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arxiv: 2605.07144 · v1 · submitted 2026-05-08 · 🪐 quant-ph

Box model of quantum annealing

Yang Wei Koh , Youjin Deng This is my paper

Pith reviewed 2026-05-11 01:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum annealingparticle-in-a-box modelresidual energydiabatic transitionsflat gapsenergy gap spectrumwave function trappingcontinuous-space annealing
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The pith

A particle-in-a-box model shows residual energy in quantum annealing depends mainly on speed, not roughness or depth.

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

The authors construct a one-dimensional particle-in-a-box model with three energy landscapes formed by modulating a sinusoidal wave with concave, convex, or flat envelopes. They solve the time-dependent Schrödinger equation to track the wave function as the potential is annealed from an initial broad state toward a final rugged landscape containing multiple local minima. The central result is that plots of residual energy versus annealing speed collapse onto nearly identical curves regardless of landscape roughness or annealing depth. This points to diabatic transitions as the dominant process and introduces flat gaps in the instantaneous energy spectrum as the feature that keeps the wave function trapped in local minima.

Core claim

In the particle-in-a-box model, three energy landscapes are built by modulating a sinusoidal wave with concave, convex, or flat envelopes. Direct numerical integration of the Schrödinger equation reveals that residual energy as a function of annealing speed is largely independent of landscape roughness and annealing depth. Flat gaps appear in the energy gap spectrum as intervals where the gap size stays nearly constant; the authors propose that these gaps allow the wave function to remain localized in a local minimum during rapid, diabatic passages through avoided crossings, explaining the observed trapping.

What carries the argument

The particle-in-a-box model with sinusoidal waves modulated by concave, convex, or flat envelopes, which produces flat gaps in the instantaneous energy gap spectrum that enable wave-function trapping during diabatic transitions.

If this is right

  • Residual energy versus annealing speed collapses across different roughness levels and annealing depths.
  • Diabatic transitions dominate the dynamics over a wide range of annealing speeds.
  • Flat gaps in the energy spectrum provide a concrete mechanism for trapping in local minima.
  • Transition probabilities deviate from simple Landau-Zener predictions in multi-minima landscapes.

Where Pith is reading between the lines

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

  • The speed-only dependence suggests that annealing schedules could be optimized without detailed knowledge of potential roughness.
  • If flat gaps persist in more realistic or higher-dimensional potentials, they may offer a general route to predict trapping without full many-body calculations.
  • The discrepancy with Landau-Zener indicates that multi-minima continuous models require broader transition formulas.

Load-bearing premise

The particle-in-a-box model with sinusoidal waves modulated by concave, convex, or flat envelopes sufficiently represents the key dynamics of continuous-space quantum annealing, including the prevalence of diabatic transitions.

What would settle it

An experimental or higher-dimensional numerical realization of continuous-space quantum annealing in which residual energy varies strongly with landscape roughness or annealing depth would contradict the reported independence.

Figures

Figures reproduced from arXiv: 2605.07144 by Yang Wei Koh, Youjin Deng.

Figure 1
Figure 1. Figure 1: FIG. 1. The three types of potential surfaces that can be [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Energy levels [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the R(T) curves for µ from 4 to 24. The parameters are Eref. = E0(sf ), and si = 1, sf = 104 for all µ’s. The choices of si and sf were made based on [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the four lowest energy levels for the case of µ = 12, where the potential surface has seven minima (including the two at the walls). The dashed lines show the graphs of E0 and E1 as a function of s, while the solid lines show those of E2 and E3. As s increases, E0,1 merge into a doubly-degenerate ground state at log s ≈ 2, and E2,3 merge to become the first-excited state at log s ≈ 2.5. At log s ≈ 4.… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Energy levels [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the R(T) curves for the case of µ = 8. In both panels, the annealing parameter starts at si = 1. In (a) the annealing ends at sf = 104.5 , while in (b) it ends at sf = 107 . For the rest of the paper, we shall refer to a small (large) sf as shallow (deep) annealing. In panel (a), the residual energy shows a crossover from exponential to polynomial decay, as seen by comparing with the dashed curve (bl… view at source ↗
Figure 8
Figure 8. Figure 8: (b) shows the R(T) curves in panel (a) replot￾ted as R(v). The curves for different sf now collapse onto a single curve. The residual energy as a function of annealing speed R(v) does not seem to depend on the annealing depth very much. Hence, in [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Effects of increasing [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Energy gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Residual energy of the convex system [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. (a) Dependence of [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Effects of increasing [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Graphs of variational energy [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
read the original abstract

A particle-in-a-box model of continuous space quantum annealing is proposed and studied numerically by solving the Schr\"odinger wave equation directly. Three types of energy landscapes with multiple local minima are considered, namely a sinusoidal wave modulated by a concave, a convex, or a flat envelope. Both static (energy spectrum) and dynamical (residual energy) behaviors are analyzed in detail, paying particular attention to the effects of landscape roughness and annealing depth. Simulation results show that the residual energy as a function of annealing speed is largely independent of these two factors. The prevalence of diabatic transitions during annealing is observed, and the discrepancy between our numerical results and the Landau-Zener formula is discussed. An interesting feature in the energy gap spectrum, which we call flat gaps, is examined. Based on it, we propose a mechanism to explain the trapping of wave function in local minima during diabatic transitions, widely observed in our data.

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

3 major / 2 minor

Summary. The manuscript proposes a particle-in-a-box model for continuous-space quantum annealing and solves the time-dependent Schrödinger equation numerically for three modulated sinusoidal potentials (concave, convex, and flat envelopes). It reports that residual energy versus annealing speed is largely independent of landscape roughness and depth, observes prevalent diabatic transitions, notes a discrepancy with the Landau-Zener formula, identifies 'flat gaps' in the instantaneous energy spectrum, and proposes a mechanism whereby these flat gaps trap the wave function in local minima during diabatic transitions.

Significance. If the independence result and flat-gap mechanism are robustly established, the work supplies a minimal, exactly solvable model that isolates the role of spectral features in diabatic trapping, a phenomenon widely seen in quantum annealing. The direct TDSE integration and the identification of flat gaps constitute concrete, falsifiable contributions that could guide both theory and experiment on continuous-space QA.

major comments (3)
  1. [§4] §4 (numerical results): the claim that residual energy is 'largely independent' of roughness and depth is presented without reported parameter ranges, grid convergence tests, or error bars on the TDSE integrations; without these, the robustness of the independence cannot be assessed.
  2. [§5.2] §5.2 (flat gaps): the proposed mechanism linking flat gaps to wave-function trapping in local minima during diabatic transitions is supported only by visual inspection of spectra and sample trajectories; no quantitative correlation is shown between the times at which flat gaps appear and the times at which probability density localizes in a given minimum across the ensemble.
  3. [§5.3] §5.3 (Landau-Zener comparison): the reported discrepancy with the Landau-Zener formula is discussed qualitatively but without a multi-level avoided-crossing analysis or controlled numerical test that isolates the contribution of flat gaps versus other spectral features.
minor comments (2)
  1. [§2] The definition of the three envelope functions (concave, convex, flat) should be given explicitly with the functional forms and the range of the modulation parameter.
  2. [§3] Figure captions for the energy-gap plots should indicate the annealing schedule parameter s at which each snapshot is taken.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions planned to improve the robustness and clarity of the results.

read point-by-point responses

  1. Referee: [§4] §4 (numerical results): the claim that residual energy is 'largely independent' of roughness and depth is presented without reported parameter ranges, grid convergence tests, or error bars on the TDSE integrations; without these, the robustness of the independence cannot be assessed.

    Authors: We agree that these supporting details are required to substantiate the independence claim. In the revised manuscript we will explicitly state the ranges of roughness parameters and annealing depths employed, present grid-convergence tests demonstrating stabilization of the residual energy for sufficiently fine spatial and temporal discretizations, and include error bars obtained from the integrator tolerance and ensemble statistics. revision: yes

  2. Referee: [§5.2] §5.2 (flat gaps): the proposed mechanism linking flat gaps to wave-function trapping in local minima during diabatic transitions is supported only by visual inspection of spectra and sample trajectories; no quantitative correlation is shown between the times at which flat gaps appear and the times at which probability density localizes in a given minimum across the ensemble.

    Authors: The mechanism is currently illustrated by representative spectra and trajectories. We will add a quantitative correlation analysis that records the instants of flat-gap appearance in the instantaneous spectrum and measures their temporal alignment with localization events in the probability density, averaged over the simulation ensemble. These statistics will be reported in the revised text and figures. revision: yes

  3. Referee: [§5.3] §5.3 (Landau-Zener comparison): the reported discrepancy with the Landau-Zener formula is discussed qualitatively but without a multi-level avoided-crossing analysis or controlled numerical test that isolates the contribution of flat gaps versus other spectral features.

    Authors: We will strengthen the comparison by including a controlled numerical test that isolates the role of flat gaps, for example by contrasting the dynamics obtained with the original potentials against those obtained with auxiliary potentials in which the flat-gap regions have been regularized. A brief multi-level avoided-crossing analysis around representative flat gaps will also be added to clarify the origin of the observed deviation from the two-level Landau-Zener prediction. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical integration of TDSE

full rationale

The paper defines a particle-in-a-box model with three envelope-modulated sinusoidal landscapes and obtains all reported results (residual energy vs. annealing speed, prevalence of diabatic transitions, flat gaps, and the proposed trapping mechanism) by direct numerical solution of the time-dependent Schrödinger equation. No parameters are fitted to subsets of the output data and then re-used as predictions; the flat-gap interpretation is offered as a post-hoc explanation of observed wavefunction localization rather than a self-definitional or self-citation-dependent step. External comparison to the Landau-Zener formula is cited but does not carry the central claims. The derivation chain is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The analysis relies on standard quantum mechanical axioms and the validity of the numerical solver for the proposed model. No explicit free parameters are mentioned, but the choice of envelope functions represents modeling assumptions.

axioms (1)
  • standard math The dynamics of the quantum system are governed by the time-dependent Schrödinger equation.
    Directly solved numerically to obtain static and dynamical behaviors.
invented entities (1)
  • Box model with modulated sinusoidal energy landscapes no independent evidence
    purpose: To simulate continuous space quantum annealing with controllable roughness and depth.
    Newly proposed model in this preprint; no independent experimental or theoretical validation provided in the abstract.

pith-pipeline@v0.9.0 · 5442 in / 1330 out tokens · 70111 ms · 2026-05-11T01:16:30.753797+00:00 · methodology

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Reference graph

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