Krylov Polynomials and Quantum Query Complexity
Kiran Adhikari
TL;DR
The paper addresses reducing quantum query complexity for preparing $f(H)|\psi_0\rangle$ by exploiting state-specific spectral structure through a state-aware Krylov/Favard framework. It shows that the minimal queries equal the $L^2(\mu)$-optimal polynomial degree, where $\mu$ is the spectral measure of $(H,|\psi_0\rangle)$, and that dynamics can be compressed to a Jacobi matrix $J$ acting on the Krylov subspace. By expanding $f(H)|\psi_0\rangle$ in Krylov polynomials $P_n$ and using Favard truncations $p_d$, one can implement the optimal polynomial via QSVT with controlled amplitude amplification overhead, achieving $q_{\min}(\varepsilon)=n_{\mu}(f,\varepsilon)$. The framework yields concrete benefits for tasks like state-aware HHL and general families of initial states, and offers a path toward reduced-depth quantum circuits when state-dependent spectral structure is favorable. The approach connects orthogonal polynomial theory, Krylov compression, and quantum query models, with implications for quantum simulation, QML, and robustness to spectral locality.
Abstract
We show that the minimal query complexity for preparing $f(H)\ket{ψ_0}$ is exactly the optimal polynomial approximation degree of $f$ in $L^2(μ)$, where $μ$ is the spectral measure of $(H,\ket{ψ_0})$. 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.