Reconstructing Strings from Substrings with Quantum Queries

  • Richard Cleve
  • Kazuo Iwama
  • François Le Gall
  • Harumichi Nishimura
  • Seiichiro Tani
  • Junichi Teruyama
  • Shigeru Yamashita
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


This paper investigates the number of quantum queries made to solve the problem of reconstructing an unknown string from its substrings in a certain query model. More concretely, the goal of the problem is to identify an unknown string S by making queries of the following form: “Is s a substring of S?”, where s is a query string over the given alphabet. The number of queries required to identify the string S is the query complexity of this problem.

First we show a quantum algorithm that exactly identifies the string S with at most \(\frac{3}{4}N + o(N)\) queries, where N is the length of S. This contrasts sharply with the classical query complexity N. Our algorithm uses Skiena and Sundaram’s classical algorithm and the Grover search as subroutines. To make them effectively work, we develop another subroutine that finds a string appearing only once in S, which may have an independent interest. We also prove two lower bounds. The first one is a general lower bound of \(\Omega(\frac{N}{\log^2{N}})\), which means we cannot achieve a query complexity of O(N 1 − ε ) for any constant ε. The other one claims that if we cannot use queries of length roughly between logN and 3 logN, then we cannot achieve a query complexity of any sublinear function in N.


quantum computing string algorithms query complexity lower bounds 


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  1. 1.
    Ambainis, A.: A note on quantum black-box complexity of almost all Boolean functions. Inf. Process. Lett. 71(1), 5–7 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ambainis, A.: A better lower bound for quantum algorithms searching an ordered list. In: Proc. 40th FOCS, pp. 352–357 (1999)Google Scholar
  3. 3.
    Ambainis, A.: Quantum lower bounds by quantum arguments. J. Comput. Syst. Sci. 64(4), 750–767 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ben-Or, M., Hassidim, A.: The Bayesian learner is optimal for noisy binary search (and pretty good for quantum as well). In: Proc. 49th FOCS, pp. 221–230 (2008)Google Scholar
  6. 6.
    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26(5), 1411–1473 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte Der Physik 46, 493–505 (1998)CrossRefGoogle Scholar
  8. 8.
    Buhrman, H., de Wolf, R.: A lower bound for quantum search of an ordered list. Inf. Process. Lett. 70(5), 205–209 (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    van Dam, W.: Quantum oracle interrogation: Getting all information for almost half the price. In: Proc. 39th FOCS, pp. 362–367 (1998)Google Scholar
  10. 10.
    Dramanac, R., Crkvenjakov, R.: DNA sequencing by hybridization. Yugoslav Patent Application 570 (1987)Google Scholar
  11. 11.
    Childs, A.M., Landahl, A.J., Parrilo, P.A.: Improved quantum algorithms for the ordered search problem via semidefinite programming. Physical Review A 75(3), 032335 (2007)Google Scholar
  12. 12.
    Childs, A.M., Lee, T.: Optimal Quantum Adversary Lower Bounds for Ordered Search. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 869–880. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Cleve, R., Iwama, K., Le Gall, F., Nishimura, H., Tani, S., Teruyama, J., Yamashita, S.: Reconstructing Strings from Substrings with Quantum Queries. arXiv:1204.4691 (2012)Google Scholar
  14. 14.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: A limit on the speed of quantum computation for insertion into an ordered list, arXiv:quant-ph/9812057Google Scholar
  15. 15.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Invariant quantum algorithms for insertion into an ordered list, arXiv:quant-ph/9901059Google Scholar
  16. 16.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. 28th STOC, pp. 212–219 (1996)Google Scholar
  17. 17.
    Høyer, P., Neerbek, J., Shi, Y.: Quantum complexities of ordered searching, sorting, and element distinctness. Algorithmica 34(4), 429–448 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Iwama, K., Nishimura, H., Raymond, R., Teruyama, J.: Quantum Counterfeit Coin Problems. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 85–96. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Lysov, Y.P., Florent’ev, V.L., Khorlin, A.A., Khrapko, K., Shik, V.V., Mirzabekov, A.D.: DNA Sequencing by hybridization with oligonucleotides. A novel method. Dokl. Acad. Sci USSR 303, 1508–1511 (1988)Google Scholar
  20. 20.
    Pevzner, P.A., Lipshutz, R.J.: Towards DNA Sequencing Chips. In: Privara, I., Ružička, P., Rovan, B. (eds.) MFCS 1994. LNCS, vol. 841, pp. 143–158. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  21. 21.
    Ramesh, H., Vinay, V.: String matching in Õ\((\sqrt{n}+\sqrt{m})\) quantum time. J. Discrete Algorithms 1(1), 103–110 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Skiena, S.S., Sundaram, G.: Reconstructing strings from substrings. J. Computational Biol. 2(2), 333–353 (1995)CrossRefGoogle Scholar
  23. 23.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Snir, M.: Lower bounds on probabilistic linear decision trees. Theoret. Comput. Sci. 38, 69–82 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Špalek, R., Szegedy, M.: All quantum adversary methods are equivalent. Theory of Computing 2(1), 1–18 (2006)MathSciNetGoogle Scholar
  26. 26.
    Zhang, S.: On the power of Ambainis lower bounds. Theoret. Comput. Sci. 339(2-3), 241–256 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Richard Cleve
    • 1
    • 2
  • Kazuo Iwama
    • 3
  • François Le Gall
    • 4
  • Harumichi Nishimura
    • 5
  • Seiichiro Tani
    • 6
  • Junichi Teruyama
    • 3
  • Shigeru Yamashita
    • 7
  1. 1.Institute for Quantum Computing and School of Computer ScienceUniversity of WaterlooCanada
  2. 2.Perimeter Institute for Theoretical PhysicsCanada
  3. 3.School of InformaticsKyoto UniversityJapan
  4. 4.Department of Computer ScienceThe University of TokyoJapan
  5. 5.School of Information ScienceNagoya UniversityJapan
  6. 6.NTT Communication Science LaboratoriesNTT CorporationAtsugiJapan
  7. 7.College of Information Science and EngineeringRitsumeikan UniversityJapan

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