Year: 2026
Type: preprint
Abstract: This paper presents an efficient quantum spectral-Galerkin method for constant-coefficient second-order linear elliptic equations. The method employs a boundary-adapted Jacobi tensor-product basis that incorporates the boundary condition into the approximation space and produces Galerkin systems with polynomially bounded condition numbers and sparsity. For analytic solutions satisfying a suitable uniform regularity condition, spectral convergence allows the normalized Galerkin coefficient state to be prepared with complexity $\mathrm{poly}(d,\log(1/ε))$, where $d$ is the spatial dimension and $ε$ is the error tolerance, under standard sparse-access and state-preparation assumptions. At the state-preparation level, this improves the precision dependence over fixed-order finite-difference and finite-element discretizations. We further show how to estimate prescribed linear functionals of the solution, with an additional $O(1/ε)$ factor arising from norm and inner-product estimation.
Source: arXiv (Cornell University)
Authors: Wang, Dongjie, Chen, Sheng
Institutions: Beijing Normal University, Beijing Normal University, Zhuhai
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: G1351894037
OpenAlex ID: W7223838721