, Volume 1, Issue 5, pp 365-388

Path Integration on a Quantum Computer

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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 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