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  • Open Access

Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra

Zhiyan Ding1, Lin Lin2,3,*, Yilun Yang2, and Ruizhe Zhang4

  • *Contact author: linlin@math.berkeley.edu
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PRX Quantum 7, 020318 – Published 29 April, 2026

DOI: https://doi.org/10.1103/jch7-734h

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 #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.

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Popular Summary

Quantum computers offer a promising path to probing the complex energy landscapes of quantum many-body systems, with applications ranging from material design to high-energy physics. While existing quantum algorithms can estimate these energy levels, they cannot resolve their degeneracies, that is, the number of distinct quantum states sharing the same energy. Access to these multiplicities is essential for characterizing exotic physical phenomena such as quantum phase transitions and topological order.

This work introduces a quantum algorithm that provably recovers both the energy levels and their degeneracies. The approach samples a collection of initial states, enabling robust identification of overlapping levels and estimation of multiplicities and observables. The algorithm is resource-efficient, requiring minimal number of qubits and shallow circuits, and is therefore well suited for early fault-tolerant quantum devices.

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Outline

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PRX Quantum 7, 020318– Published 29 April, 2026
  • Received 28 October 2025
  • Revised 3 February 2026
  • Accepted 17 March 2026
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