Filtered Quantum Phase Estimation

Accurate state preparation is a critical bottleneck in many quantum algorithms, particularly those for ground-state energy estimation. Even in fault-tolerant quantum computing, preparing a quantum state with sufficient overlap with the desired eigenstate remains a major challenge. To address this, we develop a unified cost-aware framework for filtered-state preparation that enhances the overlap of a given input state through spectral filtering. The framework covers polynomial and trigonometric realizations of filters and makes explicit the trade-off among overlap amplification, preparation success probability, and filter-implementation cost. As representative examples, we analyze Gaussian filters and introduce a modified Krylov-subspace-based filter that improves the success-probability/overlap trade-off relevant to filtered state preparation. Within this framework, we study a filtered variant of quantum phase estimation (FQPE) that mitigates the unfavorable dependence on the initial overlap present in standard QPE. Numerical experiments on Fermi-Hubbard models show that FQPE reduces the total runtime by more than two orders of magnitude in the high-precision regime, with overlap amplification exceeding a factor of one hundred.