Skip to main content
SpringerLink
Log in

Unary probabilistic and quantum automata on promise problems

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We continue the systematic investigation of probabilistic and quantum finite automata (PFAs and QFAs) on promise problems by focusing on unary languages. We show that bounded-error unary QFAs are more powerful than bounded-error unary PFAs, and, contrary to the binary language case, the computational power of Las-Vegas QFAs and bounded-error PFAs is equivalent to the computational power of deterministic finite automata (DFAs). Then, we present a new family of unary promise problems defined with two parameters such that when fixing one parameter QFAs can be exponentially more succinct than PFAs and when fixing the other parameter PFAs can be exponentially more succinct than DFAs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. See also [32] and [33] for some recent results about quantum and classical pushdown automata on promise problems.

References

  1. Ablayev, F., Gainutdinova, A., Khadiev, K., Yakaryılmaz, A.: Very narrow quantum OBDDs and width hierarchies for classical OBDDs. In: DCFS, LNCS, vol. 8614, pp. 53–64. Springer (2014)

  2. Ablayev, F., Gainutdinova, A., Khadiev, K., Yakaryılmaz, A.: Very narrow quantum OBDDs and width hierarchies for classical OBDDs. Lobachevskii J. Math. 37(6), 670–682 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ablayev, F.M., Gainutdinova, A.: On the lower bounds for one-way quantum automata. In: MFCS, LNCS, vol. 1893, pp. 132–140. Springer (2000)

  4. Ablayev, F.M., Gainutdinova, A.: Complexity of quantum uniform and nonuniform automata. In: DLT, LNCS, vol. 3572, pp. 78–87. Springer (2005)

  5. Ablayev, F.M., Gainutdinova, A., Karpinski, M., Moore, C., Pollett, C.: On the computational power of probabilistic and quantum branching program. Inf. Comput. 203(2), 145–162 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ambainis, A., Beaudry, M., Golovkins, M., Kikusts, A., Mercer, M., Thérien, D.: Algebraic results on quantum automata. Theory Comput. Syst. 39(1), 165–188 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: FOCS’98, pp. 332–341 (1998)

  8. Ambainis, A., Watrous, J.: Two-way finite automata with quantum and classical states. Theoretical Comput. Sci. 287(1), 299–311 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ambainis, A., Yakaryılmaz, A.: Superiority of exact quantum automata for promise problems. Inf. Process. Lett. 112(7), 289–291 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ambainis, A., Yakaryılmaz, A.: Automata from mathematics to applications, chap. Automata and quantum computing (to appear). (arXiv:1507.01988)

  11. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, Berlin (1976)

    MATH  Google Scholar 

  12. Bertoni, A., Carpentieri, M.: Analogies and differences between quantum and stochastic automata. Theor. Comput. Sci. 262(1–2), 69–81 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bianchi, M.P., Mereghetti, C., Palano, B.: Complexity of promise problems on classical and quantum automata. In: Computing with New Resources, LNCS, vol. 8808, pp. 161–175. Springer (2014)

  14. Condon, A., Lipton, R.J.: On the complexity of space bounded interactive proofs (extended abstract). In: FOCS’89, pp. 462–467 (1989)

  15. Doron, D., Sarid, A., Ta-Shma, A.: On approximating the eigenvalues of stochastic matrices in probabilistic logspace. Comput. Complex. 26(2), 393–420 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Freivalds, R.: Probabilistic two-way machines. In: Proceedings of the International Symposium on Mathematical Foundations of Computer Science, pp. 33–45 (1981)

  17. Gainutdinova, A., Yakaryılmaz, A.: Unary probabilistic and quantum automata on promise problems. In: Developments in Language Theory, Lecture Notes in Computer Science, vol. 9168, pp. 252–263. Springer (2015)

  18. Gainutdinova, A.F.: Comparative complexity of quantum and classical OBDDs for total and partial functions. Rus. Math. 11(59), 26–35 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Geffert, V., Yakaryılmaz, A.: Classical automata on promise problems. In: DCFS, LNCS, vol. 8614, pp. 126–137. Springer (2014). ECCC:TR14-136

  20. Geffert, V., Yakaryılmaz, A.: Classical automata on promise problems. Discret. Math. Theor. Comput. Sci. 17(2), 157–180 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Goldreich, O.: On promise problems: a survey. Essays Mem. Shimon Even, LNCS 3895, 254–290 (2006)

    MathSciNet  Google Scholar 

  22. Gruska, J., Qiu, D., Zheng, S.: Potential of quantum finite automata with exact acceptance. Int. J. Found. Comput. Sci. 26(3), 381–398 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gruska, J., Qiu, D., Zheng, S.: Generalizations of the distributed Deutsch-Jozsa promise problem. Math. Struct. Comput. Sci. 27(3), 311–331 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hirvensalo, M.: Quantum automata with open time evolution. Int. J. Nat. Comput. 1(1), 70–85 (2010)

    Article  MathSciNet  Google Scholar 

  25. Kapoutsis, C.A.: Minicomplexity. J. Autom. Lang. Comb. 17(2–4), 205–224 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Van Nostrand, Princeton (1960)

    MATH  Google Scholar 

  27. Klauck, H.: On quantum and probabilistic communication: Las Vegas and one-way protocols. In: STOC’00, pp. 644–651 (2000)

  28. Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: FOCS’97, pp. 66–75 (1997)

  29. Mereghetti, C., Palano, B., Pighizzini, G.: Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata. Theor. Inf. Appl. 35(5), 477–490 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theor. Comput. Sci. 237(1–2), 275–306 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Murakami, Y., Nakanishi, M., Yamashita, S., Watanabe, K.: Quantum versus classical pushdown automata in exact computation. IPSJ Digit. Cour. 1, 426–435 (2005)

    Article  Google Scholar 

  32. Nakanishi, M.: Quantum pushdown automata with a garbage tape. In: SOFSEM2015 Proceedings of the forty-first International Conference on Current Trends in Theory and Practice of Computer Science, LNCS, pp. 352–363. Springer (2015). (arXiv:1402.3449)

  33. Nakanishi, M., Yakaryılmaz, A.: Classical and quantum counter automata on promise problems. In: Implementation and Application of Automata, Lecture Notes in Computer Science, vol. 9223, pp. 224–237. Springer (2015)

  34. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  35. Rashid, J., Yakaryılmaz, A.: Implications of quantum automata for contextuality. In: Conference on Implementation and Application of Automata, LNCS, vol. 8587, pp. 318–331. Springer (2014). ArXiv:1404.2761

  36. Rotman, J.: An introduction to the theory of groups. number 148 in graduate texts in mathematics (1994)

  37. Salomaaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Texts and monographs in computer science. Springer, Berlin (1978)

    Book  Google Scholar 

  38. Say, A.C.C., Yakaryılmaz, A.: Quantum finite automata: A modern introduction. In: Computing with New Resources, LNCS, vol. 8808, pp. 208–222. Springer (2014)

  39. Shur, A.M., Yakaryılmaz, A.: More on quantum, stochastic, and pseudo stochastic languages with few states. Nat. Comput. 15(1), 129–141 (2016)

    Article  MathSciNet  Google Scholar 

  40. Watrous, J.: Encyclopedia of Complexity and System Science, chap. Quantum computational complexity. Springer (2009). Also available at arXiv:0804.3401

  41. Yakaryılmaz, A., Say, A.C.C.: Languages recognized by nondeterministic quantum finite automata. Quantum Inf. Comput. 10(9&10), 747–770 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Yakaryılmaz, A., Say, A.C.C.: Unbounded-error quantum computation with small space bounds. Inf. Comput. 279(6), 873–892 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zheng, S., Gruska, J., Qiu, D.: On the state complexity of semi-quantum finite automata. In: Language and Automata Theory and Applications, LNCS, vol. 8370, pp. 601–612. Springer (2014)

  44. Zheng, S., Li, L., Qiu, D., Gruska, J.: Promise problems solved by quantum and classical finite automata. Theor. Comput. Sci. 666, 48–64 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zheng, S., Qiu, D., Gruska, J., Li, L., Mateus, P.: State succinctness of two-way finite automata with quantum and classical states. Theore. Comput. Sci. 499, 98–112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Yakaryılmaz was partially supported by ERC Advanced Grant MQC and CAPES with grant 88881.030338/2013-01. We thank Viliam Geffert for his corrections on the text and the anonymous reviewers for their very helpful comments.

Author information

Authors and Affiliations

  1. Department of Theoretical Cybernetics, Institute of Computational Mathematics and Information Technologies, Kazan Federal University, Kazan, Tatarstan, Russia

    Aida Gainutdinova

  2. Center for Quantum Computer Science, University of Latvia, Rīga, Latvia

    Abuzer Yakaryılmaz

Authors
  1. Aida Gainutdinova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abuzer Yakaryılmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abuzer Yakaryılmaz.

Additional information

A preliminary version appeared as “Aida Gainutdinova and Abuzer Yakaryılmaz. Unary probabilistic and quantum automata on promise problems. In Developments in Language Theory, volume 9168 of LNCS, pages 252–263. Springer, 2015.” [17].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gainutdinova, A., Yakaryılmaz, A. Unary probabilistic and quantum automata on promise problems. Quantum Inf Process 17, 28 (2018). https://doi.org/10.1007/s11128-017-1799-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1799-0

Keywords

Search

Navigation