Advertisement

Postselection Finite Quantum Automata

  • Oksana Scegulnaja-Dubrovska
  • Lelde Lāce
  • Rūsiņš Freivalds
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6079)

Abstract

Postselection for quantum computing devices was introduced by S.Aaronson[2] as an excitingly efficient tool to solve long standing problems of computational complexity related to classical computing devices only. This was a surprising usage of notions of quantum computation. We introduce Aaronson’s type postselection in quantum finite automata.

There are several nonequivalent definitions of quantum finite automata. Nearly all of them recognize only regular languages but not all regular languages. We prove that PALINDROMES can be recognized by MM-quantum finite automata with postselection. At first we prove by a direct construction that the complement of this language can be recognized this way. This result distinguishes quantum automata from probabilistic automata because probabilistic finite automata with non-isolated cut-point 0 can recognize only regular languages but PALINDROMES is not a regular language.

Keywords

SIAM Journal Regular Language Finite Automaton Input Word Input Alphabet 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aaronson, S.: Lower bounds for local search by quantum arguments. SIAM Journal on Computing 35(4), 805–824 (2004) (See also quant-ph/0307149)MathSciNetGoogle Scholar
  2. 2.
    Aaronson, S.: Quantum Computing, Postselection, and Probabilistic Polynomial-Time. Proceedings of the Royal Society A 461(2063), 3473–3482 (2005) (See also quant-ph/0412187)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ablayev, F.M.: On Comparing Probabilistic and Deterministic Automata Complexity of Languages. In: Kreczmar, A., Mirkowska, G. (eds.) MFCS 1989. LNCS, vol. 379, pp. 599–605. Springer, Heidelberg (1989)Google Scholar
  4. 4.
    Ablayev, F.M., Freivalds, R.: Why Sometimes Probabilistic Algorithms Can Be More Effective. In: Gruska, J., Rovan, B., Wiedermann, J. (eds.) MFCS 1986. LNCS, vol. 233, pp. 1–14. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  5. 5.
    Aharonov, D., Regev, O.: Lattice problems in NP cap coNP. Journal of the ACM 52(5), 749–765 (2005)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ambainis, A., Beaudry, M., Golovkins, M., Ķikusts, A., Mercer, M., Therien, D.: Algebraic Results on Quantum Automata. Theory of Computing Systems 39(1), 165–188 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ambainis, A., Bonner, R., Freivalds, R., Ķikusts, A.: Probabilities to accept languages by quantum finite automata. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, pp. 174–183. Springer, Heidelberg (1999) (See also quant-ph/9904066)CrossRefGoogle Scholar
  8. 8.
    Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. FOCS 1998, pp. 332–341 (1998) (See also quant-ph/9802062) Google Scholar
  9. 9.
    Ambainis, A., Ķikusts, A.: Exact results for accepting probabilities of quantum automata. Theoretical Computer Science 295(1-3), 3–25 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ambainis, A., Ķikusts, A., Valdats, M.: On the class of languages recognizable by 1-way quantum finite automata. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 75–86. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Ambainis, A., Nayak, A., Ta-Shma, A., Vazirani, U.: Quantum dense coding and quantum finite automata. Journal of ACM 49, 496–511 (2002); Earlier version: STOC 1999 and quant-ph/9804043 CrossRefMathSciNetGoogle Scholar
  12. 12.
    Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P.W., Sleator, T., Smolin, J., Weinfurter, H.: Elementary gates for quantum computation. Physical Reviews A52, 3457–3467 (1995)CrossRefGoogle Scholar
  13. 13.
    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM Journal on Computing 26, 1411–1473 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Brodsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata. SIAM Journal on Computing 31(5), 1456–1478 (2002) (See also quant-ph/9903014)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bukharaev, R.G.: Probabilistic automata. Journal of Mathematical Sciences 13(3), 359–386 (1980)zbMATHCrossRefGoogle Scholar
  16. 16.
    Freivalds, R. (Freivald, R.V.): Recognition of languages with high probability on different classes of automata. Dolady Akademii Nauk SSSR 239(1), 60–62 (1978) (Russian)Google Scholar
  17. 17.
    Freivalds, R.: Projections of Languages Recognizable by Probabilistic and Alternating Finite Multitape Automata. Information Processing Letters 13(4/5), 195–198 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Freivalds, R.: Complexity of Probabilistic Versus Deterministic Automata. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 565–613. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  19. 19.
    Freivalds, R.: Non-Constructive Methods for Finite Probabilistic Automata. International Journal of Foundations of Computer Science 19(3), 565–580 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    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) (See also quant-ph/0208062)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proc. FOCS 1997, pp. 66–75 (1997)Google Scholar
  22. 22.
    Macarie, I.I.: Space-Efficient Deterministic Simulation of Probabilistic Automata. SIAM Journal on Computing 27(2), 448–465 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Moore, C., Crutchfield, J.: Quantum automata and quantum grammars. Theoretical Computer Science 237, 275–306 (2000) (Also quant-ph/9707031)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rabin, M.O.: Probabilistic automata. Information and Control 6(3), 230–245 (1963)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Oksana Scegulnaja-Dubrovska
    • 1
  • Lelde Lāce
    • 1
  • Rūsiņš Freivalds
    • 1
  1. 1.Department of Computer ScienceUniversity of LatviaRigaLatvia

Personalised recommendations