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Quantum Counterfeit Coin Problems

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Algorithms and Computation (ISAAC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6506))

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Abstract

The counterfeit coin problem requires us to find all false coins from a given bunch of coins using a balance scale. We assume that the balance scale gives us only “balanced” or “tilted” information and that we know the number k of false coins in advance. The balance scale can be modeled by a certain type of oracle and its query complexity is a measure for the cost of weighing algorithms (the number of weighings). In this paper, we study the quantum query complexity for this problem. Let Q(k,N) be the quantum query complexity of finding all k false coins from the N given coins. We show that for any k and N such that k < N/2, Q(k,N) = O(k 1/4), contrasting with the classical query complexity, Ω(klog(N/k)), that depends on N. So our quantum algorithm achieves a quartic speed-up for this problem. We do not have a matching lower bound, but we show some evidence that the upper bound is tight: any algorithm, including our algorithm, that satisfies certain properties needs Ω(k 1/4) queries.

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References

  1. Ambainis, A.: Quantum lower bounds by quantum arguments. J. Comput. Syst. Sci. 64, 750–767 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambainis, A.: Polynomial degree vs. quantum query complexity. J. Comput. Syst. Sci. 72, 220–238 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnum, H., Saks, M.E., Szegedy, M.: Quantum query complexity and semi-definite programming. In: Proc. 18th CCC, pp. 179–193 (2003)

    Google Scholar 

  4. Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26, 1411–1473 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte Der Physik 46, 493–505 (1998)

    Article  Google Scholar 

  6. Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Quantum Computation and Quantum Information: A Millennium Volume. AMS Contemporary Mathematics Series, vol. 305, pp. 53–74 (2002)

    Google Scholar 

  7. van Dam, W., Shparlinski, I.: Classical and quantum algorithms for exponential congruences. In: Kawano, Y., Mosca, M. (eds.) TQC 2008. LNCS, vol. 5106, pp. 1–10. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  8. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. 28th STOC, pp. 212–219 (1996)

    Google Scholar 

  9. Guy, R.K., Nowakowski, R.J.: Coin-weighing problems. Amer. Math. Monthly 102, 164–167 (1995)

    Article  MATH  Google Scholar 

  10. Halbeisen, L., Hungerbühler, N.: The general counterfeit coin problem. Discrete Mathematics 147, 139–150 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Høyer, P., Lee, T., Špalek, R.: Negative weights make adversaries stronger. In: Proc. 39th STOC, pp. 526–535 (2007)

    Google Scholar 

  12. Iwama, K., Nishimura, H., Raymond, R., Teruyama, J.: Quantum counterfeit coin problems. arXiv:1009.0416 (2010)

    MATH  Google Scholar 

  13. Laplante, S., Magniez, F.: Lower bounds for randomized and quantum query complexity using Kolmogorov arguments. SIAM J. Comput. 38, 46–62 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, W.A., Zhang, W.G., Nie, Z.K.: Searching for two counterfeit coins with two-arms balance. Discrete Appl. Math. 152, 187–212 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM J. Comput. 37, 413–424 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Manvel, B.: Counterfeit coin problems. Mathematics Magazine 50, 90–92 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reichardt, B.: Span programs and quantum query complexity: The general adversary bound is nearly tight for every boolean function. In: Proc. 50th FOCS, pp. 544–551 (2009)

    Google Scholar 

  18. Reichardt, B.: Reflections for quantum query algorithms. arXiv:1005.1601 (2010)

    Google Scholar 

  19. Špalek, R., Szegedy, M.: All quantum adversary methods are equivalent. Theory of Computing 2, 1–18 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Terhal, B.M., Smolin, J.A.: Single quantum querying of a database. Phys. Rev. A 58, 1822–1826 (1998)

    Article  MathSciNet  Google Scholar 

  21. Zhang, S.: On the power of Ambainis’s lower bounds. Theoret. Comput. Sci. 339, 241–256 (2005)

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Informatics, Kyoto University, Japan

    Kazuo Iwama & Junichi Teruyama

  2. School of Science, Osaka Prefecture University, Japan

    Harumichi Nishimura

  3. IBM Research, Tokyo, Japan

    Rudy Raymond

Authors
  1. Kazuo Iwama
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  2. Harumichi Nishimura
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  3. Rudy Raymond
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  4. Junichi Teruyama
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Editor information

Editors and Affiliations

  1. Department of Computer Science, KAIST, Gwahangno 335, 305-701, Yuseong-gu, Daejeon, Korea

    Otfried Cheong  & Kyung-Yong Chwa  & 

  2. School of Computer Science and Engineering, Seoul National University, 151-742, Seoul, Korea

    Kunsoo Park

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Iwama, K., Nishimura, H., Raymond, R., Teruyama, J. (2010). Quantum Counterfeit Coin Problems. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6506. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17517-6_10

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  • DOI: https://doi.org/10.1007/978-3-642-17517-6_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-17516-9

  • Online ISBN: 978-3-642-17517-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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