arxiv: 2605.04253 · v1 · submitted 2026-05-05 · 💻 cs.ET · quant-ph

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Second-Order FALQON Parameter Transfer for the Max-Cut Problem on 3-Regular Graphs

Gabriel Fernandes Thomaz , Eduarda Rodrigues Monteiro , Jerusa Marchi , Marcelo Zen Pretto , Alisson dos Passos Fumaco , Evandro Chagas Ribeiro da Rosa

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Pith reviewed 2026-05-08 17:04 UTC · model grok-4.3

classification 💻 cs.ET quant-ph

keywords FALQONMax-Cutparameter transferquantum optimization3-regular graphsapproximation ratioNISQtime step

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The pith

Transferring feedback parameters from small to large 3-regular graphs allows second-order FALQON to achieve higher Max-Cut approximation ratios by using larger time steps.

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

This paper examines how parameters for the second-order Feedback-based Algorithm for Quantum Optimization (FALQON) can be transferred across different sizes of Max-Cut problems on 3-regular graphs. It finds that using parameters tuned on small instances for larger graphs produces better approximation ratios than optimizing those parameters directly on the target large graphs. The reason is that small-instance parameters support larger time steps safely, which in turn permits shallower quantum circuits. This matters because it lowers the cost of parameter search and makes FALQON more feasible on current noisy quantum hardware by avoiding excessive circuit depths.

Core claim

The central discovery is that transferring feedback parameters optimized on small instances to larger target graphs yields significantly higher approximation ratios than natively optimizing the parameters directly on the larger graphs, because parameters trained on smaller instances can safely adopt aggressively larger time steps. Numerical experiments on circuits up to 16 layers and graphs up to 24 nodes confirm this advantageous scaling behavior, which simultaneously reduces computational overhead and enhances performance.

What carries the argument

The parameter transfer mechanism in second-order FALQON, which decouples parameter optimization on small graphs from execution on large ones to enable larger time steps.

Load-bearing premise

Parameters optimized on small graphs can be transferred directly to larger graphs and support larger time steps without causing divergence or performance degradation.

What would settle it

An experiment on a 30-node 3-regular graph where transferred parameters produce lower approximation ratios or cause instability compared to direct optimization on the same graph would falsify the claimed advantage.

Figures

Figures reproduced from arXiv: 2605.04253 by Alisson dos Passos Fumaco, Eduarda Rodrigues Monteiro, Evandro Chagas Ribeiro da Rosa, Gabriel Fernandes Thomaz, Jerusa Marchi, Marcelo Zen Pretto.

**Figure 1.** Figure 1: Average optimal time step (∆t) that maximizes the approximation ratio after 16 layers, plotted on a log-log scale as a function of the 3-regular graph size (n). The monotonically decreasing trend exhibits a strict power-law decay with a fitted regression of ∆t ≈ 0.4984 ·n−0.5110, indicating that the time step scales approximately as 1/(2√ n). The overall results were aggregated by averaging the approximat… view at source ↗

**Figure 2.** Figure 2: FALQON parameter transferability matrix. The heatmap displays view at source ↗

**Figure 3.** Figure 3: Approximation ratio of natively optimized parameters (solid red line) versus parameters transferred from smaller instances (dashed lines). The native view at source ↗

read the original abstract

The Feedback-based Algorithm for Quantum Optimization (FALQON) offers a deterministic alternative to variational quantum algorithms by bypassing classical optimization loops. However, maintaining convergence on large problem instances often requires restricting the time step, necessitating quantum circuit depths that exceed Noisy Intermediate-Scale Quantum (NISQ) hardware capabilities. This paper investigates the parameter transferability of second-order FALQON applied to the Max-Cut problem on 3-regular graphs. Through numerical experiments evaluating quantum circuits up to 16 layers on graphs up to 24 nodes, we demonstrate a highly advantageous scaling behavior: transferring feedback parameters optimized on small instances to larger target graphs yields significantly higher approximation ratios than natively optimizing the parameters directly on the larger graphs. This performance advantage arises because parameters trained on smaller instances can safely adopt aggressively larger time steps. By offloading the expensive parameter discovery phase to small-scale instances, this transfer strategy simultaneously reduces computational overhead and enhances the approximation ratio, thereby bringing FALQON closer to practical viability on near-term quantum architectures.

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

Parameter transfer from small to large 3-regular Max-Cut graphs lets second-order FALQON use bigger time steps and reach higher approximation ratios than native optimization on the targets.

read the letter

The main point is that parameters tuned on small instances transfer well enough to let second-order FALQON run with larger time steps on bigger graphs, producing better approximation ratios than optimizing the parameters directly on those larger instances. The experiments reach 24 nodes and 16 layers and show this scaling advantage comes from avoiding the conservative time-step limits that native optimization hits on the target problems. By moving the expensive search to small graphs, the approach cuts overhead while improving the final ratio, which is a practical step for NISQ-scale FALQON use on Max-Cut. The comparison is direct and numerical, so the central claim stays grounded in the reported runs rather than in any circular argument. The work takes an existing transfer idea and tests it specifically on second-order FALQON for this graph class, which is a focused empirical extension rather than a new framework. The results are consistent within the tested range and address a real circuit-depth bottleneck. The abstract gives little on graph generation, exact baselines, error bars, or statistical checks, and 24 nodes remains a modest scale, so it is not yet clear how far the advantage extends or how robust it is to different random graphs. Everything stays empirical with no supporting analysis of why the transferred parameters remain stable or where they would fail. This is the sort of targeted numerical study that people working on feedback-based quantum optimization or NISQ heuristics would want to see. It is solid enough to send for peer review, mainly because the performance comparison is concrete and the practical motivation is clear, even if reviewers will press for more methodological detail and bigger instances.

Referee Report

1 major / 1 minor

Summary. The paper investigates parameter transfer for second-order FALQON on Max-Cut for 3-regular graphs. It claims that feedback parameters optimized on small instances can be transferred to larger target graphs (up to 24 nodes, 16 layers), permitting larger time steps and producing significantly higher approximation ratios than native parameter optimization performed directly on the larger instances. The advantage is demonstrated via numerical experiments and attributed to the ability of transferred parameters to support aggressive time steps without divergence.

Significance. If the empirical performance advantage holds, the transfer strategy provides a practical route to scale FALQON by shifting expensive parameter discovery to small instances, simultaneously lowering computational cost and improving approximation ratios on NISQ-scale hardware. The reported scaling behavior—where small-instance parameters enable larger time steps on bigger graphs—is a concrete, falsifiable empirical finding that directly addresses the circuit-depth limitations of feedback-based quantum optimization.

major comments (1)

Abstract: the central claim that transferred parameters yield 'significantly higher approximation ratios' is presented without any mention of error bars, number of trials, statistical tests, the procedure for generating the 3-regular graphs, or the precise definition and implementation of the 'native optimization' baseline. These omissions are load-bearing because they prevent independent verification of whether the reported advantage is statistically robust or reproducible.

minor comments (1)

The abstract would be clearer if it stated the exact range of graph sizes and layer counts used in the experiments rather than the upper bounds alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. The single major comment is addressed point-by-point below, and we will revise the abstract accordingly to improve clarity and reproducibility.

read point-by-point responses

Referee: [—] Abstract: the central claim that transferred parameters yield 'significantly higher approximation ratios' is presented without any mention of error bars, number of trials, statistical tests, the procedure for generating the 3-regular graphs, or the precise definition and implementation of the 'native optimization' baseline. These omissions are load-bearing because they prevent independent verification of whether the reported advantage is statistically robust or reproducible.

Authors: We agree that the abstract would benefit from additional context to support independent verification. The full experimental details—including graph generation via the configuration model with rejection sampling to enforce 3-regularity, 100 independent trials per instance size, error bars as standard deviation across trials, and the native baseline as direct second-order FALQON optimization on the target graph without parameter transfer—are provided in Sections 3 and 4, along with paired t-tests confirming statistical significance (p < 0.01). To address the referee's concern, we will revise the abstract to concisely reference these elements (e.g., 'across 100 trials on randomly generated 3-regular graphs, with error bars denoting one standard deviation') while preserving the word limit and core claims. This change strengthens the presentation without altering the reported results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical comparison of parameter transfer vs. native optimization

full rationale

The paper's central claim rests on numerical experiments measuring approximation ratios for second-order FALQON on Max-Cut instances (graphs up to 24 nodes, circuits up to 16 layers). Transferred parameters from small instances are shown to permit larger time steps and higher ratios than direct optimization on target graphs. This is a direct empirical performance comparison with no derivation chain, no fitted parameters renamed as predictions, and no load-bearing self-citations or uniqueness theorems invoked to justify the result. The abstract and described experiments treat the transfer advantage as an observed outcome validated by explicit numerical comparison, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5506 in / 923 out tokens · 42504 ms · 2026-05-08T17:04:48.771827+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 3 canonical work pages · 1 internal anchor

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