Quantum Information Processing

, Volume 1, Issue 5, pp 365–388 | Cite as

Path Integration on a Quantum Computer

Article

Abstract

We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j-k with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an ε-approximation to path integrals whose integrands are at least Lipschitz. We prove:

• Path integration on a quantum computer is tractable.

• Path integration on a quantum computer can be solved roughly ε-1 times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance.

• The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most 4.46 ε-1. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved.

• The number of qubits is polynomial in ε-1. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands.

PACS: 03.67.Lx; 31.15Kb; 31.15.-p; 02.70.-c

Quantum computation quantum summation path integration quantum queries quantum speedup number of qubits 

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REFERENCES

  1. 1.
    D. S. Abrams and C. P. Williams, LANL http://arXiv.org/abs/quant-ph/9908083.Google Scholar
  2. 2.
    G. Brassard, P. Høyer, M. Mosca and A. Tapp, http://arXiv.org/abs/quant-ph/0005055.Google Scholar
  3. 3.
    F. Curbera, J. Complexity 16, 474 (2000).Google Scholar
  4. 4.
    L. Grover, Phys. Rev. Lett. 79, 325 (1996). See also http://arXiv.org/abs/quant-ph/9706033.Google Scholar
  5. 5.
    L. Grover, Proceedings 30th Annual ACM Symp. on the Theory of Computing (ACM Press, New York). See also http://arXiv.org/abs/quant-ph/9711043 and Phys. Rev. Lett. 80, 4329 (1998).Google Scholar
  6. 6.
    S. Heinrich, J. Complexity 18, 1 (2002). See also http://arXiv.org/abs/quant-ph/0105116.Google Scholar
  7. 7.
    S. Heinrich, J. Complexity 19, 1 (2003). See also http://arXiv.org/abs/quant-ph/0112153.Google Scholar
  8. 8.
    S. Heinrich and E. Novak, In: K.-T. Fang, F. J. Hickernell, and H. Niederreiter, (eds), Monte Carlo and Quasi-Monte Carlo Methods 2000 (Springer-Verlag, Berlin, 2002). See also http://arXiv.org/abs/quant-ph/0105114.Google Scholar
  9. 9.
    S. Heinrich and E. Novak, J. Complexity 19, 1 (2003). See also http://arXiv.org/abs/quantph/ 0109038.Google Scholar
  10. 10.
    M. Kwas and Y. Li, submitted for publication.Google Scholar
  11. 11.
    M. Kwas and H. Wo?niakowski, J. Complexity (to appear).Google Scholar
  12. 12.
    A. Nayak and F. Wu, STOC, May, 1999, 384-393. See also http://arXiv.org/abs/quant-ph/9804066.Google Scholar
  13. 13.
    H. Niederreiter, CBMS-NSF Reg. Conf. Series Appl. Math., 63 (SIAM, Philadelphia, 1992).Google Scholar
  14. 14.
    E. Novak, Lecture Notes in Mathematics, 1349 (Springer Verlag, Berlin, 1988).Google Scholar
  15. 15.
    E. Novak, J. Complexity, 11, 57 (1995).Google Scholar
  16. 16.
    E. Novak, J. Complexity, 17, 2 (2001). See also http://arXiv.org/abs/quant-ph/0008124.Google Scholar
  17. 17.
    E. Novak, I. H. Sloan, and H. Wo?niakowski. http://arXiv.org/abs/quant-ph/0206023.Google Scholar
  18. 18.
    L. Plaskota, G.W. Wasilkowski, and H. Wo?niakowski, J. Comp. Phys. 164, 355 (2000).Google Scholar
  19. 19.
    P. W. Shor, Proceedings of the 35th Annual Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, California) See also http://arXiv.org/abs/ quant-ph/9508027.Google Scholar
  20. 20.
    P. W. Shor, Documenta Mathematica (extra volume ICM) I, 467 (1998).Google Scholar
  21. 21.
    J. F. Traub, Physics Today, May, 39 (1999).Google Scholar
  22. 22.
    J. F. Traub, G. W. Wasilkowski, and H. Wo?niakowski, Information-based Complexity (Academic Press, New York, 1988).Google Scholar
  23. 23.
    J. F. Traub and A. G. Werschulz, Information and Complexity (Cambridge University Press, Cambridge, UK, 1998).Google Scholar
  24. 24.
    N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions on Banach Spaces (Reidel, Dordrecht, 1987).Google Scholar
  25. 25.
    G. W. Wasilkowski and H. Wo?niakowski, J. Math. Physics 37(4), 2071 (1996).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  1. 1.Computer ScienceColumbia UniversityColumbia
  2. 2.Computer Science, Columbia University and Institute of Applied MathematicsUniversity of WarsawColumbia

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