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\n This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -linear finite elements on Cartesian grids of a bounded\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -dimensional domain. An essential step in the construction is a BPX preconditioner, which transforms the linear system into a sufficiently well-conditioned one, making it amenable to quantum computation. We provide a constructive proof demonstrating that, for any fixed dimension, our quantum algorithm can compute suitable functionals of the solution to a given tolerance\n \n \n \n \n t<\/mml:mi>\n o<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n \\mathtt {tol}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with an optimal complexity of order\n \n \n \n \n \n t<\/mml:mi>\n o<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \\mathtt {tol}^{-1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n up to logarithmic terms, significantly improving over existing approaches. Notably, this approach does not rely on the regularity of the solution and achieves a quantum advantage over classical solvers already in two dimensions, whereas prior quantum methods required at least four dimensions for asymptotic benefits. We further detail the design and implementation of a quantum circuit capable of executing our algorithm, present simulator results, and report numerical experiments on current quantum hardware, confirming the feasibility of preconditioned finite element methods for near-term quantum computing.\n <\/p>","DOI":"10.1090\/mcom\/4137","type":"journal-article","created":{"date-parts":[[2025,8,14]],"date-time":"2025-08-14T13:38:23Z","timestamp":1755178703000},"page":"2637-2668","source":"Crossref","is-referenced-by-count":2,"title":["Quantum realization of the finite element method"],"prefix":"10.1090","volume":"95","author":[{"given":"M.","family":"Deiml","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Peterseim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,8,14]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1145\/1132516.1132579","article-title":"A polynomial quantum algorithm for approximating the Jones polynomial","author":"Aharonov, Dorit","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/1595931341"},{"key":"2","unstructured":"A. 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