Path Integration on a Quantum Computer
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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
- D. S. Abrams and C. P. Williams, LANL http://arXiv.org/abs/quant-ph/9908083.
- G. Brassard, P. Høyer, M. Mosca and A. Tapp, http://arXiv.org/abs/quant-ph/0005055.
- Curbera, F. (2000) J. Complexity 16: pp. 474
- Grover, L. (1996) Phys. Rev. Lett. 79: pp. 325
- Grover, L. (1998) Proceedings 30th Annual ACM Symp. on the Theory of Computing. ACM Press, New York
- Heinrich, S. (2002) J. Complexity 18: pp. 1
- Heinrich, S. (2003) J. Complexity 19: pp. 1
- Heinrich, S., Novak, E. In: Fang, K.-T., Hickernell, F. J., Niederreiter, H. eds. (2002) Monte Carlo and Quasi-Monte Carlo Methods 2000. Springer-Verlag, Berlin
- Heinrich, S., Novak, E. (2003) J. Complexity 19: pp. 1
- M. Kwas and Y. Li, submitted for publication.
- M. Kwas and H. Wo?niakowski, J. Complexity (to appear).
- A. Nayak and F. Wu, STOC, May, 1999, 384-393. See also http://arXiv.org/abs/quant-ph/9804066.
- Niederreiter, H. (1992) CBMS-NSF Reg. Conf. Series Appl. Math., 63. SIAM, Philadelphia
- Novak, E. (1988) Lecture Notes in Mathematics. Springer Verlag, Berlin
- Novak, E. (1995) J. Complexity 11: pp. 57
- Novak, E. (2001) J. Complexity 17: pp. 2
- E. Novak, I. H. Sloan, and H. Wo?niakowski. http://arXiv.org/abs/quant-ph/0206023.
- Plaskota, L., Wasilkowski, G.W., Wo?niakowski, H. (2000) J. Comp. Phys. 164: pp. 355
- 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.
- Shor, P. W. (1998) Documenta Mathematica I: pp. 467
- J. F. Traub, Physics Today, May, 39 (1999).
- Traub, J. F., Wasilkowski, G. W., Wo?niakowski, H. (1988) Information-based Complexity. Academic Press, New York
- Traub, J. F., Werschulz, A. G. (1998) Information and Complexity. Cambridge University Press, Cambridge, UK
- Vakhania, N. N., Tarieladze, V. I., Chobanyan, S. A. (1987) Probability Distributions on Banach Spaces. Reidel, Dordrecht
- Wasilkowski, G. W., Wo?niakowski, H. (1996) J. Math. Physics 37: pp. 2071
- Path Integration on a Quantum Computer
Quantum Information Processing
Volume 1, Issue 5 , pp 365-388
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- Kluwer Academic Publishers-Plenum Publishers
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- Quantum computation
- quantum summation
- path integration
- quantum queries
- quantum speedup
- number of qubits
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