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.
- F. Curbera, J. Complexity 16, 474 (2000).
- L. Grover, Phys. Rev. Lett. 79, 325 (1996). See also http://arXiv.org/abs/quant-ph/9706033.
- 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).
- S. Heinrich, J. Complexity 18, 1 (2002). See also http://arXiv.org/abs/quant-ph/0105116.
- S. Heinrich, J. Complexity 19, 1 (2003). See also http://arXiv.org/abs/quant-ph/0112153.
- 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.
- S. Heinrich and E. Novak, J. Complexity 19, 1 (2003). See also http://arXiv.org/abs/quantph/ 0109038.
- 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.
- H. Niederreiter, CBMS-NSF Reg. Conf. Series Appl. Math., 63 (SIAM, Philadelphia, 1992).
- E. Novak, Lecture Notes in Mathematics, 1349 (Springer Verlag, Berlin, 1988).
- E. Novak, J. Complexity, 11, 57 (1995).
- E. Novak, J. Complexity, 17, 2 (2001). See also http://arXiv.org/abs/quant-ph/0008124.
- E. Novak, I. H. Sloan, and H. Wo?niakowski. http://arXiv.org/abs/quant-ph/0206023.
- L. Plaskota, G.W. Wasilkowski, and H. Wo?niakowski, J. Comp. Phys. 164, 355 (2000).
- 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.
- P. W. Shor, Documenta Mathematica (extra volume ICM) I, 467 (1998).
- J. F. Traub, Physics Today, May, 39 (1999).
- J. F. Traub, G. W. Wasilkowski, and H. Wo?niakowski, Information-based Complexity (Academic Press, New York, 1988).
- J. F. Traub and A. G. Werschulz, Information and Complexity (Cambridge University Press, Cambridge, UK, 1998).
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions on Banach Spaces (Reidel, Dordrecht, 1987).
- G. W. Wasilkowski and H. Wo?niakowski, J. Math. Physics 37(4), 2071 (1996).
- Path Integration on a Quantum Computer
Quantum Information Processing
Volume 1, Issue 5 , pp 365-388
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- Quantum computation
- quantum summation
- path integration
- quantum queries
- quantum speedup
- number of qubits
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