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Complexity of Quantum Uniform and Nonuniform Automata

  • Farid Ablayev
  • Aida Gainutdinova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3572)

Abstract

We present two different types of complexity lower bounds for quantum uniform automata (finite automata) and nonuniform automata (OBDDs). We call them “metric” and “entropic” lower bounds in according to proof technique used. We present explicit Boolean functions that show that these lower bounds are tight enough.

We show that when considering “almost all Boolean functions” on n variables our entropic lower bounds gives exponential (2 c(δ)(n − − log n)) lower bound for the width of quantum OBDDs depending on the error δ allowed.

Next we consider “generalized measure-many” quantum automata. It is appeared that for uniform and nonuniform automata (for space restricted models) their measure-once and measure-many models have different computational power.

Keywords

Boolean Function Unitary Transformation Sink Node Regular Language Quantum Circuit 
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. 1.
    Ablayev, F.M.: Lower Bounds for One-way Probabilistic Communication Complexity. In: Lingas, A., Carlsson, S., Karlsson, R. (eds.) ICALP 1993. LNCS, vol. 700, pp. 241–252. Springer, Heidelberg (1993)Google Scholar
  2. 2.
    Ablayev, F., Gainutdinova, A.: On the Lower Bound for One-Way Quantum Automata. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 132–140. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Ablayev, F., Gainutdinova, A., Karpinski, M.: On the computational power of quantum branching programs. In: Freivalds, R. (ed.) FCT 2001. LNCS, vol. 2138, pp. 59–70. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Ablayev, F., Moore, C., Pollett, C.: Quantum and stochastic branching programs of bounded width. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, p. 343. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Aharonov, D., Kitaev, A., Nisan, N.: Quantum circuits with mixed states. In: Proc. of 30th STOC, pp. 20–30 (1998)Google Scholar
  6. 6.
    Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalization. In: Proc. of the 39th IEEE Conference on Foundation of Computer Science, pp. 332–342 (1998); See also quant-ph/9802062 v3Google Scholar
  7. 7.
    Ambainis, A., Schulman, L., Vazirani, U.: Computing with highly mixed states. In: Proc. 32nd ACM Symp. on Theory of Computing, pp. 697–704 (2000)Google Scholar
  8. 8.
    Brodsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata. quant-ph/9903014 (1999)Google Scholar
  9. 9.
    Hromkovich, J.: Communication complexity and parallel computing. Springer, Heidelberg (1997)Google Scholar
  10. 10.
    Kremer, I., Nisan, N., Ron, D.: On randomized one-round communication complexity. In: Proc. of the 27th annual ACM symposium on Theory of computing, pp. 596–605 (1995)Google Scholar
  11. 11.
    Klauck, H.: On quantum and probabilistic communication: La Vegas and one-way protocols. In: Proc. of the 32nd ACM Symp. Theory of Computing (2000)Google Scholar
  12. 12.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  13. 13.
    Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proc. of the 38th IEEE Conference on Foundation of Computer Science, pp. 66–75 (1997)Google Scholar
  14. 14.
    Moore, C., Crutchfield, J.: Quantum automata and quantum grammars, quantph/ 9707031Google Scholar
  15. 15.
    Nakanishi, M., Hamaguchi, K., Kashiwabara, T.: Ordered quantum branching programs are more powerful than ordered probabilistic branching programs under a bounded-width restriction. In: Du, D.-Z., Eades, P., Sharma, A.K., Lin, X., Estivill-Castro, V. (eds.) COCOON 2000. LNCS, vol. 1858, pp. 467–476. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  16. 16.
    Nayak, A.: Optimal lower bounds for quantum automata and random access codes. In: Proc. of the 40th IEEE Conference on Foundation of Computer Science, pp. 369–376 (1999); See also quant-ph/9904093Google Scholar
  17. 17.
    Rabin, M.O.: Probabilistic automata. Information and Control 6, 230–244 (1963)CrossRefGoogle Scholar
  18. 18.
    Sauerhoff, M., Sieling, D.: Quantum branching programs and space-bounded nonuniform quantum complexity. ph/0403164 (March 2004)Google Scholar
  19. 19.
    Wegener, I.: Branching Programs and Binary Decision Diagrams. SIAM Monographs on Discrete Mathematics and Applications (2000)Google Scholar
  20. 20.
    Yao, A.: Some Complexity Questions Related to Distributive Computing. In: Proc. of the 11th Annual ACM Symposium on the Theory of Computing, pp. 209–213 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Farid Ablayev
    • 1
  • Aida Gainutdinova
    • 1
  1. 1.Dept. of Theoretical CyberneticsKazan State UniversityKazanRussia

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