A structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of reaction-diffusion problems
Year: 2026
Type: preprint
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 $L^2$-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 $\varepsilon$. The expected query complexity is \[ \mathcal{O}\!\Big(dL^3\frac{\|f\|_{L^2}}{\|u\|_{L^2}} \log\!\Big(2+\frac{L\|f\|_{L^2}}{\varepsilon}\Big)\Big), \] 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 $0<\varepsilon\le\|u\|_{L^2}/2$ 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.
Source: arXiv (Cornell University)
Authors: Yang, Yin, Yu, Yue, Zhang, Long
Institution: Xiangtan University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funders: National Natural Science Foundation of China, Hunan Provincial Science and Technology Department, Natural Science Foundation of Hunan Province, Xiangtan University, Higher Education Discipline Innovation Project, Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province
OpenAlex ID: W7220919645