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arxiv: 2506.06790 · v3 · submitted 2025-06-07 · 🪐 quant-ph · cs.NE

Benchmarking Swarm Optimization Algorithms for Parameter Initialization in the Quantum Approximate Optimization Algorithm

Shashank Sanjay Bhat , Peiyong Wang , Udaya Parampalli This is my paper

Pith reviewed 2026-05-19 10:50 UTC · model grok-4.3

classification 🪐 quant-ph cs.NE
keywords QAOAswarm optimizationparameter optimizationMaxCutvariational algorithmsnoise resiliencecombinatorial optimizationquantum computing
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The pith

Swarm optimization methods achieve lower approximation gaps and more stable convergence than Adam, COBYLA, or SPSA when tuning QAOA parameters for weighted MaxCut.

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

The paper tests swarm-based optimizers including Particle Swarm Optimization, Fully Informed Particle Swarm Optimization, Quantum Particle Swarm Optimization, and an Adam-assisted variant as ways to locate good variational parameters for the Quantum Approximate Optimization Algorithm. These methods are evaluated on weighted MaxCut problems of varying sizes and depths, both in ideal conditions and under simulated noise including shot noise. The central finding is that population-based swarm searches produce smaller approximation gaps and steadier convergence than the conventional single-point optimizers Adam, COBYLA, and SPSA. The advantage persists and even stands out when measurement shots are limited and noise is present. The work therefore presents swarm search as a practical route for navigating the non-convex QAOA parameter landscape on near-term devices.

Core claim

Swarm optimization methods such as PSO, FIPSO, QPSO, and an Adam-assisted FIPSO variant locate QAOA parameters that yield lower approximation gaps and more stable convergence than standard optimizers Adam, COBYLA, and SPSA on weighted MaxCut instances, with the performance edge maintained under noisy and shot-limited regimes.

What carries the argument

Swarm optimization algorithms (PSO, FIPSO, QPSO, and Adam-assisted FIPSO) used as population-based search procedures to explore the QAOA variational parameter space.

If this is right

  • Swarm methods can reduce the number of circuit evaluations needed to reach high-quality QAOA solutions on near-term devices.
  • Population-based search remains effective even when shot noise limits the precision of each cost-function evaluation.
  • Hybrid swarm approaches that incorporate local gradient information can combine global exploration with local refinement.
  • The same initialization strategies may transfer to other variational quantum algorithms that share rugged parameter landscapes.

Where Pith is reading between the lines

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

  • If the robustness under shot noise holds, swarm initialization could become a standard pre-training step before fine-tuning with gradient methods on actual hardware.
  • Testing the same swarm variants on problems with different graph structures would reveal whether the advantage is specific to MaxCut or more general.
  • The observed stability might allow shallower circuit depths to reach comparable solution quality, reducing overall resource requirements.

Load-bearing premise

The performance advantages seen on the tested weighted MaxCut instances and simulated noise models will carry over to other combinatorial problems and to the noise characteristics of actual quantum hardware.

What would settle it

A direct comparison on a different combinatorial problem such as the traveling salesman problem or on real quantum hardware showing that swarm methods no longer produce lower gaps or more stable convergence than Adam, COBYLA, or SPSA.

Figures

Figures reproduced from arXiv: 2506.06790 by Peiyong Wang, Shashank Sanjay Bhat, Udaya Parampalli.

Figure 1
Figure 1. Figure 1: FIGURE 1: The QAOA Circuit mechanism showing the cost [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIGURE 2: QAOA Cost Landscape for a 10 node and 11 node [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIGURE 3: QAOA Circuit for a 4 node graph [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIGURE 4: Approximate ratios for Erdos Renyi Graphs with [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIGURE 5: Approximate ratios for Erdos Renyi Graphs with [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIGURE 6: Approximate ratios for Erdos Renyi Graphs with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIGURE 7: Approximation ratios for Watts–Strogatz graphs [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIGURE 8: Approximation ratios for Watts–Strogatz graphs [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIGURE 9: Approximation ratios for Watts–Strogatz graphs [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIGURE 10: QAOA parameter landscape for graphs with 3 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a prominent variational algorithm for solving combinatorial optimization problems such as the Max Cut problem. A key challenge in QAOA is the efficient identification of variational parameters ({\gamma}, \{beta}) that yield high-quality solutions. In this work, we investigate swarm optimization methods as robust strategies for exploring the QAOA parameter space. We evaluate Particle Swarm Optimization (PSO), Fully Informed Particle Swarm Optimization (FIPSO), Quantum Particle Swarm Optimization (QPSO), and an Adam-assisted FIPSO variant on weighted MaxCut instances across multiple system sizes, circuit depths, and noise regimes, including shot noise. Our results show that these methods achieve lower approximation gaps and more stable convergence compared to standard optimizers such as Adam, COBYLA, and SPSA. In particular, we observe that swarm methods maintain superior performance under noisy and shot limited conditions. These findings suggest that population based search is effective for navigating the complex QAOA landscape and is a promising approach for parameter optimization in near-term quantum algorithms.

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 manuscript benchmarks swarm-based optimizers (PSO, FIPSO, QPSO, Adam-FIPSO) against standard methods (Adam, COBYLA, SPSA) for finding QAOA variational parameters on weighted MaxCut instances. Systematic numerical tests are performed across system sizes, circuit depths p, and noise models that include shot noise; the central claim is that the swarm methods produce lower approximation gaps and more stable convergence, with particular robustness under shot-limited conditions.

Significance. If the reported advantages survive normalization of total circuit evaluations, the work would supply practical guidance for parameter search in variational quantum algorithms on NISQ hardware. The empirical scope—multiple problem sizes, depths, and explicit shot-noise simulations—is a clear strength of the study.

major comments (2)
  1. [§3 and §4] §3 (Benchmarking Protocol) and §4 (Numerical Results): the manuscript does not state whether the total number of QAOA circuit evaluations (or total shots) is equalized across optimizers. Population-based swarm methods evaluate an entire swarm per iteration while single-point methods such as COBYLA or SPSA use far fewer evaluations per step; without explicit normalization or reporting of cumulative function calls, the observed performance edge cannot be unambiguously attributed to algorithmic superiority rather than unequal resource budgets.
  2. [§4, Figures 2–4 and Tables 1–2] §4, Figures 2–4 and Tables 1–2: reported approximation gaps and convergence curves lack error bars, standard deviations across independent runs, or statistical significance tests. Given the stochastic nature of both the optimizers and the shot-noise model, it is difficult to determine whether the claimed superiority is statistically robust.
minor comments (2)
  1. [Title and Introduction] The title emphasizes “Parameter Initialization” while the body describes full optimization of the variational parameters; a brief clarifying sentence in the introduction would remove potential ambiguity.
  2. [§4] Notation for the QAOA cost function and the definition of approximation gap should be restated once in the results section for readers who skip the methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript. These suggestions have prompted us to strengthen the presentation of our benchmarking protocol and the statistical robustness of the results. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [§3 and §4] §3 (Benchmarking Protocol) and §4 (Numerical Results): the manuscript does not state whether the total number of QAOA circuit evaluations (or total shots) is equalized across optimizers. Population-based swarm methods evaluate an entire swarm per iteration while single-point methods such as COBYLA or SPSA use far fewer evaluations per step; without explicit normalization or reporting of cumulative function calls, the observed performance edge cannot be unambiguously attributed to algorithmic superiority rather than unequal resource budgets.

    Authors: We agree that a fair comparison requires explicit accounting for the total number of circuit evaluations. The original manuscript reported results primarily in terms of optimizer iterations without normalizing the cumulative QAOA circuit calls. In the revised version, Section 3 now includes a clear description of the evaluation budget for each method, and we have added a new set of plots in Section 4 that display approximation gap versus total function evaluations (rather than iterations). These normalized curves show that the performance advantage of the swarm methods remains visible even under equalized resource budgets, although the margin narrows for some problem sizes. We have also tabulated the average number of circuit evaluations required to reach a given gap threshold. revision: yes

  2. Referee: [§4, Figures 2–4 and Tables 1–2] §4, Figures 2–4 and Tables 1–2: reported approximation gaps and convergence curves lack error bars, standard deviations across independent runs, or statistical significance tests. Given the stochastic nature of both the optimizers and the shot-noise model, it is difficult to determine whether the claimed superiority is statistically robust.

    Authors: We acknowledge that the absence of variability measures and statistical tests limits the strength of the claims. In the revised manuscript we have recomputed all convergence curves and final-gap tables using 20 independent random seeds per configuration. Error bars (one standard deviation) have been added to Figures 2–4. Tables 1 and 2 now report both mean gaps and standard deviations, together with p-values from Wilcoxon rank-sum tests comparing each swarm optimizer against the classical baselines. The tests confirm that the observed improvements are statistically significant (p < 0.05) for the majority of the tested instances and noise levels. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical benchmarking with no derivation chain

full rationale

This is a direct empirical benchmarking study comparing swarm optimizers (PSO, FIPSO, QPSO, Adam-FIPSO) against Adam, COBYLA, and SPSA on weighted MaxCut QAOA instances under varying noise and shot limits. The abstract and described results consist of performance comparisons (approximation gaps, convergence stability) on fixed problem instances with no mathematical derivations, first-principles predictions, fitted parameters renamed as outputs, or load-bearing self-citations. No equations or uniqueness theorems are invoked that could reduce to inputs by construction. The central claims rest on observable algorithm outputs rather than any self-referential reduction, satisfying the default expectation of a non-circular empirical paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests entirely on numerical benchmarking experiments; no new free parameters are introduced beyond standard swarm algorithm hyperparameters, no domain axioms are invoked beyond routine quantum simulation assumptions, and no invented entities are postulated.

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