Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra

• URL: http://arxiv.org/abs/2510.07439v1
• Date: Wed, 08 Oct 2025 18:37:36 GMT
• Title: Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra
• Authors: Zhiyan Ding, Lin Lin, Yilun Yang, Ruizhe Zhang,
• Abstract summary: We introduce QFAMES, a quantum algorithm that efficiently identifies clusters of eigenvalues and determines their multiplicities.<n>QFAMES also enables the estimation of observable expectation values within targeted energy clusters.<n>We validate the effectiveness of QFAMES through numerical demonstrations.
• Score: 4.081730190778995
• License: http://creativecommons.org/licenses/by/4.0/
• 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. 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.

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