Quantum algorithms for uncertainty quantification: application to partial differential equations

• URL: http://arxiv.org/abs/2209.11220v2
• Date: Wed, 28 Sep 2022 16:22:26 GMT
• Title: Quantum algorithms for uncertainty quantification: application to partial differential equations
• Authors: Francois Golse, Shi Jin and Nana Liu
• Abstract summary: We propose new quantum algorithms for PDEs with uncertain coefficients. We show potential advantage in d, L and precision in computing ensemble averaged solutions or physical observables.
• Score: 27.175719898694073
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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