Path Integration on a Quantum Computer |
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Authors: | JF Traub H Wo?niakowski |
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Affiliation: | (1) Computer Science, Columbia University, Columbia;(2) Computer Science, Columbia University and Institute of Applied Mathematics, University of Warsaw, Columbia |
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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 -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 -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 -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 -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 |
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Keywords: | Quantum computation quantum summation path integration quantum queries quantum speedup number of qubits |
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