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Defects and their Time Scales in Quantum and Classical Annealing of the Two-Dimensional Ising Model

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Pith's one-line read Quantum annealing of the 2D Ising magnet is governed, at the end of the ramp, by three size-dependent time scales: Kibble-Zurek $L^{2.59}$ fidelity, coarsening $L^2$, and stripe-elimination $L^3$.

desk verdict A careful, honest paper with genuinely new QA results and strong SA numerics; the three-time-scale picture is probably right, but the QA L^3 claim needs a clearer data-selection criterion and the abstract overstates one result. read the letter →

arxiv 2507.09273 v1 pith:35BY7SIG submitted 2025-07-12 quant-ph cond-mat.othercond-mat.stat-mechhep-lat

Phillip Weinberg , Na Xu , Anders W. Sandvik This is my paper · ORCID
classification quant-phcond-mat.othercond-mat.stat-mechhep-lat
keywords quantumannealingKibble-Zurekscalingtransverse-fieldIsingmodelcoarseningdynamicstopologicaldefectswindingnumbersimulateddomainwall
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what defects remain when a two-dimensional transverse-field Ising ferromagnet is annealed from a high transverse field down to zero field, and on what size-dependent time scales those defects disappear. Using exact integration of the Schrödinger equation on periodic lattices up to $L=6$, the authors find that the final state is set by three distinct clocks. The chance of landing in the ferromagnetic ground state, the ground-state fidelity, freezes near the quantum critical point and scales with the Kibble-Zurek time $L^{2.59}$. The excited states then order like a classical anneal: confined defects coarsen away on $L^2$, while system-spanning horizontal or vertical stripe domains survive until the longer $L^3$ scale, which the paper detects with a new excited-state analysis. If the picture holds, quantum annealers can be tested against a three-clock benchmark using only final classical measurements.

What carries the argument

The central object is the topological winding number $W=(w_x,w_y)$ of domain walls on periodic lattices, which separates confined defects $(0,0)$, axis-spanning stripes $(1,0)/(0,1)$, and diagonal stripes $(1,1)$. The argument is carried by scaling collapse: each observable is plotted against $vL^\alpha$, and the value of $\alpha$ for which data from different $L$ fall on one curve identifies the controlling time scale, such as $\alpha=2.59$, $2$, $3$, $3.17$, or $3.42$. For quantum annealing, the decisive new tool is an excited-state analysis that subtracts the exactly known contributions of the two ferromagnetic ground states, $|a_0|^2$ and $2|a_0|^2$, and further subtracts the lowest-excited-state reference values so that the remaining signal collapses separately under $vL^3$ (stripe elimination) and $vL^2$ (coarsening).

What would settle it

On a periodic superconducting-qubit annealer with $L=10$ through $12$, measure the final $\Gamma=0$ probability of a system-spanning horizontal or vertical stripe as a function of annealing velocity $v$: if it does not collapse under $vL^{2.59}$ but instead collapses under $vL^3$, the paper's claim that KZ controls stripe survival in QA is wrong; in the same data, the excited-state-only excess energy should collapse under $vL^3$ once ground-state shots are removed.

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Extended reading notes

Core claim

The paper's central claim is that quantum annealing of the 2D transverse-field Ising model to vanishing field produces a late-time state with two coexisting layers. The ground-state probability is gap-protected after the critical point, so it retains the Kibble-Zurek signature $vL^{z+1/\nu}$ with $z=1$, $1/\nu \approx 1.59$ all the way to $\Gamma=0$; the authors demonstrate near-perfect collapse of $-\log F_{GS}$ versus $vL^{2.59}$. Everything else is carried by excited states, which the paper argues thermalize and evolve under emergent classical dynamics: ordering proceeds by coarsening of confined $(0,0)$ defects on the scale $L^2$, and the last system-spanning $(1,0)/(0,1)$ stripes are eliminated by interface fluctuations on the scale $L^3$. In classical simulated annealing the same analysis shows that diagonal $(1,1)$ stripes decay on $L^{3.42}$, longer than the classical KZ scale $L^{3.17}$, so their final $T=0$ survival probability is KZ-controlled; in quantum annealing the quantum KZ scale $L^{2.59}$ is shorter than $L^3$, so the paper concludes that horizontal and vertical stripe survival is likewise KZ-controlled even though the $L^3$ scale is still present in excited-state observables.

Load-bearing premise

The transfer of the $L^2$ and $L^3$ scales from classical to quantum annealing rests on the assumption that excitations created while crossing the quantum critical point thermalize inside the ordered phase and then move by emergent classical dynamics; if those excitations stay quantum-coherent instead, the identification of the quantum $L^2$ and $L^3$ scales as coarsening and interface fluctuations is not supported.

Editorial extensions

If this is right

  • The ground-state success probability of a quantum annealer is decided at the critical point: it obeys $vL^{2.59}$ scaling at $\Gamma=0$, so extending the ramp inside the ferromagnetic phase will not increase the fidelity.
  • Annealing times of order $L^2$ remove confined defects but leave system-spanning stripes, so intermediate-time final states of a clean annealer should be dominated by straight horizontal or vertical domain walls.
  • In simulated annealing, diagonal stripe survival at $T=0$ carries the KZ exponent $L^{3.17}$ rather than the $L^{3.42}$ fixed-temperature decay scale, because diagonal stripes decay mainly just below $T_c$.
  • Post-selecting only non-ground-state shots and subtracting the lowest-excited-state reference exposes the $L^3$ scale in quantum annealing, a measurable signature in experiments where the ground state can be identified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the emergent-classical thermalization picture is correct, the same three-clock structure should appear in any non-integrable transverse-field magnet whose ordered phase supports both confined and system-spanning defects, with exponents set by the transition's universality class and interface roughening.
  • The argument implies a practical diagnostics protocol: run the annealer at $t_{\rm QA}\sim L^2$; the final ground-state probability should then obey KZ scaling while stripe statistics still show the longer $L^3$ scale, so the two clocks measure device noise in different places.
  • By analogy with SA, periodic lattices larger than $L=6$ should show a fourth scale, $L^{3.42}$, for diagonal stripes in QA; with open boundaries that scale should disappear and only $L^3$ remain for all system-spanning walls.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies defect production and late-time ordering in the two-dimensional transverse-field Ising ferromagnet after quantum annealing to zero transverse field, using exact time evolution for L<=6, and compares with classical simulated annealing of the 2D Ising model for systems up to L=768. The central claim is that the final state of QA exhibits three distinct size-dependent time scales: Kibble-Zurek scaling of the ground-state fidelity with scale L^{2.59}, coarsening of confined defects with scale L^2, and elimination of system-spanning stripe defects with scale L^3. The SA analysis additionally identifies a longer lifetime L^{3.42} for diagonal (W=(1,1)) domain walls, whose survival in SA is argued to be controlled by the KZ scale L^{3.17} rather than by the longer fixed-temperature decay scale. The authors develop an excited-state analysis that they propose as an experimentally applicable method for detecting the L^3 scale in QA.

Significance. If the claims hold, the paper provides a concrete multi-time-scale benchmark for QA devices and a nontrivial example where KZ scaling survives deep into an ordered phase. The exact numerical solutions, with documented tight integration tolerances and parameter-free KZ collapses, are a genuine strength, as are the large-scale SA simulations with multi-spin coding. The paper also makes falsifiable predictions for QA experiments, which is valuable. The main significance, however, depends on the reliability of the L^3 detection in QA, which is the aspect that most needs scrutiny.

major comments (3)
  1. [Sec. III B, Fig. 26] The L^3 claim for QA rests primarily on the collapse of the excited-state observables in Fig. 26, but the text states that only velocities for which "no numerical anomalies are apparent" are shown, without specifying a deterministic exclusion criterion or listing the excluded runs. Because the anomalies are said to occur at the longest integration times and smallest computed differences, the excluded points are likely concentrated at the small-x=vL^3 end of the scaling plot, exactly where the collapse determines the approach to the lowest-excited-state plateau. As written, the collapse could therefore be an artifact of selective data removal. Please provide a reproducible criterion for excluding points, show the data with and without the excluded points, and demonstrate that the collapse is stable under inclusion of all data that satisfy a well-defined accuracy bound.
  2. [Abstract and Sec. IV] The statement that in QA the W=(1,0)/(0,1) stripe domains are "controlled by the KZ time scale L^{2.59}" is inferred from the comparison L^3 > L^{2.59} rather than measured directly by scaling the stripe survival probability against vL^{2.59}. The inference is plausible if the L^3 scale is established, but the abstract and conclusions present it as an observed result. Please either soften this to a prediction (as in items (i)-(iii) of the conclusions) or provide a direct test, for example by using the winding-number or stripe-identification method on the exact QA wave functions for the accessible sizes.
  3. [Sec. III B and Sec. IV] The identification of the QA L^2 and L^3 scales with coarsening and interface fluctuations relies on the assumption that the gap-protected excitations thermalize in the ordered phase and develop effectively classical dynamics. This assumption is acknowledged as "the most plausible" mechanism, but it is not directly tested. The empirical scaling collapses in Figs. 22 and 26 can stand independently of this interpretation; however, the concluding statement that "all our QA results in the ordered phase point to the same ordering mechanisms as in SA" goes beyond the numerical evidence. Please either temper this mechanistic claim or provide a diagnostic of thermalization, such as a comparison of the excited-state energy distribution with a thermal distribution at an effective temperature.
minor comments (4)
  1. [Sec. IV, first paragraph] The sentence "diagonal domains have the longer life time ∝L^{z+1/ν}≈L^{3.17}" contradicts the fixed-temperature decay exponent L^{3.42} established in Sec. II B. What is controlled by L^{3.17} is the KZ survival probability of diagonal domains in SA, not their fixed-temperature decay time. Please rephrase to avoid this inconsistency.
  2. [Sec. III A, Eq. (24a)] The scaling variable in Eq. (24a) is written as vL^{1/ν}, but it should be vL^{z+1/ν} to be consistent with Eq. (4) and with the collapse shown in Fig. 21. This appears to be a typo.
  3. [General presentation] The manuscript text contains several duplicated figure captions and repeated blocks of text, for example around Fig. 24 and in Sec. III B. Please clean these up, as they make the paper difficult to read in the present form.
  4. [References] References [52] and [53] appear to be identical (C.-W. Liu, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B 89, 054307 (2014)), and reference [36] contains the misspelling "Sicillia". Please correct these bibliographic issues.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: QA data are generated by exact evolution and tested against externally fixed KZ exponents and SA-derived scaling variables without fitted exponents; self-citations are non-load-bearing.

full rationale

The paper's derivation chain is largely self-contained. The QA results are obtained by exact numerical integration of the Schrodinger equation for L <= 6, and the KZ exponents used (z=1, 1/nu=1.587375) are imported from the external 3D-Ising/conformal-bootstrap literature, not fitted to QA data. The scaling collapses in Figs. 20 and 21 are parameter-free tests of the KZ ansatz, and the low-velocity v^2 forms are derived from adiabatic perturbation theory rather than assumed from the data. The L^2 coarsening scale in QA is tested by plotting m^2 and excess energy versus vL^2 with the exponent taken from classical SA, where it is independently established on large lattices (L up to 768); this is an external cross-check, not a fit. The L^3 scale in QA is likewise a fixed-exponent hypothesis imported from SA and tested in Figs. 25-27; the excited-state subtraction that defines Delta M and Delta E uses only the exactly known lowest-excited-state values (N-2)^2 and -2N+8, so the observables are not defined in terms of the claimed L^3 scale. The conclusion that W=(1,0)/(0,1) domains in QA obey KZ scaling follows from comparing the SA/QA L^3 elimination scale with the KZ scale L^2.59; this is a logical inference within the stated thermalization/emergent-classical-dynamics assumption, not a restatement of any fitted quantity. Self-citations (Refs. 13, 49-57) provide the critical-point value, APT scaling forms, and KZ analysis methodology, but none is load-bearing: the critical point value is stated to be insensitive to small deviations, and the scaling forms are independently re-derived and confirmed by the data. One passage in Sec. III B discloses that in Fig. 26 'we therefore only show results for those velocities for which no numerical anomalies are apparent,' which is a legitimate validity concern about selective presentation, but it is not an instance of a prediction reducing by construction to its inputs; the L^3 variable is not fitted from the retained points. Overall, the central three-time-scale claim has independent content and is not forced by definition or by a self-citation chain.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities or forces. Its free parameters are empirical scaling exponents measured from MC data; the KZ exponents for QA are taken from external conformal-bootstrap/QMC results. The main assumptions are the standard KZ scaling ansatz and the emergent classical thermalization of QA excitations.

free parameters (3)
  • a_W11_mean = 3.42 ± 0.02 (mean decay time; typical 3.56 ± 0.02)
    Exponent for elimination of W=(1,1) diagonal stripe domains in fixed-T SA. Fitted to MC data for L>=288 (Fig. 9). Used to argue that this time scale exceeds the SA KZ scale L^{3.17}, implying KZ scaling of W=(1,1) survival at T=0.
  • a_W10 = ≈3 (2.987 ± 0.013 at T=0.5)
    Exponent for elimination of W=(1,0)/(0,1) horizontal/vertical stripe domains in fixed-T SA. Confirms the L^3 scale. Fitted to MC data (Fig. 9).
  • open_boundary_log_fit = a=0.72, b=1.7 in ⟨τ_d⟩∝L^2 ln^a(L/b)
    Empirical fit for the time for an imposed diagonal domain wall to relax to a straight wall in open-boundary T=0 MC (Fig. 17). Supporting result, not central.
assumptions (5)
  • domain assumption The Kibble-Zurek scaling ansatz A(v,L)=L^{-kappa/nu} f(vL^{z+1/nu}) applies to the finite-size QA and SA processes.
    Invoked in Eqs. (2)-(4) and used throughout; standard framework, but its validity for the small L<=6 QA systems is an assumption supported by data collapses.
  • domain assumption Excitations created in QA thermalize in the ordered phase and follow emergent classical coarsening dynamics.
    Stated in Sec. III B: 'the most plausible fast ordering mechanism ... is thermalization of the gap-protected excitations in the ordered phase'. Load-bearing for transferring SA time scales to QA.
  • domain assumption For QA systems with L<=6, the infinite-size critical point s_c and the 3D Ising exponents (z=1, nu about 0.63, beta/nu about 0.518) are appropriate for scaling analysis despite finite-size shifts.
    Used in Sec. III A for Figs. 19-21; the paper notes finite-size drift but still uses s_c; the quality of collapse is the evidence.
  • standard math The winding number fully classifies system-spanning topological defects in periodic 2D Ising configurations.
    Used in Sec. II C; standard algebraic topology of domain walls on a torus.
  • standard math The lowest excited state of the classical 2D Ising model on a periodic lattice has one flipped spin with energy -2N+8 and magnetization (N-2)^2.
    Used in Eqs. (25)-(26) to construct the excited-state subtraction; exact for Gamma=0.

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Pith. "Pith review of Defects and their Time Scales in Quantum and Classical Annealing of the Two-Dimensional Ising Model." pith.science (2026). https://pith.science/paper/35BY7SIG

@misc{pith2026250709273,
  author       = {Pith},
  title        = {Pith review of: Defects and their Time Scales in Quantum and Classical Annealing of the Two-Dimensional Ising Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35BY7SIG}},
  note         = {Machine review of arXiv:2507.09273}
}
abstract

We investigate defects in the two-dimensional transverse-field Ising ferromagnet on periodic $L\times L$ lattices after quantum annealing from high to vanishing field. With exact numerical solutions for $L \le 6$, we observe the expected critical Kibble-Zurek (KZ) time scale $\propto L^{z+1/\nu}$ (with $z=1$ and $1/\nu \approx 1.59$) at the quantum phase transition. We also observe KZ scaling of the ground-state fidelity at the end of the process. The excitations evolve by coarsening dynamics of confined defects, with a time scale $\propto L^2$, and interface fluctuations of system-spanning defects, with life time $\propto L^3$. We build on analogies with classical simulated annealing, where we characterize system-spanning defects in detail and find differences in the dynamic scales of domain walls with winding numbers $W=(1,0)/(0,1)$ (horizontal/vertical) and $W=(1,1)$ (diagonal). They decay on time scales $\propto L^3$ (which applies also to system-spanning domains in systems with open boundaries) and $\propto L^{3.4}$, respectively, when imposed in the ordered phase. As a consequence of $L^{3.4}$ exceeding the classical KZ scale $L^{z+1/\nu}=L^{3.17}$ the probability of $W=(1,1)$ domains in SA scales with the KZ exponent even in the final $T=0$ state. In QA, also the $W=(1,0)/(0,1)$ domains are controlled by the KZ time scale $L^{2.59}$. The $L^3$ scale can nevertheless be detected in the excited states, using a method that we develop that should also be applicable in QA experiments.

Figures

Figures reproduced from arXiv: 2507.09273 by the authors.

Figure 1
Figure 1. FIG. 1. Phase diagram of the 2D TFIM in the plane of tem [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Spin configurations of a classical [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Excess mean Ising energy density scaled by the sys [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (18 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Data collapse of the grouped squared order parame [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Probability of reaching one of the perfectly ferro [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Snapshots of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Mean time required at fixed [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Scatter plots of the domain order parameter [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Temperature dependent probabilities of the four [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Scaling of the probability of winding numbers (1 [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: (b). This difference between the two winding sectors is consistent with the time scale L 3.42 found in Sec. II C for the elimination of the diagonal domain walls at fixed T < Tc. The KZ scale is shorter and, therefore, the W = (1, 1) domains will primarily decay very …
Figure 14
Figure 14. Figure 14: FIG. 14. Temperature dependent probability of winding num [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Configurations from MC simulations of a system o FIG. 16. Configurations from MC simulations of a system of [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Size dependence of the mean time at T 0 for sys ems with an initially imposed diagonal domain wall to evolv wall is reduced over time, thus creating a shrinking do [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. The domain-wall order parameter defined in Eq. (16) [PITH_FULL_IMAGE:figures/full_fig_p016_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Log fidelity at the infinite-size critical point [PITH_FULL_IMAGE:figures/full_fig_p018_20.png]
Figure 19
Figure 19. Figure 19: The asymptotic x → 0 behavior seems to be exponential but we have not analyzed the form further. KZ scaling deep inside the ordered phase is not ex￾pected in most observables but is unique to the gap￾protected post-critical ground-state probability (and di￾rectly rela…
Figure 22
Figure 22. Figure 22: FIG. 22. Scaling at [PITH_FULL_IMAGE:figures/full_fig_p021_22.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Scatter plots of the stripe defect order parameter der parameter mand the stripe defect order parameter m k, E(11)bd 105ifitild fth [PITH_FULL_IMAGE:figures/full_fig_p022_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Squared sublattice magnetization (a) and Ising en [PITH_FULL_IMAGE:figures/full_fig_p023_25.png]
Figure 27
Figure 27. Figure 27: FIG. 27. The data from Fig. 26 graphed versus [PITH_FULL_IMAGE:figures/full_fig_p024_27.png]

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