Path Integration on a Quantum Computer

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 is the square root of the number of function values needed on a classical computer using randomization. • 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. ∗Effort sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-01-2-0523. The U.S, Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency (DARPA), the Air Force Research Laboratory, or the U.S. Government.