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<p>Most problems in uncertainty quantification, despite their 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 <i>M</i> ≫ 1 of samples are required to obtain accurate ensemble averages. This usually involves solving the PDE <i>M</i> times. In addition, to characterise the stochasticity in a PDE, the dimension <i>L</i> of the random input variables is high in most cases, and classical algorithms suffer from the curse-of-dimensionality. We propose new quantum algorithms for PDEs with uncertain coefficients that are more efficient in <i>M</i> and <i>L</i> in various important regimes, compared to their classical counterparts. We introduce transformations that convert the original <i>d</i>-dimensional equation (with uncertain coefficients) into <i>d</i> + <i>L</i> (for dissipative equations) or <i>d</i> + 2<i>L</i> (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 <i>M</i> different initial data, so the computational cost for the quantum algorithm to obtain the ensemble average from <i>M</i> different samples is independent of <i>M</i>, while also showing potential advantage in <i>d, L</i> and precision <i>ϵ</i> in computing ensemble averaged solutions or physical observables.</p>

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Quantum algorithms for uncertainty quantification: Applications to partial differential equations

  • 期刊论文

  • 发表日期：

    2025年8月15日

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  • Francoise Golse,
  • Shi Jin,
  • Nana Liu

摘要

Most problems in uncertainty quantification, despite their 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 the 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 convert 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 independent of M, while also showing potential advantage in d, L and precision ϵ in computing ensemble averaged solutions or physical observables.