Speed up in solving Poisson and elliptic equations with perturbations via novel pseudospectral differential operator block encoding

The algorithm for Poisson and elliptic equations that uses the smallest number of elementary gates to date cite{FastInversion} relies on the efficient diagonalization in the Fourier basis of the respective discretized operators, and in this way eliminates the need for block encoding used in any of the standard matrix inversion algorithms. However, in more general cases of differential equations where these operators are perturbed (e.g. perturbed Schrödinger equation) the resulting matrix is not easily diagonalized anymore. In our work we demonstrate a block encoding technique for pseudospectral differential operators which is more efficient than the direct block encoding technique used in the QSVT version of the algorithm from cite{FastInversion}. We provide an algorithm for the perturbed inhomogeneous Schrödinger equation and apply it to Kohn-Sham equations. At the same time, we introduce an algorithm for the special cases of more general partial differential equations. We quantify the size of the discretization grid for different levels of analyticity, which also quantifies the speedup in the block-encoding of the corresponding potentials.