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Stronger methods of making quantum interactive proofs perfectly complete
 In ITCS ’13, Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science
, 2013
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Multiprover quantum MerlinArthur proof systems with small gap
, 2012
"... This paper studies multipleproof quantum MerlinArthur (QMA) proof systems in the setting when the completenesssoundness gap is small. Small means that we only lowerbound the gap with an inverseexponential function of the input length, or with an even smaller function. Using the protocol of Blie ..."
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Cited by 4 (2 self)
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This paper studies multipleproof quantum MerlinArthur (QMA) proof systems in the setting when the completenesssoundness gap is small. Small means that we only lowerbound the gap with an inverseexponential function of the input length, or with an even smaller function. Using the protocol of Blier and Tapp [BT09], we show that in this case the proof system has the same expressive power as nondeterministic exponential time (NEXP). Since singleproof QMA proof systems, with the same bound on the gap, have expressive power at most exponential time (EXP), we get a separation between single and multiprover proof systems in the ‘smallgap setting’, under the assumption that EXP 6=NEXP. This implies, among others, the nonexistence of certain operators called disentanglers (defined by Aaronson et al. [ABD+09]), with good approximation parameters. We also show that in this setting the proof system has the same expressive power if we restrict the verifier to be able to perform only Bellmeasurements, i.e., using a BellQMA verifier. This is not known to hold in the usual setting, when the gap is bounded by an inversepolynomial function of the input length. To show this we use the protocol of Chen and Drucker [CD10]. The only caveat here is that we need at least a linear amount of proofs to achieve the power of NEXP, while in the previous setting two proofs were enough. We also study the case when the prooflengths are only logarithmic in the input length and observe that in some cases the expressive power decreases. However, we show that it doesn’t decrease further if we make the proof lengths to be even shorter. 1
Onesided error QMA with shared EPR pairs – a simpler proof. arXiv:1306.5406
, 2013
"... We give a simpler proof of one of the results of Kobayashi, Le Gall, and Nishimura [KLGN13], which shows that any QMA protocol can be converted to a onesided error protocol, in which Arthur and Merlin initially share a constant number of EPR pairs and then Merlin sends his proof to Arthur. Our prot ..."
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We give a simpler proof of one of the results of Kobayashi, Le Gall, and Nishimura [KLGN13], which shows that any QMA protocol can be converted to a onesided error protocol, in which Arthur and Merlin initially share a constant number of EPR pairs and then Merlin sends his proof to Arthur. Our protocol is similar but somewhat simpler than the original. Our main contribution is a simpler and more direct analysis of the soundness property that uses wellknown results in quantum information such as properties of the trace distance and the fidelity, and the quantum de Finetti theorem. 1
Generalized Quantum ArthurMerlin Games
"... Abstract This paper investigates the role of interaction and coins in quantum ArthurMerlin games (also called publiccoin 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 ..."
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Abstract This paper investigates the role of interaction and coins in quantum ArthurMerlin games (also called publiccoin 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 ArthurMerlin 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 twoturn quantum ArthurMerlin games where both of the two messages are quantum, denoted qqQAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qqQAM 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 qqQAM: 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, qqQAM may provide a good measure in computational complexity theory. It is further proved that the class qqQAM 1 , the perfectcompleteness variant of qqQAM, gives new bounds for standard wellstudied classes of twoturn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum ArthurMerlin games: it is proved that, for any constant m ≥ 2, the class of problems having mturn quantum ArthurMerlin proof systems is either equal to PSPACE or equal to the class of problems having twoturn quantum ArthurMerlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem. ACM Subject Classification