An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations

Dongjie Wang; Sheng Chen

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arXiv · 2610.33095

An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations

Dongjie Wang·Sheng Chen

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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.

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Dongjie Wang, Sheng Chen. 2026-10-22. An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations. https://arxiv.org/abs/2610.33095

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