Quantum Algorithms for Heterogeneous PDEs: The Neutron Diffusion Eigenvalue Problem

Quantum Physics [Submitted on 6 Apr 2026] Quantum Algorithms for Heterogeneous PDEs: The Neutron Diffusion Eigenvalue Problem Andrew M. Childs, Lincoln Johnston, Brian Kiedrowski, Mahathi Vempati, Jeffery Yu We develop a hybrid classical-quantum algorithm to solve a type of linear reaction-diffusion equation, the neutron diffusion (generalized) k-eigenvalue problem that establishes nuclear criticality. The algorithm handles an equation with piecewise constant coefficients, describing a problem in a heterogeneous medium. We apply uniform finite elements and show that the quantum algorithm provides significant polynomial end-to-end speedup over its classical counterparts. This speedup leverages recent advances in quantum linear systems -- fast inversion and quantum preconditioning -- and uses Hamiltonian simulation as a subroutine. Our results suggest that quantum algorithms may provide speedups for heterogeneous PDEs, though the extent of this advantage over the fastest classical algorithm depends on the effectiveness of other classical approaches such as nonuniform or adaptive meshing for a given problem instance. Subjects: Quantum Physics (quant-ph); Analysis of PDEs (math.AP) Cite as: arXiv:2604.05098 [quant-ph]   (or arXiv:2604.05098v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.05098 Focus to learn more Submission history From: Mahathi Vempati [view email] [v1] Mon, 6 Apr 2026 18:57:21 UTC (630 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: math math.AP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)