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arXiv · 2510.11786

Krylov Polynomials and Quantum Query Complexity

Kiran Adhikari
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Abstract

We show that the minimal query complexity for preparing $f(H)\ket{\psi_0}$ is exactly the optimal polynomial approximation degree of $f$ in $L^2(\mu)$, where $\mu$ is the spectral measure of $(H,\ket{\psi_0})$. This state-aware perspective refines the worst-case bounds, unifies Krylov/Favard approximation with quantum queries, and explains how state-dependent spectral structure can yield substantial savings over uniform designs.

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Kiran Adhikari. 2025-10-13. Krylov Polynomials and Quantum Query Complexity. https://arxiv.org/abs/2510.11786

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