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.