Skip to main content
SpringerLink
Log in

From Quantum Query Complexity to State Complexity

  • Chapter
  • First Online:
Computing with New Resources

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8808))

Abstract

State complexity of quantum finite automata is one of the interesting topics in studying the power of quantum finite automata. It is therefore of importance to develop general methods how to show state succinctness results for quantum finite automata. One such method is presented and demonstrated in this paper. In particular, we show that state succinctness results can be derived out of query complexity results.

Work of the first author was supported by the Employment of Newly Graduated Doctors of Science for Scientific Excellence project/grant (CZ.1.07./2.3.00/30.0009) of Czech Republic. Work of second author was supported by the National Natural Science Foundation of China (Nos. 61272058, 61073054).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambainis, A., Freivalds, R.: One-way quantum finite automata: strengths, weaknesses and generalizations. In: Proceedings of the 39th FOCS, pp. 332–341 (1998).

    Google Scholar 

  3. Ambainis, A., Nayak, A., Ta-Shma, A., Vazirani, U.: Dense quantum coding and quantum automata. Journal of the ACM 49(4), 496–511 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambainis, A., Nahimovs, N.: Improved constructions of quantum automata. Theoretical Computer Science 410, 1916–1922 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambainis, A., Yakaryilmaz, A.: Superiority of exact quantum automata for promise problems. Information Processing Letters 112(7), 289–291 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ambainis, A.: Superlinear advantage for exact quantum algorithms. In: Proceedings of 45th STOC, pp. 891–900 (2013).

    Google Scholar 

  7. Ambainis, A., Iraids, A., Smotrovs, J.: Exact quantum query complexity of EXACT and THRESHOLD. In: Proceedings of 8th TQC, pp. 263–269 (2013)

    Google Scholar 

  8. Ambainis, A., Gruska, J., Zheng, S.G.: Exact query complexity of some special classes of Boolean functions, arXiv:1404.1684 (2014)

  9. Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. Journal of the ACM 48, 778–797 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bertoni, A., Mereghetti, C., Palano, B.: Small size quantum automata recognizing. Theoretical Computer Science 340, 394–407 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bertoni, A., Mereghetti, C., Palano, B.: Some formal tools for analyzing quantum automata. Theoretical Computer Science 356, 14–25 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brassard, G., Høyer, P.: An exact quantum polynomial-time algorithm for Simon’s problem. In: Proceedings of the Israeli Symposium on Theory of Computing and Systems, pp. 12–23 (1997).

    Google Scholar 

  13. Brodsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata. SIAM Journal on Computing 31, 1456–1478 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buhrman, H., Cleve, R., Wigderson, A.: Quantum vs. classical communication and computation. In: Proceedings of 30th STOC, pp. 63–68 (1998).

    Google Scholar 

  15. Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: a survey. Theoretical Computer Science 288, 21–43 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Buhrman, H., Cleve, R., Massar, S., de Wolf, R.: Nonlocality and Communication Complexity. Rev. Mod. Phys. 82, 665–698 (2010)

    Article  Google Scholar 

  17. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London A439, 553–558 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Frankl, P., Rödl, V.: Forbidden intersections. Trans. Amer. Math. Soc. 300(1), 259–286 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Freivalds, R., Ozols, M., Mancinska, L.: Improved constructions of mixed state quantum automata. Theoretical Computer Science 410, 1923–1931 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gruska, J.: Foundations of computing. Thomson International Computer Press (1997).

    Google Scholar 

  21. Gruska, J.: Quantum Computing. McGraw-Hill, London (1999)

    MATH  Google Scholar 

  22. Gruska, J.: Descriptional complexity issues in quantum computing. J. Automata, Languages Combin. 5(3), 191–218 (2000).

    Google Scholar 

  23. Gruska, J., Qiu, D.W., Zheng, S.G.: Generalizations of the distributed Deutsch-Jozsa promise problem, arXiv:1402.7254 (2014)

  24. Gruska, J., Qiu, D.W., Zheng, S.G.: Potential of quantum finite automata with exact acceptance, arXiv:1404.1689 (2014)

  25. Hromkovič, J., Schintger, G.: On the Power of Las Vegas for One-Way Communication Complexity. OBDDs, and Finite Automata, Information and Computation 169, 284–296 (2001)

    Article  MATH  Google Scholar 

  26. Klauck, H.: On quantum and probabilistic communication: Las Vegas and one-way protocols. In: Proceedings of the 32th STOC, pp. 644–651 (2000).

    Google Scholar 

  27. Kushilevitz, E.: Communication Complexity. Cambridge University Press (1997).

    Google Scholar 

  28. Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proceedings of the 38th FOCS, pp. 66–75 (1997).

    Google Scholar 

  29. Le Gall, F.: Exponential separation of quantum and classical online space complexity. In: Proceedings of SPAA 2006, pp. 67–73 (2006)

    Google Scholar 

  30. Mereghetti, C., Palano, B., Pighizzini, G.: Note on the Succinctness of Deterministic, Nondeterministic, Probabilistic and Quantum Finite Automata. RAIRO-Inf. Theor. Appl. 35, 477–490 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mereghetti, C., Palano, B.: On the size of one-way quantum finite automata with periodic behaviors. RAIRO-Inf. Theor. Appl. 36, 277–291 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Montanaro, A., Jozsa, R., Mitchison, G.: On exact quantum query complexity, Algorithmica (2013) doi:10.1007/s00453-013-9826-8

  33. Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theoretical Computer Science 237, 275–306 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Qiu, D.W., Li, L.Z., Zou, X.F., Mateus, P., Gruska, J.: Multi-letter quantum finite automata: decidability of the equivalence and minimization of states. Acta Inf. 48, 271–290 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Qiu, D.W., Li, L.Z., Mateus, P., Gruska, J.: Quantum finite automata. In: Finite State Based Models and Applications, CRC Handbook, pp. 113–144. CRC Press (2012).

    Google Scholar 

  36. Yao, A.C.: Some Complexity Questions Related to Distributed Computing. In: Proceedings of 11th STOC, pp. 209–213 (1979)

    Google Scholar 

  37. Yu, S.: State Complexity: Recent Results and Open Problems. Fundamenta Informaticae 64, 471–480 (2005)

    MathSciNet  MATH  Google Scholar 

  38. Yakaryilmaz, A., Cem Say, A.C.: Unbounded-error quantum computation with small space bounds. Information and Computation 209, 873–892 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yakaryilmaz, A., Cem Say, A.C.: Succinctness of two-way probabilistic and quantum finite automata. Discrete Mathematics and Theoretical Computer Science 12(4), 19–40 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Zheng, S.G., Qiu, D.W., Li, L.Z.: Some languages recognized by two-way finite automata with quantum and classical states. International Journal of Foundation of Computer Science 23(5), 1117–1129 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zheng, S.G., Qiu, D.W., Li, L.Z., Gruska, J.: One-Way Finite Automata with Quantum and Classical States. In: Bordihn, H., Kutrib, M., Truthe, B. (eds.) Languages Alive. LNCS, vol. 7300, pp. 273–290. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  42. Zheng, S.G., Qiu, D.W., Gruska, J., Li, L.Z., Mateus, P.: State succinctness of two-way finite automata with quantum and classical states. Theoretical Computer Science 499, 98–112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zheng, S.G., Gruska, J., Qiu, D.W.: On the state complexity of semi-quantum finite automata. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 601–612. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  44. Zheng, S.G., Gruska, J., Qiu, D.W.: Power of the interactive proof systems with verifiers modeled by semi-quantum two-way finite automata, arXiv:1304.3876 (2013)

Download references

Author information

Authors and Affiliations

  1. Faculty of Informatics, Masaryk University, 602 00, Brno, Czech Republic

    Shenggen Zheng

  2. Department of Computer Science, Sun Yat-sen University, Guangzhou, 510006, China

    Daowen Qiu

Authors
  1. Shenggen Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daowen Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daowen Qiu .

Editor information

Editors and Affiliations

  1. Department of Computer Science, The University of Auckland, Auckland, New Zealand

    Cristian S. Calude

  2. University of Latvia, Riga, Latvia

    Rūsiņš Freivalds

  3. School of Informatics, Kyoto University, Kyoto, Japan

    Iwama Kazuo

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Zheng, S., Qiu, D. (2014). From Quantum Query Complexity to State Complexity. In: Calude, C., Freivalds, R., Kazuo, I. (eds) Computing with New Resources. Lecture Notes in Computer Science(), vol 8808. Springer, Cham. https://doi.org/10.1007/978-3-319-13350-8_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-13350-8_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13349-2

  • Online ISBN: 978-3-319-13350-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation