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arxiv: 2602.20108 · v1 · pith:H3IZMJ5Cnew · submitted 2026-02-23 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· physics.comp-ph· quant-ph

Energy gap of quantum spin glasses: a projection quantum Monte Carlo study

L. Brodoloni , G. E. Astrakharchik , S. Giorgini , S. Pilati This is my paper

Pith reviewed 2026-05-22 11:45 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechphysics.comp-phquant-ph
keywords quantum spin glassesminimum energy gapEdwards-Anderson modelSherrington-Kirkpatrick modelquantum Monte Carlogap scalingquantum annealingdisorder distribution
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The pith

In 2D quantum spin glasses the minimum gap shrinks faster than any power law while the all-to-all model scales as N to the minus one third.

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

The paper establishes how the minimum energy gap scales with system size in two quantum spin glass models because this scaling determines whether quantum annealing can efficiently reach optimal solutions without getting trapped. In the two-dimensional Edwards-Anderson model the inverse-gap distribution develops a fat tail whose variance diverges with larger N, which forces the smallest gaps to vanish super-algebraically. The Sherrington-Kirkpatrick model keeps a finite-variance gap distribution whose average follows a power law near N to the minus one third. These contrasting behaviors are extracted with a new unbiased gap estimator inside continuous-time projection quantum Monte Carlo runs supplemented by exact sparse diagonalization on smaller instances.

Core claim

In the 2D Edwards-Anderson model the inverse-gap distribution develops a fat tail with infinite variance as N increases, indicating unfavorable super-algebraic scaling of the minimum gap. The SK model instead shows a finite-variance distribution whose disorder-averaged gap follows a slow power law close to Δ ∝ N^{-1/3}.

What carries the argument

Unbiased energy-gap estimator inside continuous-time projection quantum Monte Carlo that extracts the true minimum gap without residual finite-projection-time bias.

If this is right

  • Quantum annealing faces greater difficulty on two-dimensional spin-glass instances because of the super-algebraic gap closure.
  • All-to-all connected problems retain better gap scaling and therefore offer more promising prospects for quantum optimization.
  • The fat tail in the inverse-gap distribution is independent of whether the couplings are Gaussian or binary and appears universal for two-dimensional spin glasses.
  • The N to the minus one third scaling in the SK model is slow yet still polynomial, unlike the 2D case.

Where Pith is reading between the lines

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

  • Connectivity density may be the decisive factor separating favorable from unfavorable gap scaling in quantum spin glasses.
  • Intermediate-connectivity models could be simulated to locate any crossover between the 2D and all-to-all regimes.
  • Even the milder N to the minus one third scaling may still set practical limits on the largest instances solvable by annealing.
  • The results motivate checking whether similar fat tails appear in three-dimensional spin glasses or in other planar optimization mappings.

Load-bearing premise

The new unbiased energy-gap estimator returns the exact minimum gap without leftover bias or projection-time artifacts that would distort the tail of the inverse-gap distribution at large N.

What would settle it

A calculation showing that the variance of the inverse-gap distribution remains finite rather than diverging when system size is increased further in the 2D Edwards-Anderson model.

Figures

Figures reproduced from arXiv: 2602.20108 by G. E. Astrakharchik, L. Brodoloni, S. Giorgini, S. Pilati.

Figure 1
Figure 1. Figure 1: FIG. 1. Complementary empirical distribution function [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Hill estimator for the tail index [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Main panel: Odd energy gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Hill estimator [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Disorder averaged energy odd gap [∆ [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Power [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Odd and (offset) even imaginary-time correlation [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap $\Delta$ encountered at quantum phase transitions. We investigate the scaling of $\Delta$ with system size $N$ for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as $N$ increases. This indicates that the unfavorable super-algebraic scaling of $\Delta$, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to $\Delta \propto N^{-1/3}$. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.

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 / 2 minor

Summary. The paper claims that a new unbiased energy-gap estimator in continuous-time projection quantum Monte Carlo, cross-checked with sparse eigenvalue solvers, reveals that the inverse-gap distribution in the 2D Edwards-Anderson model develops a fat tail with infinite variance as N increases (implying super-algebraic minimum-gap scaling), while the Sherrington-Kirkpatrick model shows a finite-variance distribution whose disorder-averaged gap scales approximately as N^{-1/3}.

Significance. If the estimator accurately captures the lower tail, the results strengthen the case that rare small-gap events pose a fundamental obstacle to quantum annealing in 2D spin glasses (extending prior binary-coupling findings to Gaussian disorder) while offering a more optimistic scaling outlook for densely connected models. The direct sampling over many disorder realizations and the introduction of an unbiased estimator are positive features that avoid fitting-based circularity.

major comments (2)
  1. [Methods (gap estimator subsection)] The validation of the new unbiased gap estimator (described in the methods section on continuous-time projection QMC) is insufficient for the central claim. While consistency with sparse solvers is noted for accessible sizes, there is no explicit benchmark of the full inverse-gap distribution—including its lower tail—against exact diagonalization on small-N systems across multiple disorder realizations. This is load-bearing for the infinite-variance inference in 2D-EA, as residual projection-time bias could artificially inflate the fat-tail statistics.
  2. [2D-EA results] § on 2D-EA results: the conclusion that the inverse-gap distribution has infinite variance as N increases relies on the statistics of rare small-Δ events; without the above benchmarks or a quantitative assessment of projection-time convergence for the smallest observed gaps, the super-algebraic scaling claim remains provisional.
minor comments (2)
  1. [Abstract and SK results] The abstract states the SK scaling is 'close to Δ ∝ N^{-1/3}'; the main text should report the fitted exponent with uncertainty and the range of N used for the fit.
  2. [Figures] Figure captions for the inverse-gap histograms should explicitly state the number of disorder realizations and the projection-time cutoff employed for each N.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each of the major comments in detail below. To strengthen the manuscript, we have incorporated additional validation benchmarks and convergence analyses as requested.

read point-by-point responses

  1. Referee: [Methods (gap estimator subsection)] The validation of the new unbiased gap estimator (described in the methods section on continuous-time projection QMC) is insufficient for the central claim. While consistency with sparse solvers is noted for accessible sizes, there is no explicit benchmark of the full inverse-gap distribution—including its lower tail—against exact diagonalization on small-N systems across multiple disorder realizations. This is load-bearing for the infinite-variance inference in 2D-EA, as residual projection-time bias could artificially inflate the fat-tail statistics.

    Authors: We acknowledge the referee's concern regarding the validation of our gap estimator. Although the manuscript notes consistency with sparse eigenvalue solvers for accessible system sizes, we agree that an explicit comparison of the full inverse-gap distribution, with emphasis on the lower tail, against exact diagonalization would provide more robust support for our claims. Accordingly, we have extended our analysis to include such benchmarks for small-N systems (N ≤ 20) across numerous disorder realizations. The results confirm that our unbiased estimator accurately reproduces the distribution, including the rare small-gap events, with negligible bias at the projection times employed. A new figure and accompanying discussion will be added to the Methods section in the revised manuscript. revision: yes

  2. Referee: [2D-EA results] § on 2D-EA results: the conclusion that the inverse-gap distribution has infinite variance as N increases relies on the statistics of rare small-Δ events; without the above benchmarks or a quantitative assessment of projection-time convergence for the smallest observed gaps, the super-algebraic scaling claim remains provisional.

    Authors: We appreciate this observation, as the reliability of the lower tail is indeed critical to our inference of infinite variance. In the revised manuscript, we include a quantitative assessment of projection-time convergence specifically for the smallest gaps encountered in our simulations. By systematically increasing the projection time for representative disorder realizations exhibiting small gaps, we demonstrate that the gap estimates converge to stable values well within the range used in our production data. This analysis, combined with the exact diagonalization benchmarks, substantiates that the observed fat tail is not an artifact of insufficient projection time. We believe these additions address the provisional nature of the claim and solidify the evidence for super-algebraic scaling in the 2D-EA model. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical sampling of disorder realizations

full rationale

The paper's central claims about gap distributions and scaling in the 2D-EA and SK models are obtained via direct continuous-time projection QMC simulations across many disorder realizations, using a newly introduced unbiased gap estimator whose performance is presented as an independent methodological contribution. No load-bearing step reduces a claimed prediction or scaling result to a fitted parameter, self-citation chain, or definitional equivalence; the reported fat-tail behavior and power-law exponents emerge from the sampled statistics rather than from any algebraic or fitting closure within the paper's own equations. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The study is a numerical characterization that relies on standard quantum Monte Carlo methodology and linear-algebra libraries rather than new analytic assumptions or fitted constants.

axioms (1)
  • domain assumption Continuous-time projection quantum Monte Carlo with the proposed unbiased estimator converges to the true ground-state gap in the limit of infinite projection time and sufficient disorder sampling.
    Invoked when the authors state that the estimator is unbiased and use it to extract scaling for large N.

pith-pipeline@v0.9.0 · 5777 in / 1361 out tokens · 47076 ms · 2026-05-22T11:45:10.369144+00:00 · methodology

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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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
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    Relation between the paper passage and the cited Recognition theorem.

    We investigate the scaling of ∆ with system size N for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations

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.

Reference graph

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