eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-06-06
488
511
10.4230/LIPIcs.CCC.2015.488
article
Generalized Quantum Arthur-Merlin Games
Kobayashi, Hirotada
Le Gall, Francois
Nishimura, Harumichi
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol033-ccc2015/LIPIcs.CCC.2015.488/LIPIcs.CCC.2015.488.pdf
interactive proof systems
Arthur-Merlin games
quantum computing
complete problems
entanglement