Quantum algorithms for uncertainty quantification: application to partial differential equations

TL;DR

This paper introduces quantum algorithms that significantly improve the efficiency of uncertainty quantification in PDEs by reducing computational costs related to sample size and stochastic dimensions, outperforming classical methods.

Contribution

The authors develop quantum algorithms that transform PDEs with uncertain coefficients into higher-dimensional equations, enabling efficient ensemble averaging independent of sample size M.

Findings

Quantum algorithms reduce dependence on sample size M.

Transformations lower the effective dimensionality of PDEs.

Potential quantum advantage in high-dimensional uncertainty quantification.

Abstract

Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations (PDEs), large numbers M>>1 of samples are required to obtain accurate ensemble averages. This usually involves solving the PDE M times. In addition, to characterise the stochasticity in a PDE, the dimension L of the random input variables is high in most cases, and classical algorithms suffer from curse-of-dimensionality. We propose new quantum algorithms for PDEs with uncertain coefficients that are more efficient in M and L in various important regimes, compared to their classical counterparts. We introduce transformations that transfer the original d-dimensional equation (with uncertain coefficients) into d+L (for dissipative equations) or d+2L (for wave type equations) dimensional equations (with certain coefficients) in which the uncertainties appear only in the initial data. These transformations also allow one to superimpose the M different initial data, so the computational cost for the quantum algorithm to obtain the ensemble average from M different samples is then independent of M, while also showing potential advantage in d, L and precision in computing ensemble averaged solutions or physical observables.