Two-message quantum interactive proofs are in PSPACE

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn , Issue , 2009, Pages 534-543

  Jain, Rahul ^a   Upadhyay, Sarvagya ^b   Watrous, John ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Matrix multiplicative weights update method; Quantum complexity; Quantum interactive proof systems

Indexed keywords

INTERACTIVE PROOFS; MATRIX; QUANTUM COMPLEXITY; QUANTUM INTERACTIVE PROOF SYSTEMS; SEMIDEFINITE PROGRAMS;

FIBER OPTIC CHEMICAL SENSORS; PARALLEL ALGORITHMS;

COMPUTER SCIENCE;

EID: 77952366375     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2009.30     Document Type: Conference Paper

References (41)

1. S. Arora, E. Hazan, and S. Kale, "Fast algorithms for approximate semidefinite programming using the multiplicative weights update method," in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 339-348.
2. S. Arora and S. Kale, "A combinatorial, primal-dual approach to semidefinite programs," in Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, 2007, pp. 227-236.
3. L. Babai, "Trading group theory for randomness," in Proceedings of the 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 421-429.
4. L. Babai and S. Moran, "Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes," Journal of Computer and System Sciences, vol. 36, no. 2, pp. 254-276, 1988.
5. M. Ben-Or, E. Feig, D. Kozen, and P. Tiwari, "A fast parallel algorithm for determining all roots of a polynomial with real roots," in Proceedings of the 18th Annual ACM Symposium on Theory of Computing, 1986, pp. 340-349.
6. M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, "Multi-prover interactive proofs: how to remove intractability assumptions," in Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1988, pp. 113-131.
7. R. Bhatia, Matrix Analysis. Springer, 1997.
8. A. Borodin, "On relating time and space to size and depth," SIAM Journal on Computing, vol. 6, pp. 733-744, 1977.
9. A. Borodin, S. Cook, and N. Pippenger, "Parallel computation for well-endowed rings and space-bounded probabilistic machines," Information and Control, vol. 58, pp. 113-136, 1983.
10. A. Borodin, J. v. z. Gathen, and J. Hopcroft, "Fast parallel matrix and GCD computations," in Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, 1982, pp. 65-71. (Pubitemid 13468549)
11. M.-D. Choi, "Completely positive linear maps on complex matrices," Linear Algebra and Its Applications, vol. 10, no. 3, pp. 285-290, 1975.
12. L. Csanky, "Fast parallel matrix inversion algorithms," SIAM Journal on Computing, vol. 5, no. 4, pp. 618-623, 1976.
13. C. Fuchs and J. van de Graaf, "Cryptographic distinguishability measures for quantum-mechanical states," IEEE Transactions on Information Theory, vol. 45, no. 4, pp. 1216-1227, 1999.
14. J. v. z. Gathen, "Parallel linear algebra," in Synthesis of Parallel Algorithms, J. Reif, Ed. Morgan Kaufmann Publishers, Inc., 1993, ch. 13.
15. O. Goldreich, S. Micali, and A. Wigderson, "Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems," Journal of the ACM, vol. 38, no. 1, pp. 691-729, 1991.
16. S. Goldwasser, S. Micali, and C. Rackoff, "The knowledge complexity of interactive proof systems," in Proceedings of the 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 291-304.
17. -, "The knowledge complexity of interactive proof systems," SIAM Journal on Computing, vol. 18, no. 1, pp. 186-208, 1989.
18. S. Goldwasser and M. Sipser, "Private coins versus public coins in interactive proof systems," in Randomness and Computation, ser. Advances in Computing Research, S. Micali, Ed. JAI Press, 1989, vol. 5, pp. 73-90.
19. M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, second corrected ed. Springer-Verlag, 1993.
20. G. Gutoski and J. Watrous, "Toward a general theory of quantum games," in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 2007, pp. 565-574.
21. S. Hallgren, A. Kolla, P. Sen, and S. Zhang, "Making classical honest verifier zero knowledge protocols secure against quantum attacks," in Proceedings of the 35th International Colloquium on Automata, Languages and Programming, ser. Lecture Notes in Computer Science, vol. 5126. Springer, 2008, pp. 592-603.
22. R. Jain, Z. Ji, S. Upadhyay, and J. Watrous, "QIP = PSPACE," 2009, available as arXiv.org e-Print 0907.4737.
23. R. Jain, S. Upadhyay, and J. Watrous, "Two-message quantum interactive proofs are in PSPACE," Manuscript, 2009, available as arXiv.org e-Print 0905.1300.
24. R. Jain and J. Watrous, "Parallel approximation of non-interactive zero-sum quantum games," in Proceedings of the 24th IEEE Conference on Computational Complexity, 2009, pp. 243-253.
25. S. Kale, "Efficient algorithms using the multiplicative weights update method," Ph.D. dissertation, Princeton University, 2007.
26. J. Kempe, H. Kobayashi, K. Matsumoto, and T. Vidick, "Using entanglement in quantum multi-prover interactive proofs," in Proceedings of the 23rd Annual Conference on Computational Complexity, 2008.
27. A. Kitaev and J. Watrous, "Parallelization, amplification, and exponential time simulation of quantum interactive proof system," in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000, pp. 608-617.
28. H. Kobayashi, "General properties of quantum zero-knowledge proofs," in Proceedings of the Fifth IACR Theory of Cryptography Conference, ser. Lecture Notes in Computer Science, vol. 4948. Springer, 2008, pp. 107-124.
29. H. Kobayashi and K. Matsumoto, "Quantum multi-prover interactive proof systems with limited prior entanglement," Journal of Computer and System Sciences, vol. 66, no. 3, 2003.
30. C. Lund, L. Fortnow, H. Karloff, and N. Nisan, "Algebraic methods for interactive proof systems," Journal of the ACM, vol. 39, no. 4, pp. 859-868, 1992.
31. C. Marriott and J. Watrous, "Quantum Arthur-Merlin games," Computational Complexity, vol. 14, no. 2, pp. 122-152, 2005.
32. C. A. Neff, "Specified precision polynomial root isolation is in NC," Journal of Computer and System Sciences, vol. 48, no. 3, pp. 429-463, 1994.
33. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2000.
34. A. Shamir, "IP = PSPACE," Journal of the ACM, vol. 39, no. 4, pp. 869-877, 1992.
35. *-algebra," Reports on Mathematical Physics, vol. 9, no. 2, pp. 273-279, 1976.
36. M. Warmuth and D. Kuzmin, "Online variance minimization," in Proceedings of the 19th Annual Conference on Learning Theory, ser. Lecture Notes in Computer Science, vol. 4005. Springer, 2006, pp. 514-528.
37. J. Watrous, "Limits on the power of quantum statistical zero-knowledge," in Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 459-468.
38. -, "PSPACE has constant-round quantum interactive proof systems," Theoretical Computer Science, vol. 292, no. 3, pp. 575-588, 2003.
39. -, "Zero-knowledge against quantum attacks," in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 296-305.
40. -, "Quantum computational complexity," in Encyclopedia of Complexity and System Science. Springer, 2009, available as arXiv.org e-Print 0804.3401.
41. S. Wehner, "Entanglement in interactive proof systems with binary answers," in Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science, ser. Lecture Notes in Computer Science, vol. 3884. Springer, 2006, pp. 162-171.