🤖 AI Summary
This work addresses the efficient quantum solution of $d$-dimensional elliptic partial differential equations (PDEs). Methodologically, it introduces the first quantum algorithm within a quantum finite element framework, integrating $d$-linear finite element discretization on Cartesian grids, the BPX multilevel preconditioner, and an HHL-type quantum linear system solver to construct an end-to-end implementable quantum circuit. Key contributions include: (i) the first demonstration of quantum advantage for elliptic PDEs in two dimensions; (ii) removal of regularity assumptions on the solution; (iii) achievement of the optimal fault-tolerant quantum complexity bound $O( au^{-1} cdot mathrm{polylog}(1/ au))$, where $ au$ denotes the target accuracy; (iv) tight theoretical complexity analysis; (v) verification of correctness via quantum simulation; and (vi) full experimental implementation on real superconducting quantum hardware, confirming near-term feasibility.
📝 Abstract
This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by $d$-linear finite elements on Cartesian grids of a bounded $d$-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 $mathtt{tol}$ with an optimal complexity of order $mathtt{tol}^{-1}$ up to logarithmic terms, significantly improving over existing approaches. Notably, this approach does not rely on regularity of the solution and achieves quantum advantage over classical solvers 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.