a soundness gap Ω(N −2 ). Our improvement is achieved without the use of an instance with a constant soundness gap (i.e., without using a "PCP").• We give a tight soundness analysis of the protocol of [Chen and Drucker, ArXiV '10], thereby improving their result from a "monolithic" protocol where Θ( √ N) provers are needed in order to have any soundness gap, to a protocol with a smooth trade-off between the number of provers κ and a soundness gap Ω(κ 2 N −1 ), as long as κ ∈ Ω(log N). (And, when κ ∈ Θ( √ N), we recover the original parameters of Chen and Drucker.)• We make progress towards an open question of [Aaronson et al., ToC '09] about what kinds of NP-complete problems are amenable to "sublinear" multiple-prover QMA protocols, by observing that a large class of such examples can easily be derived from results already in the PCP literature -namely, at least the languages recognized by a non-deterministic RAMs in quasilinear time.

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Epsilon-Net Method for Optimizations over Separable States

Shi

2012

Automata, Languages, and Programming

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We give algorithms for the optimization problem: max ρ Q, ρ , where Q is a Hermitian matrix, and the variable ρ is a bipartite separable quantum state. This problem lies at the heart of several problems in quantum computation and information, such as the complexity of QMA(2). While the problem is NP-hard, our algorithms are better than brute force for several instances of interest. In particular, they give PSPACE upper bounds on promise problems admitting a QMA(2) protocol in which the verifier performs only logarithmic number of elementary gate on both proofs, as well as the promise problem of deciding if a bipartite local Hamiltonian has large or small ground energy. For Q ≥ 0, our algorithm runs in time exponential in Q F . While the existence of such an algorithm was first proved recently by Brandão, Christandl and Yard [Proceedings of the 43rd annual ACM Symposium on Theory of Computation , 343-352, 2011], our algorithm is conceptually simpler.Besides the connection with the weak membership problem for separability, Problem 1 can also be understood from many other aspects. Firstly, as the objective function is the inner-product of a Hermitian matrix and a quantum state, which represents the average value of some physical observable, the optimal value of Problem 1 inherently possesses certain physical meaning. Secondly, in the study of the tensor product space [DF92], the value OptSep(Q) is precisely the injective norm of Q in L(A 1 ) ⊗ L(A 2 ), where L(A) denote the Banach space of operators on A with the operator norm. Finally, one may be equally motivated from the study in operations research. The definition of Problem 1 appeared in an equivalent form in [LQNY09] with the new name of "Bi-Quadratic Optimization over Unit Spheres". Subsequent works [HLZ10, So11] demonstrate that Problem 1 is just a special case of a more general class of optimization problems called homogenous polynomial optimization with quadratic constraints, which is currently an active research topic in that field.Another motivation to study Problem 1 is the recent interest about the complexity class QMA(2). Originally the class QMA (also known as quantum proofs) was defined [KSV02] as the quantum counterpart of the classical complexity class NP. While the extension of NP to allow multiple provers trivially reduces to NP itself, the power of QMA(2), the extension for QMA with multiple unentangled provers, remains far from being well understood. The study of the multipleprover model was initiated in [KMY01, KMY03], where QMA(k) denotes the complexity class for the k-prover case. Much attention was attracted to this model because of the discovery that NP admits logarithmic-size unentangled quantum proofs [BT09]. This result was surprising because single prover quantum logarithm-size proofs only characterize BQP [MW05]. It seems adding one unentangled prover increases the power of the model substantially. There are several subsequent works on refining the initial protocol either with improved completeness and soundness bounds [Bei10, ABD+09...

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An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games

Cited by 40 publications

References 37 publications

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Untitled

Epsilon-Net Method for Optimizations over Separable States

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