Efficient quantum loading of probability distributions through Feynman propagators

We present quantum algorithms for the loading of probability distributions using Hamiltonian simulation for one dimensional Hamiltonians of the form ${\hat H}= \Delta + V(x) \mathbb{I}$. We consider the potentials $V(x)$ for which the Feynman propagator is known to have an analytically closed form and utilize these Hamiltonians to load probability distributions including the normal, Laplace and Maxwell-Boltzmann into quantum states. We also propose a variational method for probability distribution loading based on constructing a coarse approximation to the distribution in the form of a `ladder state' and then projecting onto the ground state of a Hamiltonian chosen to have the desired probability distribution as ground state. These methods extend the suite of techniques available for the loading of probability distributions, and are more efficient than general purpose data loading methods used in quantum machine learning.