Generalized Quantum Arthur-Merlin Games

Authors Hirotada Kobayashi, Francois Le Gall, Harumichi Nishimura

Thumbnail PDF


  • Filesize: 0.58 MB
  • 24 pages

Document Identifiers

Author Details

Hirotada Kobayashi
Francois Le Gall
Harumichi Nishimura

Cite AsGet BibTex

Hirotada Kobayashi, Francois Le Gall, and Harumichi Nishimura. Generalized Quantum Arthur-Merlin Games. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 488-511, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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.
  • interactive proof systems
  • Arthur-Merlin games
  • quantum computing
  • complete problems
  • entanglement


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Scott Aaronson, Salman Beigi, Andrew Drucker, Bill Fefferman, and Peter Shor. The power of unentanglement. Theory of Computing, 5:1-42 (article 1), 2009. Google Scholar
  2. Dorit Aharonov. A simple proof that Toffoli and Hadamard are quantum universal. e-Print archive, arXiv:quant-ph/0301040, 2003. Google Scholar
  3. Dorit Aharonov, Alexei Kitaev, and Noam Nisan. Quantum circuits with mixed states. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pages 20-30, 1998. Google Scholar
  4. László Babai. Trading group theory for randomness. In Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, pages 421-429, 1985. Google Scholar
  5. Salman Beigi, Peter Shor, and John Watrous. Quantum interactive proofs with short messages. Theory of Computing, 7:101-117 (article 7), 2011. Google Scholar
  6. Avraham Ben-Aroya, Oded Schwartz, and Amnon Ta-Shma. Quantum expanders: Motivation and constructions. Theory of Computing, 6:47-79 (article 3), 2010. Google Scholar
  7. Jin-Yi Cai. Lectures in computational complexity, August 2012. Available at\string jyc/710/book.pdf. Google Scholar
  8. André Chailloux, Dragos Florin Ciocan, Iordanis Kerenidis, and Salil Vadhan. Interactive and noninteractive zero knowledge are equivalent in the help model. In Theory of Cryptography, Fifth Theory of Cryptography Conference, TCC 2008, volume 4948 of Lecture Notes in Computer Science, pages 501-534, 2008. A full version available as Cryptology ePrint Archive, Report 2007/467, 2007. Google Scholar
  9. Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM Journal on Computing, 18(1):186-208, 1989. Google Scholar
  10. Shafi Goldwasser and Michael Sipser. Private coins versus public coins in interactive proof systems. In Silvio Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 73-90. JAI Press, 1989. Google Scholar
  11. Gustav Gutoski. Quantum Strategies and Local Operations. PhD thesis, David R. Cheriton School of Computer Science, University of Waterloo, 2009. e-Print archive, arXiv:1003.0038 [quant-ph]. Google Scholar
  12. Patrick Hayden, Kevin Milner, and Mark M. Wilde. Two-message quantum interactive proofs and the quantum separability problem. Quantum Information and Computation, 14(5-6):0384-0416, 2014. Google Scholar
  13. Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, and John Watrous. QIP = PSPACE. Journal of the ACM, 58(6):article 30, 2011. Google Scholar
  14. Rahul Jain, Sarvagya Upadhyay, and John Watrous. Two-message quantum interactive proofs are in PSPACE. In 50th Annual Symposium on Foundations of Computer Science, pages 534-543, 2009. Google Scholar
  15. Stephen P. Jordan, Hirotada Kobayashi, Daniel Nagaj, and Harumichi Nishimura. Achieving perfect completeness in classical-witness quantum Merlin-Arthur proof systems. Quantum Information and Computation, 12(5-6):0461-0471, 2012. Google Scholar
  16. Alexei Kitaev and John Watrous. Parallelization, amplification, and exponential time simulation of quantum interactive proof systems. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 608-617, 2000. Google Scholar
  17. Alexei \relaxYu. Kitaev, Alexander H. Shen, and Mikhail N. Vyalyi. Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. American Mathematical Society, 2002. Google Scholar
  18. Hirotada Kobayashi. Non-interactive quantum perfect and statistical zero-knowledge. In Algorithms and Computation, 14th International Symposium, ISAAC 2003, volume 2906 of Lecture Notes in Computer Science, pages 178-188, 2003. Google Scholar
  19. Hirotada Kobayashi, François Le Gall, and Harumichi Nishimura. Generalized quantum Arthur-Merlin games. e-Print archive, arXiv:1312.4673v2 [quant-ph], 2014. Google Scholar
  20. Hirotada Kobayashi, François Le Gall, and Harumichi Nishimura. Stronger methods of making quantum interactive proofs perfectly complete. SIAM Journal on Computing, 44(2):243-289, 2015. Google Scholar
  21. Hirotada Kobayashi, Keiji Matsumoto, and Tomoyuki Yamakami. Quantum Merlin-Arthur proof systems: Are multiple Merlins more helpful to Arthur? Chicago Journal of Theoretical Computer Science, 2009:article 3, 2009. Google Scholar
  22. Clemens Lautemann. BPP and the polynomial hierarchy. Information Processing Letters, 17(4):215-217, 1983. Google Scholar
  23. Carsten Lund, Lance Fortnow, Howard Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859-868, 1992. Google Scholar
  24. Chris Marriott and John Watrous. Quantum Arthur-Merlin games. Computational Complexity, 14(2):122-152, 2005. Google Scholar
  25. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. Google Scholar
  26. Christos H. Papadimitriou. Games against nature. Journal of Computer and System Sciences, 31(2):288-301, 1985. Google Scholar
  27. Bill Rosgen and John Watrous. On the hardness of distinguishing mixed-state quantum computations. In Twentieth Annual IEEE Conference on Computational Complexity, pages 344-354, 2005. Google Scholar
  28. Adi Shamir. IP = PSPACE. Journal of the ACM, 39(4):869-877, 1992. Google Scholar
  29. Alexander Shen. IP = PSPACE: Simplified proof. Journal of the ACM, 39(4):878-880, 1992. Google Scholar
  30. Yaoyun Shi. Both Toffoli and Controlled-NOT need little help to do universal quantum computing. Quantum Information and Computation, 3(1):084-092, 2003. Google Scholar
  31. Peter W. Shor. Fault-tolerant quantum computation. In 37th Annual Symposium on Foundations of Computer Science, pages 56-65, 1996. Google Scholar
  32. John Watrous. Capturing quantum complexity classes via quantum channels. Talk at the 6th Workshop on Quantum Information Processing, December 2002. Google Scholar
  33. John Watrous. Limits on the power of quantum statistical zero-knowledge. In 43rd Annual Symposium on Foundations of Computer Science, pages 459-468, 2002. Google Scholar
  34. John Watrous. PSPACE has constant-round quantum interactive proof systems. Theoretical Computer Science, 292(3):575-588, 2003. Google Scholar
  35. John Watrous. Zero-knowledge against quantum attacks. SIAM Journal on Computing, 39(1):25-58, 2009. Google Scholar
  36. Stephanie Wehner. Entanglement in interactive proof systems with binary answers. In STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, volume 3884 of Lecture Notes in Computer Science, pages 162-171, 2006. Google Scholar
  37. Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2013. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail