Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra

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.

💡 Research Summary

The paper introduces QFAMES (Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra), a quantum algorithm designed to extract fine‑grained spectral information—both the locations of eigenvalues and their multiplicities—from a Hamiltonian under realistic physical assumptions. While estimating eigenvalues alone is a well‑studied problem (quantum phase estimation, QPE), determining eigenvalue degeneracies is #BQP‑complete in the worst case and cannot be achieved with a single initial state. QFAMES overcomes this barrier by employing multiple, possibly non‑orthogonal, initial states on the left and right (denoted {Uℓ|0⟩} and {Vr|0⟩}), a generalized Hadamard‑test circuit with a single ancilla qubit, and Gaussian‑distributed evolution times.

The algorithm proceeds in two stages. First, it collects complex amplitudes Zℓ,r(t) = ⟨0|Uℓ† e^{-iHt} Vr|0⟩ for a set of times {tₙ} drawn from a Gaussian distribution. This sampling yields a data matrix whose dimension depends only on the number of initial states (L·R) and not on the desired energy precision, thereby avoiding the large time‑grid overhead typical of subspace methods. The Gaussian time filter produces an effective energy filter that respects Heisenberg‑limited scaling (ΔE·T≈1).

Second, classical post‑processing analyzes the matrix to (i) locate clusters of dominant eigenvalues that are separated by at least a gap Δ, and (ii) determine the exact multiplicity of each cluster. The multiplicity extraction reduces to a singular‑value decomposition of the overlap matrices Φ and Ψ (which encode ⟨ϕℓ|Eₘ⟩ and ⟨ψᵣ|Eₘ⟩). Crucially, the authors assume a “uniform overlap condition”: every normalized vector in a degenerate eigenspace must have a non‑negligible overlap with at least one left and one right initial state. Under this condition, the singular‑value spectrum reveals the dimension of the eigenspace, and the algorithm can exactly recover the multiplicity. The paper proves that if the condition fails, an information‑theoretic barrier prevents any algorithm from resolving the degeneracy (Theorem B.1).

Complexity analysis shows that the maximum evolution time T_max scales as O(e^{O(p_tail)} ε^{-1}) and the total evolution time T_total scales similarly, where p_tail is the total overlap weight of non‑dominant eigenstates. When the dominant subspace carries a constant fraction of the total weight (the regime of interest for many physical systems), both times are essentially linear in 1/ε. The algorithm thus achieves Heisenberg‑limited precision with a data matrix whose size is independent of ε, a significant improvement over traditional subspace diagonalization techniques whose matrix dimensions grow inversely with the target precision.

The authors benchmark QFAMES on three representative models: (1) the 1‑D transverse‑field Ising model, using variational trial states and spin‑flip ansätze; (2) the 2‑D toric code, where the ground‑state manifold is four‑fold degenerate, demonstrating exact recovery of the topological degeneracy; and (3) the XXZ spin chain, employing matrix‑product‑state (MPS) initial states. In all cases, QFAMES accurately identifies eigenvalue clusters within a prescribed ε and determines multiplicities without numerical instability, even when eigenvalues are closely spaced (δ ≪ Δ).

A notable extension addresses mixed initial states, which arise in dissipative state‑preparation or low‑temperature thermalization protocols. By introducing an auxiliary register, the authors construct a control‑free, ancilla‑free variant of the generalized Hadamard test that works with density matrices rather than pure states. This broadens the applicability of QFAMES to near‑term devices where controlled state‑preparation or long coherent evolution may be unavailable.

Compared with prior work, QFAMES offers three distinct advantages: (i) Gaussian time sampling achieves Heisenberg‑limited scaling; (ii) an explicit energy filter separates nearby eigenvalues, improving the condition number of the subproblems; and (iii) the data matrix size depends only on the number of initial states, not on the inverse precision, leading to lower sample complexity. Moreover, the algorithm does not require the initial states to be linearly independent, allowing practitioners to use physically motivated, possibly redundant trial states without risking ill‑conditioned overlap matrices.

In summary, QFAMES provides the first provably efficient quantum algorithm for simultaneously locating dominant eigenvalues and exactly determining their multiplicities under realistic assumptions. Its ability to resolve degeneracies, handle mixed states, and operate with modest quantum resources makes it a powerful tool for studying quantum phase transitions, topological order, and other phenomena where spectral multiplicities carry essential physical meaning. The work opens new avenues for quantum spectral analysis beyond energy estimation, with potential impact on quantum chemistry, condensed‑matter simulation, and fault‑tolerant quantum algorithm design.