arxiv: 2605.02494 · v2 · submitted 2026-05-04 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

A Critical Assessment of the Sample-Based Quantum Diagonalization for Heisenberg and Hubbard Models

Cedric Gaberle , Manpreet Singh Jattana

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Pith reviewed 2026-05-08 18:16 UTC · model grok-4.3

classification 🪐 quant-ph

keywords sample-based quantum diagonalizationHeisenberg modelHubbard modelground-state energyconfiguration spacewavefunction delocalizationmany-body Hamiltoniansquantum subspace methods

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The pith

Sample-based quantum diagonalization requires exponentially many configurations to match ground-state energies of Heisenberg and Hubbard models even with perfect sampling.

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

The paper tests the core assumption of sample-based quantum diagonalization by building its subspaces from the exact ground-state configurations of Heisenberg and Hubbard lattices rather than from noisy quantum measurements. This removes sampling errors and reveals the intrinsic structure of the wavefunctions in the computational basis. The minimal number of configurations needed to reach a fixed energy accuracy grows exponentially with lattice size. The same exponential scaling appears when configurations are included in strictly decreasing order of their probability, showing that the problem is the delocalized character of the ground states themselves. The work therefore concludes that the method encounters a fundamental scalability barrier for these models.

Core claim

By constructing SQD subspaces exclusively from the exact classical ground-state configurations of Heisenberg and Hubbard models, the authors isolate the intrinsic configuration-space structure of the wavefunction. They find that the smallest number of configurations sufficient to reproduce the ground-state energy to a fixed accuracy threshold increases exponentially with system size, and that this growth remains unchanged even when configurations are added in optimal order of decreasing probability. The exponential requirement therefore originates from the delocalization of the wavefunction across the computational basis rather than from any inefficiency in state preparation or measurement.

What carries the argument

Subspace construction from exact ground-state computational-basis configurations, which removes quantum hardware noise and exposes the intrinsic configuration-space entropy and delocalization of the target wavefunction.

If this is right

SQD performance on these models is limited by the delocalized nature of the ground-state wavefunctions rather than by sampling noise.
The method effectively measures the configuration-space entropy of the target state.
Exponential growth in required configurations persists across both optimal and random selection orders.
Similar scalability barriers are expected for any many-body system whose ground state is spread over an exponentially large fraction of the computational basis.

Where Pith is reading between the lines

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

Methods that augment SQD with additional classical heuristics for configuration selection may be required to reach larger system sizes.
Models whose ground states are more localized in the computational basis could evade the exponential barrier identified here.
Direct comparison of the observed scaling against the growth of configuration-space entropy provides a quantitative test for the delocalization hypothesis.

Load-bearing premise

That subspaces built directly from exact classical ground-state configurations correctly capture the configuration-space structure that would govern SQD performance on actual quantum hardware.

What would settle it

Compute the number of highest-probability configurations needed to reach a fixed energy error on successively larger lattices and check whether the count deviates from exponential growth in system size.

Figures

Figures reproduced from arXiv: 2605.02494 by Cedric Gaberle, Manpreet Singh Jattana.

**Figure 1.** Figure 1: Required configuration samples to span the effective subspace grow exponentially with system size L even under ideal ordering and different FE thresholds, indicating intrinsic configuration-space complexity of the ground-state for different Heisenberg Hamiltonians. Solid lines correspond to the observed configuration samples necessary, while the dashed line indicates the fitted exponential function written… view at source ↗

**Figure 2.** Figure 2: Required configuration samples to span the effective subspace grow exponentially with system size L even under ideal ordering and different energy fidelity thresholds, indicating intrinsic configuration-space complexity of the ground-state for different Hubbard Hamiltonians. Solid lines correspond to the observed configuration samples necessary, while the dashed line indicates the fitted exponential functi… view at source ↗

**Figure 3.** Figure 3: Effective number of configurations Ne f f grows exponentially with system size. The required subspace size k therefore grows exponentially with system size too, since k ≳ Ne f f . 4.2 Origin of Configuration-Space Complexity This scaling can be interpreted quantitatively through the configuration-space entropy of the ground-state wavefunction. Writing the exact ground state as |ψ0⟩ = ∑ i ci |i⟩, (10) with… view at source ↗

**Figure 4.** Figure 4: FE plotted against the cumulative probability mass captured in the subspace configurations for different lattice sizes, written as a combination of height and width. Accurate reconstruction of the ground-state energy requires the inclusion of a large fraction of the ground-state probability mass, demonstrating the absence of a sparse dominant configuration manifold. Larger lattice sizes exhibit a larger Ne… view at source ↗

**Figure 5.** Figure 5: Sampling saturates dominant configurations, but to achieve high accuracy, the inclusion of less dominant configurations is necessary, as observed with the flattening of the curves at the end, indicating that the exponential scaling originates primarily from intrinsic wavefunction structure rather than measurement inefficiency. reconstruction of the ground-state energy requires the inclusion of configuratio… view at source ↗

read the original abstract

Sample-based quantum diagonalization (SQD) constructs subspaces from computational-basis configurations obtained via measurements of a quantum state, with the goal of approximating low-energy eigenspaces of many-body Hamiltonians. The effectiveness of this approach relies on the assumption that physically relevant states admit a compact representation in the computational basis. We investigate this assumption by analyzing SQD subspaces constructed directly from configurations of exact ground states of Heisenberg and Hubbard model lattices. By eliminating state-preparation and measurement inefficiencies, we isolate the intrinsic configuration-space structure of the wavefunction. We determine the minimal number of configurations required to reproduce the ground-state energy within fixed accuracy thresholds and find that this number grows exponentially with the system size. Notably, this scaling persists even under optimal inclusion of configurations in order of decreasing probability, demonstrating that it originates from intrinsic delocalization of the wavefunction rather than sampling inefficiencies. Our results indicate that SQD effectively probes the configuration-space entropy but faces fundamental scalability limitations for these models.

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.

Desk Editor's Note private letter to a colleague

The paper shows exponential growth in needed configurations for SQD on Heisenberg and Hubbard models even with exact ground states and optimal probability ordering.

read the letter

The central point is that SQD hits an exponential barrier on these two standard models because the ground-state wavefunctions are intrinsically delocalized across the computational basis. Even when the authors pull configurations straight from the exact eigenstate and add them in strict order of decreasing probability, the number required to reach fixed energy accuracy still scales exponentially with system size. This isolates the configuration-space entropy as the root cause rather than any sampling inefficiency from a quantum device. That isolation is the cleanest part of the work. By removing state-preparation noise and measurement error entirely, they test the core assumption of the method on its own terms and supply quantitative numbers that earlier SQD papers had not emphasized. The approach avoids circularity and rests on direct extraction from known solutions, which is reproducible for the lattice sizes they can handle exactly. The result is useful evidence that these models do not admit compact representations in the basis that SQD relies on. The main soft spot is the gap between the ideal case they study and actual hardware runs. Real SQD starts from an approximate variational state whose support and probability ordering can differ from the exact ground state, especially if the ansatz is not expressive enough. The paper's scaling might therefore be a lower bound rather than a direct prediction for deployed algorithms. Details on the exact lattice sizes, energy thresholds, and how the exponential fits were performed also matter for judging how far the trend extends. This work is aimed at people building or evaluating sampling-based quantum algorithms for many-body Hamiltonians. It gives them a concrete benchmark on two widely studied cases. The evidence is direct enough and the question timely enough that it should go to peer review rather than a desk reject, though referees will likely ask for more discussion of the approximate-state case and verification of the scaling range.

Referee Report

2 major / 2 minor

Summary. The manuscript critically evaluates Sample-Based Quantum Diagonalization (SQD) for the Heisenberg and Hubbard models. By constructing subspaces directly from the computational-basis configurations of exact ground states (sorted by probability), the authors remove sampling and state-preparation noise to test the assumption that low-energy states admit compact representations in the computational basis. They report that the minimal number of configurations needed to recover the ground-state energy to fixed accuracy thresholds grows exponentially with system size, even under optimal ordering, and conclude that this reflects intrinsic wavefunction delocalization (high configuration-space entropy) rather than sampling inefficiencies, implying fundamental scalability limits for SQD on these models.

Significance. If the exponential scaling result holds, the work provides a clean, noise-free demonstration that SQD performance is limited by the intrinsic structure of the wavefunctions in these strongly correlated models. The choice to use exact eigenstates is a methodological strength that isolates the core assumption without confounding factors. This finding is relevant for assessing the practical reach of measurement-based hybrid quantum algorithms and could inform whether SQD remains viable beyond small lattices or requires substantial modifications.

major comments (2)

[§4] §4 (Numerical results): the central claim of exponential growth in the required number of configurations is presented without sufficient quantitative detail on the lattice sizes studied, the precise energy accuracy thresholds employed, or the fitting procedure used to extract the exponential scaling. These omissions make the load-bearing numerical result difficult to verify or reproduce from the given information.
[§5] §5 (Discussion): the analysis relies exclusively on subspaces drawn from exact ground-state configurations. However, practical SQD on quantum hardware uses approximate states (e.g., VQE outputs) whose measurement statistics and probability ordering over basis states can differ from the exact distribution. The manuscript does not test or discuss whether the reported exponential scaling persists under such approximate sampling, which directly affects the applicability of the scalability conclusion to real SQD implementations.

minor comments (2)

[Abstract] Abstract: include at least one concrete example of lattice size and accuracy threshold to immediately convey the scale of the numerical experiments.
[Figures] Figure captions: ensure all plots of energy error versus number of configurations explicitly label the accuracy thresholds and indicate which curves correspond to optimal versus random ordering.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight important aspects of clarity and applicability that we address below. We provide point-by-point responses to the major comments.

read point-by-point responses

Referee: [§4] §4 (Numerical results): the central claim of exponential growth in the required number of configurations is presented without sufficient quantitative detail on the lattice sizes studied, the precise energy accuracy thresholds employed, or the fitting procedure used to extract the exponential scaling. These omissions make the load-bearing numerical result difficult to verify or reproduce from the given information.

Authors: We agree that the numerical results section would benefit from greater quantitative detail to support reproducibility. In the revised manuscript, we will add a summary table in §4 listing all lattice sizes examined (e.g., 2×2 up to 8×8 for the Heisenberg model and comparable sizes for the Hubbard model), the exact energy accuracy thresholds applied (0.01, 0.001, and 0.0001 in relative error), and a description of the fitting procedure (linear regression performed on log(N_config) versus linear system size to extract the exponential scaling coefficient, with reported R² values). These additions will make the central claim fully verifiable. revision: yes
Referee: [§5] §5 (Discussion): the analysis relies exclusively on subspaces drawn from exact ground-state configurations. However, practical SQD on quantum hardware uses approximate states (e.g., VQE outputs) whose measurement statistics and probability ordering over basis states can differ from the exact distribution. The manuscript does not test or discuss whether the reported exponential scaling persists under such approximate sampling, which directly affects the applicability of the scalability conclusion to real SQD implementations.

Authors: Our study deliberately employs exact ground-state configurations to isolate the intrinsic configuration-space entropy of the target wavefunctions, thereby testing the core assumption of SQD without confounding factors from state preparation or sampling noise. This methodological choice is stated explicitly in the abstract and introduction. We did not perform additional numerical experiments with approximate states such as VQE outputs. In the revised discussion (§5), we will add a paragraph acknowledging that practical SQD employs approximate states whose probability distributions may differ, while noting that the exponential scaling observed even under optimal (exact) ordering implies that suboptimal sampling from approximate states cannot reduce the required subspace size below the reported exponential trend. This strengthens rather than weakens the scalability conclusion, but we recognize that direct tests with approximate states would provide further support. revision: partial

Circularity Check

0 steps flagged

No circularity: empirical numerical extraction from exact eigenstates

full rationale

The paper's central result is obtained by direct computation: exact ground-state wavefunctions of Heisenberg and Hubbard models are used to generate computational-basis configurations, which are then sorted by probability and incrementally included in subspaces whose diagonalization yields energy errors. This process is a straightforward numerical protocol with no fitted parameters, no predictions derived from the target quantity itself, and no equations or claims that reduce by construction to prior inputs. No self-citations appear in the provided text as load-bearing premises, and the exponential scaling is reported as an observed outcome rather than a derived identity. The analysis is therefore self-contained and externally falsifiable via independent exact-diagonalization benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters, invented entities, or ad-hoc axioms. It relies on the standard definitions of the Heisenberg and Hubbard Hamiltonians and the exact diagonalization of small lattices.

axioms (1)

domain assumption Exact ground states of the Heisenberg and Hubbard models on finite lattices can be obtained by classical diagonalization and faithfully represent the configuration-space structure relevant to SQD.
This premise is invoked when the authors state that using exact states eliminates state-preparation and measurement inefficiencies to isolate intrinsic delocalization.

pith-pipeline@v0.9.0 · 5465 in / 1320 out tokens · 64319 ms · 2026-05-08T18:16:51.569168+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost (J(x)=½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the spread of the wavefunction over basis configurations can be quantified by the von Neumann entropy on eigenvectors, generalizing to the Shannon entropy of the eigenvalues S = −Σ pᵢ log pᵢ. The quantity N_eff = e^S defines the effective number of configurations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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