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Generalized Quantum Arthur-Merlin Games

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Abstract

This paper investigates the role of interaction and coins in quantum Arthur-Merlin games (also called public-coin 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 Arthur-Merlin 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 two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM 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 qq-QAM: 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, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM_1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m >= 2, the class of problems having $m$-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.

BibTeX - Entry

@InProceedings{kobayashi_et_al:LIPIcs:2015:5069,
  author =	{Hirotada Kobayashi and Francois Le Gall and Harumichi Nishimura},
  title =	{{Generalized Quantum Arthur-Merlin Games}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{488--511},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{David Zuckerman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5069},
  URN =		{urn:nbn:de:0030-drops-50697},
  doi =		{10.4230/LIPIcs.CCC.2015.488},
  annote =	{Keywords: interactive proof systems, Arthur-Merlin games, quantum computing, complete problems, entanglement}
}

Keywords: interactive proof systems, Arthur-Merlin games, quantum computing, complete problems, entanglement
Seminar: 30th Conference on Computational Complexity (CCC 2015)
Issue date: 2015
Date of publication: 2015


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