Schrödingerization for quantum linear systems problems
Abstract
We develop a Schrödingerization algorithm for quantum linear systems problems in two higher dimensions. The earlier LC-Schrödingerization approach represents the solution as a time integral of a homogeneous convection response and implements this integral by a linear combination of unitaries (LCU) over evolution times. We instead use Duhamel's principle to incorporate the integral into an inhomogeneous convection equation with zero initial data. Fourier projection in the convection variable and Schrödingerization in a second auxiliary variable then realize the solution without the separate LCU step. A joint choice of the kernel and recovery procedure gives an evolution time independent of the target accuracy, a uniformly bounded L2 kernel normalization, and accurate recovery on a fixed interval. We establish the periodization and discretization bounds and analyze the interval recovery probability. With block preconditioning, we retain a single logarithmic precision factor and obtain linear condition-number dependence without variable-time amplitude amplification. Under exact oracle access and given a valid constant-factor solution-norm estimate, the algorithm uses 𝒪(κAlog(1/ε)) queries to each original input oracle, with constant success probability and ℓ2 state error at most ε. The matrix-query bound matches the standard worst-case scaling when the supplied norm bounds are tight. UnitaryLab simulations on positive-definite and indefinite systems demonstrate the solution accuracy and probability gain from interval recovery.
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BibTeX
@misc{yang2026schrödingerization,
title = {{Schrödingerization for quantum linear systems problems}},
author = {Yin Yang and Yue Yu and Long Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25660},
eprint = {2609.25660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25660},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Schrödingerization for quantum linear systems problems
AU - Yin Yang
AU - Yue Yu
AU - Long Zhang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.25660
UR - https://arxiv.org/abs/2609.25660
AB - We develop a Schrödingerization algorithm for quantum linear systems problems in two higher dimensions. The earlier LC-Schrödingerization approach represents the solution as a time integral of a homogeneous convection response and implements this integral by a linear combination of unitaries (LCU) over evolution times. We instead use Duhamel's principle to incorporate the integral into an inhomogeneous convection equation with zero initial data. Fourier projection in the convection variable and Schrödingerization in a second auxiliary variable then realize the solution without the separate LCU step. A joint choice of the kernel and recovery procedure gives an evolution time independent of the target accuracy, a uniformly bounded L^2 kernel normalization, and accurate recovery on a fixed interval. We establish the periodization and discretization bounds and analyze the interval recovery probability. With block preconditioning, we retain a single logarithmic precision factor and obtain linear condition-number dependence without variable-time amplitude amplification. Under exact oracle access and given a valid constant-factor solution-norm estimate, the algorithm uses 𝒪(κ_Alog(1/ε)) queries to each original input oracle, with constant success probability and ℓ^2 state error at most ε. The matrix-query bound matches the standard worst-case scaling when the supplied norm bounds are tight. UnitaryLab simulations on positive-definite and indefinite systems demonstrate the solution accuracy and probability gain from interval recovery.
ER -