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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