Quantum Sampling and Moment Estimation for Transformed Gaussian Random Fields

TL;DR

This paper introduces a quantum algorithm for efficiently sampling and estimating moments of transformed Gaussian random fields, enabling faster computation of statistical properties relevant in modeling microstructures in PDEs.

Contribution

It develops a quantum method for sampling transformed Gaussian fields directly from statistical parameters, bypassing input bottlenecks and enabling efficient moment estimation.

Findings

Quantum sampling achieves accuracy $ ext{tol}$ in polylogarithmic time.

Algorithms for estimating moments have complexity $ ext{tol}^{-1}$ times polylog factors.

Numerical experiments demonstrate the method's feasibility on simulated quantum hardware.

Abstract

We present a quantum algorithm for efficiently sampling transformed Gaussian random fields on $d$-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 approximating it to accuracy $\mathtt{tol} > 0$ in time $\mathcal{O}(\operatorname{polylog} \mathtt{tol}^{-1})$. 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 $\mathcal{O}(\mathtt{tol}^{-1} \operatorname{polylog} \mathtt{tol}^{-1})$. We illustrate the theoretical findings through numerical experiments on simulated quantum hardware.