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arxiv: 2510.07439 · v3 · submitted 2025-10-08 · 🪐 quant-ph · cs.DS

Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra

Zhiyan Ding , Lin Lin , Yilun Yang , Ruizhe Zhang This is my paper

Pith reviewed 2026-05-18 09:02 UTC · model grok-4.3

classification 🪐 quant-ph cs.DS
keywords quantum algorithmeigenvalue multiplicityspectral analysisquantum filteringmany-body systemstopological orderphase transitionsenergy clusters
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The pith

A quantum algorithm identifies clusters of closely spaced dominant eigenvalues and determines their multiplicities under physical assumptions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents QFAMES, a quantum method designed to extract fine details about the eigenvalues of many-body Hamiltonians and how many times each value repeats. It targets groups of nearly equal dominant eigenvalues rather than resolving every individual level. This focus on clusters, which often appear in real physical systems, lets the procedure avoid the general computational hardness that blocks exact multiplicity counting on quantum computers. The same filtering step also supports direct estimation of observable averages restricted to states inside a chosen energy window. Tests on the transverse-field Ising model for phase identification and the toric code for degeneracy counting illustrate the approach in practice.

Core claim

QFAMES 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. The method supplies rigorous theoretical guarantees and shows advantages in sample complexity compared with prior subspace techniques.

What carries the argument

QFAMES, a quantum filtering procedure that isolates and counts multiplicities inside targeted energy clusters rather than across the full spectrum.

If this is right

  • Observable expectation values can be estimated inside specific energy clusters to characterize quantum phase transitions.
  • Ground-state degeneracy can be extracted for topologically ordered phases such as the two-dimensional toric code.
  • The procedure generalizes directly to mixed initial states.
  • Sample complexity improves over existing subspace-based spectral methods while retaining rigorous guarantees.
  • Numerical demonstrations confirm utility for both the transverse-field Ising model and topologically ordered systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The filtering idea could be combined with existing ground-state preparation routines to focus resources on a chosen energy window.
  • Application to noisy intermediate-scale devices would test whether the cluster assumption survives realistic decoherence.
  • Similar filtering steps might resolve degeneracies in other spectral problems such as those arising in open quantum systems.

Load-bearing premise

The assumption that dominant eigenvalues in physical Hamiltonians form tight clusters is what makes accurate multiplicity extraction feasible.

What would settle it

A concrete Hamiltonian whose dominant eigenvalues are well-separated rather than clustered, yet on which QFAMES still reports incorrect multiplicity counts, would disprove the central claim.

Figures

Figures reproduced from arXiv: 2510.07439 by Lin Lin, Ruizhe Zhang, Yilun Yang, Zhiyan Ding.

Figure 1
Figure 1. Figure 1: Illustration of the QFAMES algorithm. Given the Hamiltonian and two sets of initial states prepared by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagram of the QFAMES algorithm for post-processing the 3-tensor generated from quantum data of the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the QFAMES algorithm on the illustrative example. The dashed vertical lines are the [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: QFAMES (Algorithm 1) vs QMEGS in the illustrative example as a function of (left): max evolution time Tmax = σT, and (right): total evolution time Ttotal. The dashed line stands for the fitted error proportional to 1/T, as predicted by Eq. (27). A. Illustrative example To illustrate the effectiveness of QFAMES, we consider a simple Hamiltonian with three eigenvalues: λ0 = λ1 = 0 and λ2 = 0.1. There are onl… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the QFAMES algorithm on the TFIM model. The upper row shows the Frobenius norm of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of Algorithm 2 on the TFIM model. The observable is S z = 1 2L PL i=1 Zi . In the manifold of near-ground-state cluster, the possible ranges of observable expectation values are shown by the gray lines. A transition from ferromagnetic to paramagnetic phase can be observed when g increases from 0.5 to 1.5. where Zi and Xi are the Pauli operators acting on the i-th qubit and g is the coupling co… view at source ↗
Figure 7
Figure 7. Figure 7: 2D Toric codes on a 2-by-4 lattice with (left) torus, and (right) cylinder boundary conditions. Dashed lines stand for periodic boundary conditions in the corresponding directions. The models consist of 16 and 18 qubits, respectively, represented by the small circles. The Toric code ground state is known to exhibit Z2 topological order, which has been realized on a quantum computer based on the prior knowl… view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of GSD estimation for 2D Toric code models with [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Hadamard test circuit. W is either the identity or the phase gate S † . Here INIT |0⟩ = |ϕ⟩. For each n, the Hadamard test circuit guarantees that E[Zn] = ⟨ϕ|e −iHtn |ϕ⟩ = X m∈[M] |⟨ϕ|Em⟩|2 | {z } :=pm ·e −iλmtn . (A1) Thus, the dataset {(tn, Zn)}n∈[N] corresponds to the unbiased noisy samples of the Fourier signal x(t) = X m∈[M] pm exp(−iλmt). (A2) Estimating the dominant eigenvalues {λm}m∈D from these sa… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces QFAMES, a quantum algorithm that identifies clusters of closely spaced dominant eigenvalues and determines their multiplicities under physically motivated assumptions on the spectrum. This is claimed to bypass worst-case #BQP-completeness of fine-grained spectral analysis while also enabling estimation of observable expectation values within targeted energy clusters. Numerical validation is provided on the transverse-field Ising model for quantum phases and the 2D toric code for ground-state degeneracy, with a generalization to mixed initial states and claims of rigorous theoretical guarantees plus advantages in sample complexity over subspace methods.

Significance. If the efficiency claims and complexity bypass hold under the stated assumptions, the work would provide a practically relevant tool for quantum many-body spectral analysis, particularly for studying phase transitions and topological order where clusters of dominant eigenvalues are physically natural. The numerical demonstrations on standard models like Ising and toric code add concrete support for applicability, and the mixed-state generalization broadens the scope.

major comments (2)
  1. [Abstract] Abstract: The central claim that physically motivated assumptions on clusters of closely spaced dominant eigenvalues suffice to bypass #BQP-completeness requires explicit quantitative conditions (e.g., lower bounds on inter-cluster gaps relative to 1/poly(n) or intra-cluster spacing thresholds that guarantee polynomial scaling of the filtering subroutine). No such thresholds or gap assumptions are stated, leaving open whether the algorithm remains efficient when gaps are sub-polynomial as noted in the hardness reduction for multiplicity counting.
  2. [Abstract] Abstract and claims of rigorous guarantees: The manuscript asserts 'rigorous theoretical guarantees' and efficiency under the assumptions, but the provided text contains no derivation details, error bounds, or explicit assumption statements for the quantum filtering and multiplicity estimation steps. This makes it impossible to verify that the approach avoids the #BQP-hardness reduction without additional exponential resources.
minor comments (1)
  1. [Abstract] The abstract mentions 'significant advantages over existing subspace-based quantum spectral analysis methods' in sample complexity and degeneracy resolution, but does not cite or compare against specific prior works (e.g., quantum phase estimation variants or subspace diagonalization algorithms); adding these references would improve context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive feedback on the abstract and the strength of our claims. We address each major comment below and will revise the manuscript to improve clarity and explicitness while preserving the core contributions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that physically motivated assumptions on clusters of closely spaced dominant eigenvalues suffice to bypass #BQP-completeness requires explicit quantitative conditions (e.g., lower bounds on inter-cluster gaps relative to 1/poly(n) or intra-cluster spacing thresholds that guarantee polynomial scaling of the filtering subroutine). No such thresholds or gap assumptions are stated, leaving open whether the algorithm remains efficient when gaps are sub-polynomial as noted in the hardness reduction for multiplicity counting.

    Authors: We agree that the abstract would benefit from explicit quantitative gap conditions to strengthen the complexity claim. The main text (Section 3 and Theorem 1) assumes inter-cluster gaps of at least 1/poly(n) and intra-cluster eigenvalue spacings sufficiently small to ensure the filtering operator achieves polynomial sample complexity via standard quantum signal processing techniques. These conditions are physically motivated for dominant clusters in many-body spectra and suffice to bypass the #BQP-hardness reduction, which requires resolving multiplicities across sub-polynomial gaps. We will revise the abstract to state these thresholds explicitly. revision: yes

  2. Referee: [Abstract] Abstract and claims of rigorous guarantees: The manuscript asserts 'rigorous theoretical guarantees' and efficiency under the assumptions, but the provided text contains no derivation details, error bounds, or explicit assumption statements for the quantum filtering and multiplicity estimation steps. This makes it impossible to verify that the approach avoids the #BQP-hardness reduction without additional exponential resources.

    Authors: The full manuscript contains the derivations, error bounds (additive error O(1/poly(n)) with high probability), and explicit assumption statements in Sections 3–4 and the appendix proofs for the filtering and multiplicity estimation subroutines. The abstract summarizes these results. To address the concern about verifiability from the abstract alone, we will add a concise statement of the key assumptions and polynomial scaling guarantee in the revised abstract, ensuring readers can see that no exponential overhead is introduced beyond the stated conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; QFAMES claims rest on external physical assumptions and new algorithm design

full rationale

The paper introduces QFAMES as a quantum algorithm for identifying eigenvalue clusters and multiplicities under physically motivated assumptions on spectra (e.g., clusters of closely spaced dominant eigenvalues). This allows bypassing worst-case #BQP-completeness for fine-grained analysis. The abstract and description tie efficiency and guarantees directly to these external assumptions and the algorithm's filtering steps, without reducing claims to fitted parameters, self-citations, or internal definitions by construction. Numerical demonstrations on Ising and toric code models are presented as validation, not as the source of the core claims. No load-bearing self-citation chains or ansatz smuggling are indicated. The derivation is self-contained against external benchmarks, consistent with a standard algorithm paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on unspecified physically motivated assumptions about eigenvalue clustering that are invoked to evade #BQP-completeness; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Physically motivated assumptions on the structure of eigenvalue spectra (clusters of closely spaced dominant eigenvalues) hold for the target Hamiltonians.
    Abstract states these assumptions allow bypassing worst-case #BQP-completeness for multiplicity determination.

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Reference graph

Works this paper leans on

94 extracted references · 94 canonical work pages · 3 internal anchors

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