Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra
Abstract
Fine-grained spectral properties of quantum Hamiltonians, including both eigenvalues and their multiplicities, provide useful information for characterizing many-body quantum systems as well as for understanding phenomena such as topological order. Extracting such information with small additive error is -complete in the worst case. In this work, we introduce QFAMES (Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra), a quantum algorithm that efficiently identifies clusters of closely spaced dominant eigenvalues and determines their multiplicities under physically motivated assumptions, which allows us to bypass worst-case complexity barriers. QFAMES also enables the estimation of observable expectation values within targeted energy clusters, providing a powerful tool for studying quantum phase transitions and other physical properties. We validate the effectiveness of QFAMES through numerical demonstrations, including its applications to characterizing quantum phases in the transverse-field Ising model and estimating the ground-state degeneracy of a topologically ordered phase in the two-dimensional toric code model. We also generalize QFAMES to the setting of mixed initial states. Our approach offers rigorous theoretical guarantees and significant advantages over existing subspace-based quantum spectral analysis methods, particularly in terms of the sample complexity and the ability to resolve degeneracies.
Cite
@article{arxiv.2510.07439,
title = {Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra},
author = {Zhiyan Ding and Lin Lin and Yilun Yang and Ruizhe Zhang},
journal= {arXiv preprint arXiv:2510.07439},
year = {2026}
}