Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity
Abstract
We connect different degrees of smoothness of real-valued periodic functions to the number of qubits required for their high-precision Fourier-basis amplitude encodings. Our central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We use Gevrey regularity to analyze a quantum algorithm for solving general linear partial differential equations (PDEs) with periodic boundary conditions. We then extend the solver to PDEs with Dirichlet boundary conditions using thin Gevrey collars and a regularized inverse. Our resource bounds specify precision-dependent Fourier cutoffs and account for restricted discrete stability, elementary gate costs, and restriction to the Dirichlet domain. Finally, we apply our approach to isotropic and anisotropic Poisson equations with periodic and Dirichlet boundary conditions.
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BibTeX
@misc{ronagh2026breaking,
title = {{Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity}},
author = {Pooya Ronagh and Mariia Sobchuk and Xiaoran Li and Grecia Castelazo and Ala Shayeghi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.07893},
eprint = {2608.07893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.07893},
}
RIS
TY - UNPB
TI - Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity
AU - Pooya Ronagh
AU - Mariia Sobchuk
AU - Xiaoran Li
AU - Grecia Castelazo
AU - Ala Shayeghi
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.07893
UR - https://arxiv.org/abs/2608.07893
AB - We connect different degrees of smoothness of real-valued periodic functions to the number of qubits required for their high-precision Fourier-basis amplitude encodings. Our central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We use Gevrey regularity to analyze a quantum algorithm for solving general linear partial differential equations (PDEs) with periodic boundary conditions. We then extend the solver to PDEs with Dirichlet boundary conditions using thin Gevrey collars and a regularized inverse. Our resource bounds specify precision-dependent Fourier cutoffs and account for restricted discrete stability, elementary gate costs, and restriction to the Dirichlet domain. Finally, we apply our approach to isotropic and anisotropic Poisson equations with periodic and Dirichlet boundary conditions.
ER -