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 e-approximation to path integrals whose integrands are at least Lipschitz. We prove: 1. Path integration on a quantum computer is tractable. 2. Path integration on a quantum computer can be solved roughly e^{-1} times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. 3. The number of quantum queries is the square root of the number of function values needed on a classical computer using randomization. 4.The number of qubits is polynomial in e^{-1}. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands.
