Skip to main content

Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies

Abstract

In this paper, we study quantum Ordered Binary Decision Diagrams(\(\mathrm {OBDD}\)) model; it is a restricted version of read-once quantum branching programs, with respect to “width” complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique (“reordering”) for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function \(\mathrm {REQ}\) such that the deterministic \(\mathrm {OBDD}\) complexity of it is at least \(2^{\varOmega (n / \log n)}\), and the quantum \(\mathrm {OBDD}\) complexity of it is at most \(O(n^2/\log n)\). It is the biggest known gap for explicit functions not representable by \(\mathrm {OBDD}\)s of a linear width. Another function(shifted equality function) allows us to obtain a gap \(2^{\varOmega (n)}\) vs \(O(n^2)\). Moreover, we prove the bounded error quantum and probabilistic \(\mathrm {OBDD}\) width hierarchies for complexity classes of Boolean functions. Additionally, using “reordering” method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (\({\textit{k}}\text {-}\mathrm {OBDD}\)) of polynomial width, for \(k = o(n / \log ^3 n)\). We prove a similar hierarchy for bounded error probabilistic \({\textit{k}}\text {-}\mathrm {OBDD}\)s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017 Khadiev and Khadieva (2017)

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

Notes

  1. We use \(\log\) to denote logarithms base 2.

References

  • Ablayev F, Ablayev M, Huang JZ, Khadiev K, Salikhova N, Wu D (2019) On quantum methods for machine learning problems part i: quantum tools. Big Data Min Anal 3(1):41–55

    Article  Google Scholar 

  • 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: International Conference on Current Trends in Theory and Practice of Informatics. LNCS, vol. 10706, pp. 197–211. Springer

  • Ablayev F, Gainutdinova A, Khadiev K, Yakaryılmaz A (2016) Very narrow quantum OBDDs and width hierarchies for classical OBDDs. Lobachevskii J Math 37(6):670–682. https://doi.org/10.1134/S199508021606007X

    MathSciNet  Article  MATH  Google Scholar 

  • Ablayev F, Gainutdinova A (2005) Complexity of quantum uniform and nonuniform automata. In: de Felice, C., Restivo, A. (eds.) Developments in language theory, 9th international conference, DLT 2005, Palermo, Italy, July 4-8, 2005, Proceedings. Lecture Notes in Computer Science, vol. 3572, pp. 78–87. Springer . https://doi.org/10.1007/11505877_7

  • Ablayev F, Gainutdinova A, Karpinski M (2001) On Computational Power of Quantum Branching Programs. In: Freivalds, R. (ed.) Fundamentals of Computation Theory, 13th International Symposium, FCT 2001, Riga, Latvia, August 22-24, 2001, Proceedings. Lecture Notes in Computer Science. vol. 2138, pp. 59–70. Springer https://doi.org/10.1007/3-540-44669-9_8

  • Ablayev F, Gainutdinova A, Karpinski M, Moore C, Pollett C (2005) On the computational power of probabilistic and quantum branching program. Inf Comput 203(2):145–162. https://doi.org/10.1016/j.ic.2005.04.003

    MathSciNet  Article  MATH  Google Scholar 

  • Ablayev F, Gainutdinova A, Khadiev K, Yakaryilmaz A (2014) Very narrow quantum OBDDs and width hierarchies for classical OBDDs. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) Descriptional Complexity of Formal Systems - 16th International Workshop, DCFS 2014, Turku, Finland, August 5-8, 2014. Proceedings. Lecture Notes in Computer Science, vol. 8614, pp. 53–64. Springer

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

    MathSciNet  Article  Google Scholar 

  • Ablayev F, Khasianov A, Vasiliev A (2010) On complexity of quantum branching programs computing equality-like boolean functions. In: ECCC 286

  • Ablayev F, Vasiliev A (2008) On the computation of boolean functions by quantum branching programs via fingerprinting. Elect Coll Comput Comp (ECCC) 15(059):1–9.  http://eccc.hpi-web.de/eccc-reports/2008/TR08-059/index.html

  • Ablayev FM, Vasiliev A (2013) Algorithms for quantum branching programs based on fingerprinting. Int J Softw Inform 7(4):485–500

    MATH  Google Scholar 

  • Ablayev FM, Vasilyev A (2009) On quantum realisation of boolean functions by the fingerprinting technique. Discret Math Appl 19(6):555–572

    Article  Google Scholar 

  • Ambainis A, Nahimovs N (2008) Improved constructions of quantum automata. TQC. Springer, Berlin, pp 47–56

    Google Scholar 

  • Ambainis A, Nahimovs N (2009) Improved constructions of quantum automata. Theor Comput Sci 410(20):1916–1922

    MathSciNet  Article  Google Scholar 

  • Ambainis A, Freivalds R (1998) 1-way quantum finite automata: strengths, weaknesses and generalizations. In: FOCS’98: Proceedings of the 39th Annual Symposium on Foundations of Computer Science. pp. 332–341 (http://arxiv.org/abs/quant-ph/9802062)

  • Ambainis A, Yakaryılmaz A (2015) Automata and quantum computing. Tech Rep arXiv:1507.01988

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

    Article  Google Scholar 

  • Gainutdinova AF (2015) Comparative complexity of quantum and classical OBDDs for total and partial functions. Russ Math 59(11):26–35. https://doi.org/10.3103/S1066369X15110031

    MathSciNet  Article  MATH  Google Scholar 

  • Gainutdinova A, Yakaryılmaz A (2015) Unary probabilistic and quantum automata on promise problems. Developments in language theory. Springer, Cham, pp 252–263

    Chapter  Google Scholar 

  • Gainutdinova A, Yakaryılmaz A (2017) Nondeterministic unitary obdds pp. 126–140

  • Gainutdinova A, Yakaryılmaz A (2018) Unary probabilistic and quantum automata on promise problems. Quantum Inform Proc 17(2):28

    MathSciNet  Article  Google Scholar 

  • Hromkovič J, Sauerhoff M (2003) The power of nondeterminism and randomness for oblivious branching programs. Theory Comput Syst 36(2):159–182

    MathSciNet  Article  Google Scholar 

  • Ibrahimov R, Khadiev K, Prūsis K, Yakaryılmaz A (2021) Error-free affine, unitary, and probabilistic obdds. Int J Found Comput Sci 32(7):849–860

    MathSciNet  Article  Google Scholar 

  • Ibrahimov R, Khadiev K, Prūsis K, Yakaryılmaz A (2017) Zero-error affine, unitary, and probabilistic OBDDs. arXiv preprint arXiv:1703.07184

  • JáJá J, Prasanna VK, Simon J (1984) Information transfer under different sets of protocols. SIAM J Comput 13(4):840–849. https://doi.org/10.1137/0213052

    MathSciNet  Article  MATH  Google Scholar 

  • Khadiev K (2015) Width hierarchy for \(k\)-OBDD of small width. Lobachevskii J Math 36(2):178–83

    MathSciNet  Article  Google Scholar 

  • Khadiev K, Ibrahimov R, Yakaryılmaz A (2018) New size hierarchies for two way automata. arXiv: 1801.10483

  • Khadiev K, Khadieva A (2019) Two-way quantum and classical machines with small memory for online minimization problems. In: International conference on micro- and nano-electronics 2018. Proc. SPIE, vol. 11022, p. 110222T . https://doi.org/10.1117/12.2522462

  • Khadiev K, Khadieva A (2021) Quantum online streaming algorithms with logarithmic memory. Int J Theor Phys 60:608–616. https://doi.org/10.1007/s10773-019-04209-1

    MathSciNet  Article  MATH  Google Scholar 

  • Khadiev K, Khadieva A, Kravchenko D, Rivosh A, Yamilov R, Mannapov, I (2017) Quantum versus classical online algorithms with advice and logarithmic space. arXiv:1710.09595

  • Khadiev K, Khadieva A, Mannapov I (2017) Quantum online algorithms with respect to space complexity. arXiv:1709.08409

  • Khadiev K, Khadieva A, Mannapov I (2018) Quantum online algorithms with respect to space and advice complexity. Lobachevskii J Math 39(9):1210–1220

    MathSciNet  Article  Google Scholar 

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

  • Khadiev K, Khadieva A (2017) Reordering method and hierarchies for quantum and classical ordered binary decision diagrams. computer science - theory and applications: 12th International Computer Science Symposium in Russia. CSR 2017, Kazan, Russia, June 8–12, 2017, Proceedings. Springer International Publishing, Cham, pp 162–175

  • Khadiev K, Khadieva A (2022) Quantum and classical log-bounded automata for the online disjointness problem. Mathematics 10(1):143. https://doi.org/10.3390/math10010143

    Article  Google Scholar 

  • Khadiev K, Khadieva A, Ziatdinov M, Mannapov I, Kravchenko D, Rivosh A, Yamilov R (2022) Two-way and one-way quantum and classical automata with advice for online minimization problems. Theor Comput Sci 920:76–94. https://doi.org/10.1016/j.tcs.2022.02.026

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  • Krajícek J (2008) An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams. J Symb Log 73(1):227–237. https://doi.org/10.2178/jsl/1208358751

    MathSciNet  Article  MATH  Google Scholar 

  • Krause M, Meinel C, Waack S (1991) Separating the eraser turing machine classes Le, NLe, co-NLe and Pe. Theor Comput Sci 86(2):267–275

    Article  Google Scholar 

  • Kushilevitz E, Nisan N (1997) Communication complexity. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Le Gall F (2006) Exponential separation of quantum and classical online space complexity. In: SPAA ’06, ACM pp. 67–73

  • Le Gall F (2009) Exponential separation of quantum and classical online space complexity. Theory Comput Syst 45(2):188–202

    MathSciNet  Article  Google Scholar 

  • Nakanishi M, Hamaguchi K, Kashiwabara, T (2000) Ordered quantum branching programs are more powerful than ordered probabilistic branching programs under a bounded-width restriction. In: COCOON. LNCS, vol. 1858, pp. 467–476. Springer

  • Nisan N, Wigderson A (1993) Rounds in communication complexity revisited. SIAM J Comput 22(1):211–219. https://doi.org/10.1137/0222016

    MathSciNet  Article  MATH  Google Scholar 

  • Qiu D, Yu S (2009) Hierarchy and equivalence of multi-letter quantum finite automata. Theor Comput Sci 410(30–32):3006–3017

    MathSciNet  Article  Google Scholar 

  • Sauerhoff M (2005) Quantum vs. classical read-once branching programs. arXiv preprint quant-ph/0504198

  • Sauerhoff M (2006) Quantum vs. classical read-once branching programs. In: Krause, M., Pudlák, P., Reischuk, R., van Melkebeek, D. (eds.) Complexity of Boolean Functions. No. 06111 in Dagstuhl Seminar Proceedings, Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany, Dagstuhl, Germany , http://drops.dagstuhl.de/opus/volltexte/2006/616

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

    MathSciNet  Article  Google Scholar 

  • Savickỳ P, Žák S (2000) A read-once lower bound and a (1,+ k)-hierarchy for branching programs. Theor Comput Sci 238(1):347–362

    MathSciNet  Article  Google Scholar 

  • Say AC, Yakaryılmaz A (2014) Quantum finite automata: a modern introduction. Comput New Resour. Springer, Cham, pp 208–222

    Chapter  Google Scholar 

  • Wegener I (2000) Branching Programs and Binary Decision Diagrams, SIAM. https://doi.org/10.1137/1.9780898719789

Download references

Acknowledgements

This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program (“PRIORITY-2030”). We thank Alexander Vasiliev and Aida Gainutdinova from Kazan Federal University and Andris Ambainis from University of Latvia for their helpful comments and discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kamil Khadiev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khadiev, K., Khadieva, A. & Knop, A. Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies. Nat Comput (2022). https://doi.org/10.1007/s11047-022-09904-3

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11047-022-09904-3

Keywords

  • Quantum computing
  • Quantum OBDD
  • OBDD
  • Branching programs
  • Quantum vs classical
  • Quantum models
  • Hierarchy
  • Computational complexity
  • Probabilistic OBDD