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arxiv: 2306.11060 · v1 · submitted 2023-06-19 · 🪐 quant-ph

PCA and t-SNE analysis in the study of QAOA entangled and non-entangled mixing operators

Brian Garc\'ia Sarmina , Guo-Hua Sun , Shi-Hai Dong This is my paper

Pith reviewed 2026-05-24 07:56 UTC · model grok-4.3

classification 🪐 quant-ph
keywords QAOAentanglementmixing operatorsPCAt-SNEmax-cutparameter optimizationquantum algorithms
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The pith

PCA and t-SNE applied to QAOA parameters separate entangled mixing operators from non-entangled ones at depths 2L and 3L.

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

The paper collects final RZ, RX, and RY parameters from QAOA optimizations of max-cut problems using Stochastic Hill Climbing with Random Restarts. It then runs PCA and t-SNE on those parameter sets for circuits of depth 1L, 2L, and 3L, once with and once without an entanglement stage inside the mixing operator. The resulting low-dimensional maps show that entangled models at the higher depths retain more variance under PCA and lower Kullback-Leibler divergence under optimized t-SNE, with visible clustering in some entangled cases. A sympathetic reader would care because the work supplies a concrete, classical way to detect whether adding entanglement inside the mixer changes the structure of the optimized parameter space.

Core claim

Processing the final gate parameters of entangled and non-entangled QAOA models through PCA and t-SNE produces mappings in which entangled versions at 2L and 3L depths preserve a greater fraction of the original information, quantified by higher explained variance, while certain entangled graphs form visible clusters; the separation between the two model classes is also measured by Kullback-Leibler divergence after t-SNE optimization.

What carries the argument

Dimensionality reduction via PCA and t-SNE applied directly to the collection of optimized RZ, RX, RY parameters from entangled versus non-entangled QAOA mixing operators.

If this is right

  • Entangled QAOA models at depths 2L and 3L exhibit higher explained variance in their PCA mappings than non-entangled counterparts.
  • Certain entangled QAOA configurations produce visible clusters in both the PCA and t-SNE visualizations.
  • Kullback-Leibler divergence after t-SNE optimization is lower for the entangled parameter sets, indicating better structure preservation.
  • The numerical and visual differences between entangled and non-entangled models become more pronounced as depth increases from 1L to 3L.

Where Pith is reading between the lines

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

  • The same visualization pipeline could be used to compare other variational quantum algorithms when entanglement is added or removed from their ansatz.
  • If the separation persists across optimizers, the method offers a low-cost classical diagnostic for deciding when entanglement inside the mixer is likely to be beneficial.
  • Observed clusters may correspond to distinct families of max-cut graphs or to different quality levels of the final cuts obtained.

Load-bearing premise

The final parameters reached by Stochastic Hill Climbing with Random Restarts reflect intrinsic properties of the entangled versus non-entangled QAOA models rather than artifacts of that particular optimizer or its initialization procedure.

What would settle it

Repeating the entire parameter collection and subsequent PCA/t-SNE analysis on the same max-cut instances but with a different optimizer such as gradient descent, and finding that the explained-variance gap and the clustering disappear, would falsify the claim that the observed separation is produced by the presence or absence of entanglement.

Figures

Figures reproduced from arXiv: 2306.11060 by Brian Garc\'ia Sarmina, Guo-Hua Sun, Shi-Hai Dong.

Figure 1
Figure 1. Figure 1: Individual rotations in mixing operators. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Entangled rotations in mixing operators. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Levels of depth, with one pair of phase and [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Non-entangled and entangled mixing operators. The dataset of solved max-cut problems used in this study was generated specifically for our forth￾coming research, which includes a comprehensive analysis of the QAOA. For detailed results and information on all the experiments performed, including the optimized parameters, we refer the reader to reference [14]. It is important to note that for the 10n and 15n… view at source ↗
Figure 5
Figure 5. Figure 5: PCA individual graphs for 4n cyclic configuration max-cut problem solved using QAOA, first 3 components. Red corresponds to the 3p parameter 1L non-entangled, blue 3p parameter 1L entangled, green 6p parameter 2L non-entangled and purple 6p parameter 2L entangled model. two graphs (PCA 1 vs PCA 2 and PCA 1 vs PCA 3) exhibit distinct line patterns. These patterns suggest correlations and structure in the da… view at source ↗
Figure 6
Figure 6. Figure 6: PCA pair graphs for 4n cyclic configuration [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: t-SNE individual graphs for 4n cyclic configuration max-cut problem solved using QAOA, with different perplexity values 3, 30 and 99. Red corresponds to the 3p parameter 1L non-entangled, blue 3p parameter 1L entangled, green 6p parameter 2L non￾entangled and purple 6p parameter 2L entangled model. perplexity level, which has a linear pattern similar to the one obtained in the PCA graph for the model and p… view at source ↗
Figure 9
Figure 9. Figure 9: PCA individual graphs for 4n complete configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: PCA pair graphs for 4n complete configuration max-cut problem solved using QAOA, first 3 components. [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: PCA individual graphs for 10n cyclic configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: PCA individual graphs for 10n cyclic configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: PCA pair graphs for 10n cyclic configuration max-cut problem solved using QAOA, first 3 components. [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: PCA individual graphs for 10n complete configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: PCA individual graphs for 10n complete configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: PCA pair graphs for 10n complete configuration max-cut problem solved using QAOA, first 3 components. [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: PCA individual graphs for 15n cyclic configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: PCA individual graphs for 15n cyclic configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: PCA pair graphs for 15n cyclic configuration max-cut problem solved using QAOA, first 3 components. [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: PCA individual graphs for 15n complete configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: PCA individual graphs for 15n complete configuration max-cut problem solved using QAOA, first 3 [PITH_FULL_IMAGE:figures/full_fig_p025_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: PCA pair graphs for 15n complete configuration max-cut problem solved using QAOA, first 3 components. [PITH_FULL_IMAGE:figures/full_fig_p026_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: t-SNE individual graphs for 4n complete configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p027_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: t-SNE pair graphs for 4n pair complete configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p028_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: t-SNE individual graphs for 10n cyclic configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p029_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: t-SNE individual graphs for 10n cyclic configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p030_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: t-SNE pair graphs for 10n cyclic configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p031_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: t-SNE individual graphs for 10n complete configuration max-cut problem solved using QAOA, with [PITH_FULL_IMAGE:figures/full_fig_p032_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: t-SNE individual graphs for 10n complete configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p033_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: t-SNE pair graphs for 10n complete configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p034_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: t-SNE individual graphs for 15n cyclic configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p035_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: t-SNE individual graphs for 15n cyclic configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p036_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: t-SNE pair graphs for 15n cyclic configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p036_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: t-SNE individual graphs for 15n complete configuration max-cut problem solved using QAOA, with [PITH_FULL_IMAGE:figures/full_fig_p037_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: t-SNE individual graphs for 15n complete configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p037_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: t-SNE pair graphs for 15n complete configuration max-cut problem solved using QAOA, with different [PITH_FULL_IMAGE:figures/full_fig_p038_36.png] view at source ↗
read the original abstract

In this paper, we employ PCA and t-SNE analysis to gain deeper insights into the behavior of entangled and non-entangled mixing operators within the Quantum Approximate Optimization Algorithm (QAOA) at varying depths. Our study utilizes a dataset of parameters generated for max-cut problems using the Stochastic Hill Climbing with Random Restarts optimization method in QAOA. Specifically, we examine the $RZ$, $RX$, and $RY$ parameters within QAOA models at depths of $1L$, $2L$, and $3L$, both with and without an entanglement stage inside the mixing operator. The results reveal distinct behaviors when we process the final parameters of each set of experiments with PCA and t-SNE, where in particular, entangled QAOA models with $2L$ and $3L$ present an increase in the amount of information that can be preserved in the mapping. Furthermore, certain entangled QAOA graphs exhibit clustering effects in both PCA and t-SNE. Overall, the mapping results clearly demonstrate a discernible difference between entangled and non-entangled models, quantified numerically through explained variance in PCA and Kullback-Leibler divergence (after optimization) in t-SNE, where some of these differences are also visually evident in the mapping data produced by both methods.

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 / 3 minor

Summary. The manuscript applies PCA and t-SNE to the final RZ/RX/RY parameters obtained from QAOA circuits (depths 1L, 2L, 3L) with and without an entanglement stage in the mixing operator, optimized via Stochastic Hill Climbing with Random Restarts on max-cut instances. It reports that the resulting low-dimensional mappings exhibit discernible differences, with entangled models at 2L and 3L showing higher explained variance in PCA, clustering effects, and differences in post-optimization KL divergence under t-SNE.

Significance. If the reported differences can be shown to arise from the intrinsic structure of the mixing operators rather than from optimizer-specific convergence behavior, the work would provide concrete empirical evidence that entanglement alters the geometry of optimized QAOA parameter sets, which could inform variational ansatz design. The current analysis supplies no such controls.

major comments (3)
  1. [experimental setup / results] The experimental dataset is never quantified: neither the number of max-cut instances nor the total number of optimized parameter vectors fed to PCA/t-SNE is stated, so the reliability of the reported explained-variance differences and clustering cannot be assessed.
  2. [results / discussion] No comparison of achieved approximation ratios (or final objective values) is given between the entangled and non-entangled families. Because the entangled circuits are higher-dimensional, differential convergence of SHC+RR could produce the observed parameter-structure differences even if both families reach comparable solution quality; this confound is load-bearing for the central claim.
  3. [t-SNE subsection] The t-SNE analysis reports KL divergence after optimization but supplies neither the perplexity values used, nor any sensitivity check to hyper-parameters or random seed, nor a statistical test that the KL differences between entangled and non-entangled cases are significant.
minor comments (3)
  1. [methods] The depth notation “1L, 2L, 3L” is introduced without explicit definition; clarify whether it denotes number of layers or something else.
  2. [figures] Figures displaying the PCA and t-SNE embeddings would benefit from consistent color or marker legends that explicitly label entangled versus non-entangled runs.
  3. [introduction] A brief reference to prior QAOA literature on entangling versus non-entangling mixers would help situate the empirical observations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important aspects of clarity, controls, and statistical rigor in our analysis. We address each major comment below and commit to revisions that will strengthen the manuscript without altering its core findings.

read point-by-point responses

  1. Referee: The experimental dataset is never quantified: neither the number of max-cut instances nor the total number of optimized parameter vectors fed to PCA/t-SNE is stated, so the reliability of the reported explained-variance differences and clustering cannot be assessed.

    Authors: We agree that this information is missing from the manuscript. In the revised version we will explicitly report the number of max-cut instances and the total number of optimized parameter vectors supplied to PCA and t-SNE, thereby allowing readers to assess the statistical reliability of the reported differences. revision: yes

  2. Referee: No comparison of achieved approximation ratios (or final objective values) is given between the entangled and non-entangled families. Because the entangled circuits are higher-dimensional, differential convergence of SHC+RR could produce the observed parameter-structure differences even if both families reach comparable solution quality; this confound is load-bearing for the central claim.

    Authors: The referee correctly identifies a potential confound. We will add a direct comparison of final objective values and approximation ratios between the two families in the revised manuscript. This addition will help readers evaluate whether the observed parameter-space differences are accompanied by comparable solution quality. revision: yes

  3. Referee: The t-SNE analysis reports KL divergence after optimization but supplies neither the perplexity values used, nor any sensitivity check to hyper-parameters or random seed, nor a statistical test that the KL differences between entangled and non-entangled cases are significant.

    Authors: We acknowledge these omissions. The revised t-SNE subsection will report the perplexity values, include sensitivity checks across hyper-parameters and random seeds, and add a statistical test for the significance of the KL-divergence differences. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical analysis of optimizer outputs

full rationale

The paper applies standard PCA and t-SNE dimensionality reduction to parameter vectors (RZ/RX/RY) obtained from Stochastic Hill Climbing runs on QAOA instances. Explained variance and post-optimization KL divergence are direct algorithmic outputs of these methods and do not reduce to any fitted quantity defined inside the paper. No derivation chain, first-principles prediction, or uniqueness claim is asserted; the work contains no equations that equate a result to its own inputs by construction and makes no load-bearing use of self-citations. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the Stochastic Hill Climbing optimizer produces parameter distributions whose structure is meaningfully captured by linear and nonlinear projections; no free parameters are introduced beyond the standard choices of PCA components and t-SNE perplexity.

axioms (1)
  • domain assumption The final parameters obtained after Stochastic Hill Climbing with Random Restarts faithfully represent the optimization landscape of the QAOA model.
    All analysis is performed on the output of this optimizer; if the optimizer systematically misses important regions, the observed separation may be an artifact.

pith-pipeline@v0.9.0 · 5765 in / 1368 out tokens · 35661 ms · 2026-05-24T07:56:30.753669+00:00 · methodology

discussion (0)

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Reference graph

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