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Parallel Repetition for Entangled k-player Games via Fast Quantum Search

Authors Kai-Min Chung, Xiaodi Wu, Henry Yuen



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Kai-Min Chung
Xiaodi Wu
Henry Yuen

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Kai-Min Chung, Xiaodi Wu, and Henry Yuen. Parallel Repetition for Entangled k-player Games via Fast Quantum Search. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 512-536, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.CCC.2015.512

Abstract

We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a k-player free game G with entangled value val^*(G) = 1 - epsilon, the n-fold repetition of G has entangled value val^*(G^(\otimes n)) at most (1 - epsilon^(3/2))^(Omega(n/sk^4)), where s is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G^(\otimes n)) <= (1 - epsilon^2)^(Omega(n/s)), due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games G where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1 - epsilon^2)^(Omega(n/sk^2)). We also show that the dependence of the exponent on k is necessary: we exhibit a k-player free game G and n >= 1 such that val^*(G^(\otimes n)) >= val^*(G)^(n/k). Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: in particular, our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic Grover speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.
Keywords
  • Parallel repetition
  • quantum entanglement
  • communication complexity

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References

  1. Scott Aaronson and Andris Ambainis. Quantum search of spatial regions. In Foundations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on, pages 200-209. IEEE, 2003. Google Scholar
  2. Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. Non-signalling parallel repetition using de finetti reductions. arXiv preprint arXiv:1411.1582, 2014. Google Scholar
  3. Boaz Barak, Anup Rao, Ran Raz, Ricky Rosen, and Ronen Shaltiel. Strong parallel repetition theorem for free projection games. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 352-365. Springer, 2009. Google Scholar
  4. Harry Buhrman, Richard Cleve, and Avi Wigderson. Quantum vs. classical communication and computation. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 63-68. ACM, 1998. Google Scholar
  5. Harry Buhrman, Serge Fehr, and Christian Schaffner. On the parallel repetition of multi-player games: The no-signaling case. arXiv preprint arXiv:1312.7455, 2013. Google Scholar
  6. André Chailloux and Giannicola Scarpa. Parallel repetition of entangled games with exponential decay via the superposed information cost. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 296-307. Springer, 2014. Google Scholar
  7. André Chailloux and Giannicola Scarpa. Parallel repetition of free entangled games: Simplification and improvements. arXiv preprint arXiv:1410.4397, 2014. Google Scholar
  8. Kai-Min Chung, Xiaodi Wu, and Henry Yuen. Parallel repetition for entangled k-player games via fast quantum search. arXiv preprint arXiv:1501.00033, 2015. Google Scholar
  9. Irit Dinur, David Steurer, and Thomas Vidick. A parallel repetition theorem for entangled projection games. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 197-208, 2014. Google Scholar
  10. Uriel Feige and Oleg Verbitsky. Error reduction by parallel repetition: a negative result. Combinatorica, 22(4):461-478, 2002. Google Scholar
  11. Joseph Fitzsimons and Thomas Vidick. A multiprover interactive proof system for the local hamiltonian problem. arXiv preprint arXiv:1409.0260, 2014. Google Scholar
  12. Thomas Holenstein. Parallel repetition: simplifications and the no-signaling case. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 411-419. ACM, 2007. Google Scholar
  13. Rahul Jain, Attila Pereszlényi, and Penghui Yao. A parallel repetition theorem for entangled two-player one-round games under product distributions. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 209-216, 2014. Google Scholar
  14. Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A lower bound for the bounded round quantum communication complexity of set disjointness. In Foundations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on, pages 220-229. IEEE, 2003. Google Scholar
  15. Julia Kempe and Thomas Vidick. Parallel repetition of entangled games. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pages 353-362. ACM, 2011. Google Scholar
  16. Robert König, Renato Renner, and Christian Schaffner. The operational meaning of min-and max-entropy. IEEE Transactions on Information Theory, 55(9):4337-4347, 2009. Google Scholar
  17. Ashwin Nayak and Julia Salzman. Limits on the ability of quantum states to convey classical messages. Journal of the ACM (JACM), 53(1):184-206, 2006. Google Scholar
  18. Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge university press, 2010. Google Scholar
  19. Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763-803, 1998. Google Scholar
  20. Ran Raz. A counterexample to strong parallel repetition. SIAM Journal on Computing, 40(3):771-777, 2011. Google Scholar
  21. Ricky Rosen. A k-provers parallel repetition theorem for a version of no-signaling model. Discrete Math., Alg. and Appl., 2(4):457-468, 2010. Google Scholar
  22. Mark M. Wilde. Quantum information theory. Cambridge University Press, 2013. Google Scholar
  23. Dong Yang, Karol Horodecki, Michal Horodecki, Pawel Horodecki, Jonathan Oppenheim, and Wei Song. Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof. IEEE Transactions on Information Theory, 55(7):3375-3387, 2009. Google Scholar
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