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arXiv · 2609.19676

Linear combination of Schrödingerization for quantum linear systems with optimal matrix-query complexity

Yin Yang·Yue Yu·Long Zhang
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Abstract

Quantum linear systems algorithms (QLSAs) aim to solve linear systems $A\bb{x}=\bb{b}$ exponentially faster than classical methods under certain conditions. In this work, we develop quantum algorithms for solving linear algebraic equations from an ODE-based perspective. Inspired by the linear combination of Hamiltonian simulation (LCHS) representation in the Fourier approach \cite{Childs2017QLSA}, we express the solution $\bb{x}$ as a linear combination of solutions to a system of linear convection equations, which become Schrödinger-type equations with unitary evolutions in the Fourier domain. We refer to this representation as LC-Schrödingerization. Based on this result, we construct an LCHS-based quantum algorithm with two LCHS instances: one for time-marching and one for numerical integration. The key construction uses the derivative of a Gaussian-smoothed hat function and recovers the solution over a fixed auxiliary interval. This permits a truncation time independent of the target accuracy and avoids the loss in success probability from selecting a single grid point. Periodization and explicit Fourier coefficients provide the corresponding projection error bounds. Under the stated oracle assumptions and given a constant-factor estimate of the solution norm, direct simulation of the select operators and block preconditioning achieve the optimal matrix-query complexity $\mathcal{O}(κ_A\log\frac1\varepsilon)$ without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.

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Yin Yang, Yue Yu, Long Zhang. 2026-09-17. Linear combination of Schrödingerization for quantum linear systems with optimal matrix-query complexity. https://arxiv.org/abs/2609.19676

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