Quantum preconditioning method for finite difference discretizations of the Poisson equation via Schrödingerization
This work provides a theoretically efficient quantum algorithm for a fundamental PDE problem, but the results are theoretical and not yet demonstrated experimentally.
The paper presents a quantum preconditioning framework for solving Poisson equation linear systems, achieving near-optimal query complexity of O(poly(d) ε^{-1} polylog(ε^{-1})) by combining Schrödingerization with a BPX multilevel preconditioner and a novel structure-aware block-encoding that avoids unfavorable normalization scaling.
We present a quantum preconditioning framework for solving linear systems arising from a finite difference discretization of the Poisson equation. It is based on the combination of the Schrödingerization technique \cite{JLY22b,JLYPRL24} and the BPX multilevel preconditioner in order to achieve near-optimal complexity. The Schrödingerization technique transforms linear partial and ordinary differential equations into Schrödinger-type systems with unitary evolution in one higher dimension, making them suitable for quantum simulation. A key contribution is a structure-aware construction of the block-encoding for the symmetrically preconditioned matrix $A_S = S^\top A S$, where $A$ is the stiffness matrix and $S$ encodes the BPX preconditioner in factored form. By establishing a novel commuting identity, we avoid the unfavorable normalization scaling that would otherwise arise from naive multiplication of block-encodings. This yields an exact block-encoding of $A_S$ with normalization $\mathcal{O}(d^2(L+1))$, where $d$ is the spatial dimension and $L$ is the number of levels. Combined with the Schrödingerization-based Hamiltonian simulation, the overall quantum algorithm achieves a query complexity of $\mathcal{O}\big(\mathrm{poly}(d)\varepsilon^{-1} \mathrm{polylog}(\varepsilon^{-1}) \big)$ for estimating linear functionals of the solution to a given tolerance $\varepsilon$.