Advertisement

Optimal Parallel Quantum Query Algorithms

  • Stacey Jeffery
  • Frederic Magniez
  • Ronald de Wolf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

Abstract

We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. We show tight bounds for a number of problems, specifically Θ((n/p)2/3) p-parallel queries for element distinctness and Θ((n/p) k/(k + 1)) for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to f’s block sensitivity.

Keywords

Boolean Function Orthogonal Array Quantum Algorithm Query Complexity Quantum Circuit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and the element distinctness problems. J. of the ACM 51(4), 595–605 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ambainis, A.: Quantum lower bounds by quantum arguments. J. of Computer and System Sciences 64(4), 750–767 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. on Computing 37(1), 210–239 (2007), Earlier version in FOCS 2004. quant-ph/0311001CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Ambainis, A., Bačkurs, A., Smotrovs, J., de Wolf, R.: Optimal quantum query bounds for almost all Boolean functions. In: Proc. 30th STACS, pp. 446–453 (2013)Google Scholar
  5. 5.
    Beals, R., Brierley, S., Gray, O., Harrow, A., Kutin, S., Linden, N., Shepherd, D., Stather, M.: Efficient distributed quantum computing. Proc. of the Royal Society A469, 2153 (2013)Google Scholar
  6. 6.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. of the ACM 48(4), 778–797 (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    de Beaudrap, N., Cleve, R., Watrous, J.: Sharp quantum vs. classical query complexity separations. Algorithmica 34(4), 449–461 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Belovs, A.: Learning-graph-based quantum algorithm for k-distinctness. In: Proc. of 53rd IEEE FOCS, pp. 207–216 (2012)Google Scholar
  9. 9.
    Belovs, A.: Span programs for functions with constant-sized 1-certificates. In: Proc. of 43rd ACM STOC, pp. 77–84 (2012)Google Scholar
  10. 10.
    Belovs, A.: Adversary lower bound for element distinctness, arXiv:1204.5074 (2012)Google Scholar
  11. 11.
    Belovs, A., Childs, A.M., Jeffery, S., Kothari, R., Magniez, F.: Time-efficient quantum walks for 3-distinctness. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 105–122. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Belovs, A., Lee, T.: Quantum algorithm for k-distinctness with prior knowledge on the input. Technical Report arXiv:1108.3022, arXiv (2011)Google Scholar
  13. 13.
    Belovs, A., Rosmanis, A.: On the power of non-adaptive learning graphs. In: Proc. of 28th IEEE CCC, pp. 44–55 (2013) References are to arXiv:1210.3279v2Google Scholar
  14. 14.
    Belovs, A., Špalek, R.: Adversary lower bound for the k-sum problem. In: Proc. of 4th ITCS, pp. 323–328 (2013)Google Scholar
  15. 15.
    Buhrman, H., de Wolf, R.: Complexity measures and decision tree complexity: A survey. Theoretical Computer Science 288(1), 21–43 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Cleve, R., Watrous, J.: Fast parallel circuits for the quantum Fourier transform. In: Proc. of 41st IEEE FOCS, pp. 526–536 (2000)Google Scholar
  17. 17.
    van Dam, W.: Quantum oracle interrogation: Getting all information for almost half the price. In: Proc. of 39th IEEE FOCS, pp. 362–367 (1998)Google Scholar
  18. 18.
    Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. of the Royal Society of London A439, 553–558 (1992)Google Scholar
  19. 19.
    Green, F., Homer, S., Moore, C., Pollett, C.: Counting, fanout and the complexity of quantum ACC. Quantum Inf. and Comp. 2(1), 35–65 (2002)MathSciNetGoogle Scholar
  20. 20.
    Grover, L., Rudolph, T.: How significant are the known collision and element distinctness quantum algorithms? Quantum Inf. and Comp. 4(3), 201–206 (2004)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. of 28th ACM STOC, pp. 212–219 (1996)Google Scholar
  22. 22.
    Høyer, P., Lee, T., Špalek, R.: Negative weights make adversaries stronger. In: Proc. of 39th ACM STOC, pp. 526–535 (2007)Google Scholar
  23. 23.
    Høyer, P., Špalek, R.: Quantum fan-out is powerful. Th. Comp. 1, 81–103 (2005)Google Scholar
  24. 24.
    Jeffery, S., Magniez, F., de Wolf, R.: Optimal parallel quantum query algorithms. arXiv:1309.6116 (2013)Google Scholar
  25. 25.
    Jozsa, R.: An introduction to measurement based quantum computation. In: Angelakis, D.G., Christandl, M., Ekert, A. (eds.) Quantum Information Processing, pp. 137–158. IOS Press (2006) arXiv:0508124Google Scholar
  26. 26.
    Lee, T., Mittal, R., Reichardt, B., Špalek, R., Szegedy, M.: Quantum query complexity of state conversion. In: Proc. of 52nd IEEE FOCS, pp. 344–353 (2011) References are to arXiv:1011.3020v2Google Scholar
  27. 27.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. SIAM J. on Computing 40(1), 142–164 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Montanaro, A.: Nonadaptive quantum query complexity. Information Processing Letters 110(24), 1110–1113 (2010)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Moore, C., Nilsson, M.: Parallel quantum computation and quantum codes. SIAM J. on Computing 31(3), 799–815 (2002)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Nisan, N.: CREW PRAMs and decision trees. SIAM J. on Computing 20(6), 999–1007 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Nisan, N., Szegedy, M.: On the degree of Boolean functions as real polynomials. Computational Complexity 4(4), 301–313 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Reichardt, B.: Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function. In: Proc. of 50th IEEE FOCS, pp. 544–551 (2009)Google Scholar
  33. 33.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. on Computing 26(5), 1484–1509 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Simon, D.: On the power of quantum computation. SIAM J. on Computing 26(5), 1474–1483 (1997)CrossRefzbMATHGoogle Scholar
  35. 35.
    Takahashi, Y., Tani, S.: Collapse of the hierarchy of constant-depth exact quantum circuits. In: Proc. of 28th IEEE CCC (2013)Google Scholar
  36. 36.
    Zalka, C.: Grover’s quantum searching algorithm is optimal. Physical Review A 60, 2746–2751 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Stacey Jeffery
    • 1
  • Frederic Magniez
    • 2
  • Ronald de Wolf
    • 3
  1. 1.IQCUniversity of WaterlooCanada
  2. 2.CNRS, LIAFA, Univ Paris Diderot, Sorbonne Paris-CitéParisFrance
  3. 3.CWI and University of AmsterdamThe Netherlands

Personalised recommendations