Purified phase estimation samples spectra efficiently
Quantum phase estimation (QPE) is a cornerstone algorithm for extracting Hamiltonian eigenvalues, but its standard, eigenstate-centric form relies on carefully prepared coherent inputs that are costly or impractical for many strongly correlated systems. We overcome this bottleneck via DOS-QPE, an incoherent, purification-based variant of QPE that works directly with mixed-state probes and estimates the density of states (DOS) of the Hamiltonian. By adding a purification register and simple entangling layers, we turn standard QPE into an ensemble-based DOS sampler without modifying the core phase-estimation block. Conceptually, this purification closely aligns with the recent random purification channel framework from quantum learning theory, but instantiated here as a concrete phase-estimation circuit. We further equip DOS-QPE with symmetry-adapted input ensembles and a compressed-sensing reconstruction pipeline, and demonstrate on fermionic and nuclear Hamiltonians that a single experimental setup can recover rich spectral information relevant to thermodynamics, spectroscopy, and many-body structure.