Quantum linear system algorithm with optimal queries to initial state preparation

G. H. Low, Y. Su · 2024 · arXiv（期刊版 Quantum, DOI 10.22331/q-2026-03-23-2041）

| 编号 QC019 | 主题 量子算法工具箱 | 链接 arXiv:2410.18178 |

摘要

摘要保留原文（英文）

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We present a quantum linear system algorithm with query complexity Θ $(1/\sqrt{p})$ to $O_b$ that is optimal, and query complexity O (κ log $(1/p)$ (log log $\left(1/p\right)+\log (1/ϵ)))$ to $O_A$ that is nearly optimal in all parameters including the condition number $\kappa =\Vert A\Vert \Vert A^{-1}\Vert$ , success amplitude $\sqrt{p}=\Vert A^{-1}|b⟩\Vert /\Vert A^{-1}\Vert$ , and accuracy ϵ. In various applications to solving differential equations, preparing ground states of operators with real spectra, estimating and transforming eigenvalues of non-normal matrices, we can further improve the dependence on p to nearly match or outperform best previous results based on other methods. As κ can be arbitrarily larger than $1/\sqrt{p}$ our algorithm contrasts with recent results that have $\mathbf{O}\left(\kappa \log \left(1/ϵ\right)\right)$ complexity to both oracles, which, while optimal in $O_A.$ , is highly suboptimal in $O_b$

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