A structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of reaction-diffusion problems
Abstract
We present a structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of constant-coefficient reaction--diffusion equations on the unit cube, including the Poisson equation. Building on existing quantum finite element and finite difference methods, we express the preconditioned operator through explicit one-dimensional maps, tensor products, and norm-preserving interlevel transfers. Symmetric preconditioning gives multilevel coefficients, and its favorable conditioning alone does not ensure efficient recovery of a state in the original finite element nodal basis. We instead use the local L2-orthonormal basis underlying the construction as the output representation of the same finite element function. Its coefficient norm equals the continuous L2 norm. Assuming direct access to the preconditioned right-hand side, we prepare this orthonormal coefficient state with a represented function error at most ε. The expected query complexity is 𝒪 (dL3|f|L2/|u|L2 log (2+L|f|L2/ε)), where u is the exact solution, f is the right-hand side, d is the spatial dimension, and L=log2(1/h) for mesh width h. The bound holds for 0<ε≤|u|L2/2 and depends only polylogarithmically on mesh refinement for fixed continuous data. We also analyze linear-functional estimation and illustrate the construction and continuous error bounds with a one-dimensional UnitaryLab experiment.
Related
- Two-level additive Schwarz preconditioners with GDSW coarse spaces for high-order discretizations in 2D
- Two-level additive Schwarz preconditioners with GDSW coarse spaces for high-order discretizations in 3D
- A mesh- and horizon-robust preconditioner for finite element discretizations of volume-constrained nonlocal diffusion problems
- Fast and scalable multi-GPU sliced FFT solvers for asymptotically compatible discretizations of 3D nonlocal diffusion problems
- Structure-Preserving Detailed-Balance Master-Equation Discretizations for Fokker--Planck Equations
BibTeX
@misc{yang2026structureaware,
title = {{A structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of reaction-diffusion problems}},
author = {Yin Yang and Yue Yu and Long Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06987},
eprint = {2610.06987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06987},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of reaction-diffusion problems
AU - Yin Yang
AU - Yue Yu
AU - Long Zhang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.06987
UR - https://arxiv.org/abs/2610.06987
AB - We present a structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of constant-coefficient reaction--diffusion equations on the unit cube, including the Poisson equation. Building on existing quantum finite element and finite difference methods, we express the preconditioned operator through explicit one-dimensional maps, tensor products, and norm-preserving interlevel transfers. Symmetric preconditioning gives multilevel coefficients, and its favorable conditioning alone does not ensure efficient recovery of a state in the original finite element nodal basis. We instead use the local L^2-orthonormal basis underlying the construction as the output representation of the same finite element function. Its coefficient norm equals the continuous L^2 norm. Assuming direct access to the preconditioned right-hand side, we prepare this orthonormal coefficient state with a represented function error at most ε. The expected query complexity is 𝒪 (dL^3|f|_L^2/|u|_L^2 log (2+L|f|_L^2/ε)), where u is the exact solution, f is the right-hand side, d is the spatial dimension, and L=log_2(1/h) for mesh width h. The bound holds for 0<ε≤|u|_L^2/2 and depends only polylogarithmically on mesh refinement for fixed continuous data. We also analyze linear-functional estimation and illustrate the construction and continuous error bounds with a one-dimensional UnitaryLab experiment.
ER -