This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by d d -linear finite elements on Cartesian grids of a bounded d d -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 t o l \mathtt {tol} with an optimal complexity of order t o l − 1 \mathtt {tol}^{-1} 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.