{"ID":2858970,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.07439","arxiv_id":"2510.07439","title":"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 $\\#\\textsf{BQP}$-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.","short_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 $\\#\\textsf{BQP...","url_abs":"https://arxiv.org/abs/2510.07439","url_pdf":"https://arxiv.org/pdf/2510.07439v3","authors":"[\"Zhiyan Ding\",\"Lin Lin\",\"Yilun Yang\",\"Ruizhe Zhang\"]","published":"2025-10-08T18:37:36Z","proceeding":"quant-ph","tasks":"[\"quant-ph\",\"cs.DS\"]","methods":"[]","has_code":false}