Quantum preconditioning method for linear systems problems via Schr\"odingerization

We present a quantum computational framework that systematically converts classical linear iterative algorithms with fixed iteration operators into their quantum counterparts using the Schr\"odingerization technique [Shi Jin, Nana Liu and Yue Yu, Phys. Rev. Lett., vol. 133 No. 230602,2024]. This is achieved by capturing the steady state of the associated differential equations. The Schr\"odingerization technique transforms linear partial and ordinary differential equations into Schr\"odinger-type systems, making them suitable for quantum computing. This is accomplished through the so-called warped phase transformation, which maps the equation into a higher-dimensional space. Building on this framework, we develop a quantum preconditioning algorithm that leverages the well-known BPX multilevel preconditioner for the finite element discretization of the Poisson equation. The algorithm achieves a near-optimal dependence on the number of queries to our established input models, with a complexity of $\mathscr{O}(\text{polylog} \frac{1}{\varepsilon})$ for a target accuracy of $\varepsilon$ when the dimension $d\geq 2$. This improvement results from the Hamiltonian simulation strategy applied to the Schr\"odingerized preconditioning dynamics, coupled with the smoothing of initial data in the extended space.