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Complexity Upper Bounds for Classical Locally Random Reductions Using a Quantum Computational Argument

  • Rahul Tripathi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

We use a quantum computational argument to prove, for any integer k ≥ 2, a complexity upper bound for nonadaptive k-query classical locally random reductions (LRRs) that allow bounded-errors. Extending and improving a recent result of Pavan and Vinodchandran [PV], we prove that if a set L has a nonadaptive 2-query classical LRR to functions g and h, where both g and h can output O(logn) bits, such that the reduction succeeds with probability at least Open image in new window , then Open image in new window . Previous complexity upper bound for nonadaptive 2-query classical LRRs was known only for much restricted LRRs: LRRs in which the target functions can only take values in {0,1,2} and the error probability is zero [PV]. For k > 2, we prove that if a set L has a nonadaptive k-query classical LRR to boolean functions g 1, g 2, ..., g k such that the reduction succeeds with probability at least 2/3 and the distribution on \((k/2+\sqrt{k})\)-element subsets of queries depends only on the input length, then Open image in new window . Previously, for no constant k > 2, a complexity upper bound for nonadaptive k-query classical LRRs was known even for LRRs that do not make errors.

Our proofs follow a two stage argument: (1) simulate a nonadaptive k-query classical LRR by a 1-query quantum weak LRR, and (2) upper bound the complexity of this quantum weak LRR. To carry out the two stages, we formally define nonadaptive quantum weak LRRs, and prove that if a set L has a 1-query quantum weak LRR to a function g, where g can output polynomial number of bits, such that the reduction succeeds with probability at least Open image in new window , then Open image in new window .

Keywords

Boolean Function Density Operator Random Oracle Input Length Pure Quantum State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Aar04]
    Aaronson, S.: Limitations of quantum advice and one-way communication. In: Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pp. 320–332. IEEE Computer Society Press, Los Alamitos (2004)CrossRefGoogle Scholar
  2. [Aar05]
    Aaronson, S.: Quantum computing, postselection, and probabilistic polynomial-time. Technical Report 05-003, Electronic Colloquium on Computational Complexity (ECCC) (January 2005), http://www.eccc.uni-trier.de/eccc/
  3. [Aar06]
    Aaronson, S.: Lower bounds for local search by quantum arguments. SIAM Journal on Computing 35(4), 804–824 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [AFK89]
    Abadi, M., Feigenbaum, J., Kilian, J.: On hiding information from an oracle. Journal of Computer and System Sciences 39(1), 21–50 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [AR03]
    Aharonov, D., Regev, O.: A lattice problem in quantum NP. In: Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, pp. 210–219. IEEE Computer Society Press, Los Alamitos (2003)CrossRefGoogle Scholar
  7. [AR05]
    Aharonov, D., Regev, O.: Lattice problems in NP ∩ coNP. Journal of the ACM 52(5), 749–765 (2005)CrossRefMathSciNetGoogle Scholar
  8. [AS98]
    Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Bab87]
    Babai, L.: A random oracle separates PSPACE from the Polynomial Hierarchy. Information Processing Letters 26(1), 51–53 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [BF90]
    Beaver, D., Feigenbaum, J.: Hiding instances in multioracle queries. In: Choffrut, C., Lengauer, T. (eds.) STACS 1990. LNCS, vol. 415, pp. 37–48. Springer, Heidelberg (1990)Google Scholar
  11. [BFG06]
    Beigel, R., Fortnow, L., Gasarch, W.: A tight lower bound for restricted PIR protocols. Computational Complexity 15, 82–91 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [BFKR97]
    Beaver, D., Feigenbaum, J., Kilian, J., Rogaway, P.: Locally random reductions: Improvements and applications. Journal of Cryptology 10(1), 17–36 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  13. [BFL91]
    Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity 1(1), 3–40 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [BK95]
    Blum, M., Kannan, S.: Designing programs that check their work. Journal of the ACM 42(1), 269–291 (1995)zbMATHCrossRefGoogle Scholar
  15. [BLR93]
    Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences 47(3), 549–595 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [BM84]
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing 13(4), 850–864 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [dW06]
    de Wolf, R.: Lower bounds on matrix rigidity via a quantum argument. In: Proceedings of the 33rd International Colloquium on Automata, Languages, and Programming, pp. 62–71 (2006)Google Scholar
  18. Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43, 268–292 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  19. [FS92]
    Fortnow, L., Szegedy, M.: On the power of two-local random reductions. Information Processing Letters 44(6), 303–306 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [GM84]
    Goldwasser, S., Micali, S.: Probabilistic encryption. Journal of Computer Security 28, 270–299 (1984)zbMATHMathSciNetGoogle Scholar
  21. [KdW04]
    Kerenidis, I., de Wolf, R.: Exponential lower bound for 2-query locally decodable codes via a quantum argument. Journal of Computer and System Sciences 69(3), 395–420 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  22. [Ker05]
    Kerenidis, I.: Quantum multiparty communication complexity and circuit lower bounds. Technical Report quant-ph/0504087, Los Alamos e-Print Quantum Physics Technical Report Archive (April 12, 2005)Google Scholar
  23. [LFKN92]
    Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. Journal of the ACM 39(4), 859–868 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Lip91]
    Lipton, R.: New directions in testing. In: Feigenbaum, J., Merritt, M. (eds.) Distributed Computing and Cryptography. DIMACS series in Discrete Mathematics and Theoretical Computer Science, pp. 191–202. American Mathematical Society (1991)Google Scholar
  25. [LLS05]
    Laplante, S., Lee, T., Szegedy, M.: The quantum adversary method and classical formula size lower bounds. In: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pp. 76–90. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  26. [Nay99]
    Nayak, A.: Optimal lower bounds for quantum automata and random access codes. In: Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, pp. 369–377. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  27. [NC00]
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  28. [PV]
    Pavan, A., Vinodchandran, N.: 2-local random reductions to 3-valued functions. Computational Complexity (to appear)Google Scholar
  29. [Riv86]
    Rivest, R.: Workshop on communication and computing. MIT, Cambridge (1986)Google Scholar
  30. [Sha92]
    Shamir, A.: IP=PSPACE. Journal of the ACM 39(4), 869–877 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  31. [WdW05]
    Wehner, S., de Wolf, R.: Improved lower bounds for locally decodable codes and private information retrieval. In: Proceedings of the 32nd International Colloquium on Automata, Languages, and Programming. LNCS, pp. 1424–1436. Springer, Heidelberg (2005)Google Scholar
  32. [Yao90]
    Yao, A.: An application of communication complexity to cryptography. In: Lecture at DIMACS Workshop on Structural Complexity and Cryptography (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Rahul Tripathi
    • 1
  1. 1.Department of Computer Science and Engineering, University of South Florida, Tampa, FL 33620USA

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