发表机构
Stony Brook University（石溪大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究量子 Krylov 线性求解中超越全局条件数的情形，通过谱压缩与方向稳定性证明归一化解复杂度不随全局条件数发散，并给出不依赖小特征值逆的端到端复杂度界限。
AI 中文摘要
全局条件数依赖可能显著高估制备归一化量子线性系统解的难度。许多现有的超越条件数方法在全局病态方向与目标解相关性有限时表现良好。本文研究了时间演化量子 Krylov 线性求解（QKS）的一个更困难情形，其中消失的特征值仍与解相关且必须保留。我们识别出一个 QKS 情形，其中与解相关的谱被压缩到一个紧凑的约化空间中，病态性沿与解对齐的软方向集中，因此归一化消除了沿该方向的发散放大，仅留下有界的横向响应。因此，归一化态的复杂度不必继承全局条件数的发散性。我们的贡献有三方面：（1）我们证明与解相关的信息具有紧凑且稳定的 QKS 表示，当相关特征值消失时，子空间维度、演化时间和重构开销均有界。（2）我们证明归一化态的难度由逆放大是否改变解方向决定，而非仅由小特征值决定；即使任意病态的约化系统，当放大与解对齐时，仍可保持方向稳定。（3）我们证明该稳定性在有限投影系统误差下持续存在，并传递到实际 QKS 输出，从而得到不依赖于保留小特征值之逆的端到端复杂度界限。一个显式的分离族进一步表明，完整的逆放大准则可能发散，而相应的 QKS 量保持有界。
英文摘要
Global condition-number dependence can substantially overestimate the difficulty of preparing normalized quantum linear-system solutions. Many existing beyond-conditioning approaches are favorable when globally ill-conditioned directions have limited relevance to the target solution. Here we study a harder regime for time-evolution quantum Krylov linear solving (QKS), where a vanishing eigenvalue remains solution-relevant and must be retained. We identify a QKS regime in which the solution-relevant spectrum is compressed into a compact reduced space where the ill-conditioning is concentrated along a soft direction aligned with the solution, so that normalization removes the divergent amplification along that direction and leaves only a bounded transverse response. Consequently, the normalized-state complexity need not inherit the divergence of the global condition number. Our contributions are threefold: (1) We show that the solution-relevant information admits a compact and stable QKS representation, with bounded subspace dimension, evolution time, and reconstruction overhead as the relevant eigenvalue vanishes. (2) We show that normalized-state difficulty is governed by whether inverse amplification changes the solution direction, rather than by the small eigenvalue alone; even an arbitrarily ill-conditioned reduced system can remain directionally stable when the amplification is solution-aligned. (3) We prove that this stability persists under finite projected-system errors and transfers to the actual QKS output, yielding an end-to-end complexity bound without inverse dependence on the retained small eigenvalue. An explicit separation family further shows that a full inverse-amplification criterion can diverge while the corresponding QKS quantities remain bounded.
