Quantum Sampling and Moment Estimation for Transformed Gaussian Random Fields

Abstract

Abstract. We present a quantum algorithm for efficiently sampling transformed Gaussian random fields in [Formula: see text]-dimensional domains, based on an enhanced version of the classical moving average method. Pointwise transformations enforcing boundedness are essential for using Gaussian random fields in quantum computation and arise naturally, for example, in modeling coefficient fields representing microstructures in partial differential equations. Generating this microstructure from its few statistical parameters directly on the quantum device bypasses the input bottleneck. Our method enables an efficient quantum representation of the resulting random field and prepares a quantum state that approximates it to accuracy [Formula: see text] in time [Formula: see text]. Combined with amplitude estimation and a quantum pseudorandom number generator, this leads to algorithms for estimating linear and nonlinear observables, including mixed and higher-order moments, with total complexity [Formula: see text]. We illustrate the theoretical findings through numerical experiments on simulated quantum hardware.