Krylov Polynomials and Quantum Query Complexity

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.