An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations
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 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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BibTeX
@misc{wang2026efficient,
title = {{An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations}},
author = {Dongjie Wang and Sheng Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33095},
eprint = {2610.33095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33095},
}
RIS
TY - UNPB
TI - An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations
AU - Dongjie Wang
AU - Sheng Chen
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.33095
UR - https://arxiv.org/abs/2610.33095
AB - 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 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.
ER -