Quantum Krylov Linear Solving Beyond Global Conditioning via Spectral Compression and Directional Stability
Abstract
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.
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BibTeX
@misc{yu2026quantum,
title = {{Quantum Krylov Linear Solving Beyond Global Conditioning via Spectral Compression and Directional Stability}},
author = {Sijia Yu and Yifan Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06620},
eprint = {2610.06620},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06620},
}
RIS
TY - UNPB
TI - Quantum Krylov Linear Solving Beyond Global Conditioning via Spectral Compression and Directional Stability
AU - Sijia Yu
AU - Yifan Zhou
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06620
UR - https://arxiv.org/abs/2610.06620
AB - 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.
ER -