Skip to main content
SpringerLink
Log in

Lower Bounds and Hierarchies for Quantum Memoryless Communication Protocols and Quantum Ordered Binary Decision Diagrams with Repeated Test

  • Conference paper
  • First Online:
SOFSEM 2018: Theory and Practice of Computer Science (SOFSEM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10706))

Abstract

We explore multi-round quantum memoryless communication protocols. These are restricted version of multi-round quantum communication protocols. The “memoryless” term means that players forget history from previous rounds, and their behavior is obtained only by input and message from the opposite player. The model is interesting because this allows us to get lower bounds for models like automata, Ordered Binary Decision Diagrams and streaming algorithms. At the same time, we can prove stronger results with this restriction. We present a lower bound for quantum memoryless protocols. Additionally, we show a lower bound for Disjointness function for this model. As an application of communication complexity results, we consider Quantum Ordered Read-k-times Branching Programs (k-QOBDD). Our communication complexity result allows us to get lower bound for k-QOBDD and to prove hierarchies for sublinear width bounded error k-QOBDDs, where \(k=o(\sqrt{n})\). Furthermore, we prove a hierarchy for polynomial size bounded error k-QOBDDs for constant k. This result differs from the situation with an unbounded error where it is known that an increase of k does not give any advantage.

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 EPUB and 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

References

  1. 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 

  2. Ablayev, F., Ambainis, A., Khadiev, K., Khadieva, A.: Lower bounds and hierarchies for quantum memoryless communication protocols and quantum ordered binary decision diagrams with repeated test. arXiv preprint arXiv:1703.05015 (2017)

  3. Ablayev, F., Gainutdinova, A.: Complexity of quantum uniform and nonuniform automata. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 78–87. Springer, Heidelberg (2005). https://doi.org/10.1007/11505877_7

    Chapter  Google Scholar 

  4. Ablayev, F., Gainutdinova, A., Karpinski, M.: On computational power of quantum branching programs. In: Freivalds, R. (ed.) FCT 2001. LNCS, vol. 2138, pp. 59–70. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44669-9_8

    Chapter  Google Scholar 

  5. Ablayev, F., 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. Ablayev, F., Gainutdinova, A., Khadiev, K., Yakaryılmaz, A.: Very narrow quantum OBDDs and width hierarchies for classical OBDDs. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) DCFS 2014. LNCS, vol. 8614, pp. 53–64. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09704-6_6

    Google Scholar 

  7. Ablayev, F., Khadiev, K.: Extension of the hierarchy for k-OBDDs of small width. Russ. Math. 53(3), 46–50 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ablayev, F., Khasianov, A., Vasiliev, A.: On complexity of quantum branching programs computing equality-like boolean functions. In: ECCC (2008, to appear in 2010)

    Google Scholar 

  9. Ambainis, A.: A new protocol and lower bounds for quantum coin flipping. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 134–142. ACM (2001)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC\(^1\). J. Comput. Syst. Sci. 38(1), 150–164 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bollig, B., Sauerhoff, M., Sieling, D., Wegener, I.: Hierarchy theorems for kOBDDs and kIBDDs. Theoret. Comput. Sci. 205(1), 45–60 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chailloux, A., Kerenidis, I., Laurière, M.: The information cost of quantum memoryless protocols. arXiv preprint arXiv:1703.01061 (2017)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gainutdinova, A., Yakaryılmaz, A.: Nondeterministic unitary OBDDs. In: Weil, P. (ed.) CSR 2017. LNCS, vol. 10304, pp. 126–140. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58747-9_13. arXiv:1612.07015

    Chapter  Google Scholar 

  16. Homeister, M., Waack, S.: Quantum ordered binary decision diagrams with repeated tests. arXiv preprint arXiv:quant-ph/0507258 (2005)

  17. Khadiev, K.: Width hierarchy for k-OBDD of small width. Lobachevskii J. Math. 36(2), 178–183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khadiev, K.: On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs. Lobachevskii J. Math. 37(6), 682–703 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khadiev, K., Ibrahimov, R.: Width hierarchies for quantum and classical ordered binary decision diagrams with repeated test. In: Proceedings of the Fourth Russian Finnish Symposium on Discrete Mathematics. TUCS Lecture Notes, no. 26. Turku Centre for Computer Science (2017)

    Google Scholar 

  20. Khadiev, K., Khadieva, A.: Reordering method and hierarchies for quantum and classical ordered binary decision diagrams. In: Weil, P. (ed.) CSR 2017. LNCS, vol. 10304, pp. 162–175. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58747-9_16

    Chapter  Google Scholar 

  21. Klauck, H.: On quantum and probabilistic communication: Las Vegas and one-way protocols. In: STOC 2000: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 644–651 (2000)

    Google Scholar 

  22. Klauck, H.: Quantum communication complexity. arXiv preprint arXiv:quant-ph/0005032 (2000)

  23. Klauck, H.: Lower bounds for quantum communication complexity. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 288–297. IEEE (2001)

    Google Scholar 

  24. Klauck, H.: Lower bounds for quantum communication complexity. SIAM J. Comput. 37(1), 20–46 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Klauck, H., Nayak, A., Ta-Shma, A., Zuckerman, D.: Interaction in quantum communication and the complexity of set disjointness. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 124–133. ACM (2001)

    Google Scholar 

  26. Klauck, H., Nayak, A., Ta-Shma, A., Zuckerman, D.: Interaction in quantum communication. IEEE Trans. Inf. Theory 53(6), 1970–1982 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. Random Struct. Algorithms 34(3), 368–394 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. 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.-Z., Eades, P., Estivill-Castro, V., Lin, X., Sharma, A. (eds.) COCOON 2000. LNCS, vol. 1858, pp. 467–476. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44968-X_46

    Chapter  Google Scholar 

  30. Nisan, N., Widgerson, A.: Rounds in communication complexity revisited. In: Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pp. 419–429. ACM (1991)

    Google Scholar 

  31. Raz, R.: Exponential separation of quantum and classical communication complexity. In: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pp. 358–367. ACM (1999)

    Google Scholar 

  32. Sauerhoff, M.: Quantum vs. classical read-once branching programs. In: Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, no. 06111. Internationales Begegnungs- und Forschungszentrum für Informatik (2006)

    Google Scholar 

  33. Sauerhoff, M., Sieling, D.: Quantum branching programs and space-bounded nonuniform quantum complexity. Theoret. Comput. Sci. 334(1–3), 177–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Vasiliev, A.V.: Functions computable by boolean circuits of logarithmic depth and branching programs of a special type. J. Appl. Ind. Math. 2(4), 585–590 (2008). https://doi.org/10.1134/S1990478908040145

    Article  MathSciNet  Google Scholar 

  35. Wegener, I.: Branching Programs and Binary Decision Diagrams: Theory and Applications. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  36. Yakaryılmaz, A., Say, A.C.C.: Succinctness of two-way probabilistic and quantum finite automata. Discret. Math. Theoret. Comput. Sci. 12(2), 19–40 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Zheng, S., Gruska, J.: Time-space tradeoffs for two-way finite automata. arXiv preprint arXiv:1507.01346 (2015)

Download references

Acknowledgements

The work is partially supported by ERC Advanced Grant MQC. The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.

Author information

Authors and Affiliations

  1. Faculty of Computing, Center for Quantum Computer Science, University of Latvia, Rīga, Latvia

    Andris Ambainis & Kamil Khadiev

  2. Kazan Federal University, Kazan, Russia

    Farid Ablayev, Kamil Khadiev & Aliya Khadieva

Authors
  1. Farid Ablayev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andris Ambainis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kamil Khadiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Aliya Khadieva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kamil Khadiev .

Editor information

Editors and Affiliations

  1. Vienna University of Technology , Vienna, Austria

    A Min Tjoa

  2. ISAE-ENSMA, Chasseneuil-du-Poitou, France

    Ladjel Bellatreche

  3. Vienna University of Technology, Vienna, Austria

    Stefan Biffl

  4. Utrecht University, Utrecht, The Netherlands

    Jan van Leeuwen

  5. Academy of Sciences, Prague, Czech Republic

    Jiří Wiedermann

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ablayev, F., Ambainis, A., Khadiev, K., Khadieva, A. (2018). Lower Bounds and Hierarchies for Quantum Memoryless Communication Protocols and Quantum Ordered Binary Decision Diagrams with Repeated Test. In: Tjoa, A., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds) SOFSEM 2018: Theory and Practice of Computer Science. SOFSEM 2018. Lecture Notes in Computer Science(), vol 10706. Edizioni della Normale, Cham. https://doi.org/10.1007/978-3-319-73117-9_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-73117-9_14

  • Published:

  • Publisher Name: Edizioni della Normale, Cham

  • Print ISBN: 978-3-319-73116-2

  • Online ISBN: 978-3-319-73117-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation