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REVIEW 3 major objections 4 minor 51 references

Hamiltonian formulations of centroid-based clustering

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Centroid-based clustering can be written as a spin-Hamiltonian ground-state search, and the combined intracluster/intercluster objective turns out to be equivalent to maximizing the size-weighted squared centroid distance.

desk verdict Clean QUBO trick and a neat combined objective, but the reported Intra* vs weighted MaxCut results contradict their proved equivalence, so the experiments need correction before the empirical claims can be trusted. read the letter →

arxiv 2502.06542 v1 pith:D4BOGEL3 submitted 2025-02-10 quant-ph cs.ET

Myeonghwan Seong , Daniel K. Park This is my paper · ORCID
classification quant-phcs.ET
keywords Hamiltonianclusteringcentroid-basedQUBOquantumannealingk-meansintraclusterdistanceinterclusterconstrained
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 tries to show that centroid-based clustering—grouping points around their means—can be reformulated as finding the ground state of a spin Hamiltonian, so that quantum computers and annealers can solve it. The central move is to treat the number of points in each cluster, which normally appears as an awkward denominator in centroid formulas, as variables inside the objective and then multiply them away, converting rational objective functions into quadratic binary Hamiltonians. The strongest result is for the objective that combines intracluster tightness with intercluster separation: the paper proves this combined objective is equivalent to maximizing the cluster-size-weighted squared distance between the two centroids, and the resulting pairwise Hamiltonian ran on D-Wave Advantage with zero chain breaks, achieving Rand Index values at or above k-means on Gaussian, Iris, Wine, and 0-1 MNIST data. The wider claim is that this gives a flexible unified framework for clustering—combining multiple objectives and constraints such as must-link/cannot-link and cardinality—that is compatible with both gate-based quantum algorithms and quantum annealing.

What carries the argument

The central object is the size-weighted centroid-separation identity, Eq. (12) = Eq. (13): the combined intracluster/intercluster objective equals $-N N_+^2 N_-^2 \|\mu_+-\mu_-\|_2^2$. The machinery that carries the argument is the substitution of the cluster-size variables $N_\pm = \sum_i (1\pm z_i)/2$ and centroid formulas into the distance function $l(\mu,z,s)$, followed by multiplication by $N_+^2 N_-^2$ to clear denominators. For the combined objective, every denominator cancels and all higher-order terms telescope, leaving only quadratic terms $Z_i Z_j$; this is the property that makes the Hamiltonian embeddable on the D-Wave clique sampler, a complete graph of up to 175 logical qubits, without slack variables. The same machinery produces the intra-only quadratic variant and the inter-only variant, which contains higher-order terms and requires slack variables, making it less hardware-friendly.

What would settle it

Take a data set of two compact, well-separated Gaussian clusters plus one distant outlier, and compute the exact ground state of the pairwise Hamiltonian in Eq. (A27) by brute force. If the minimizing assignment puts the outlier alone in one cluster, a near-singleton, while standard k-means assigns it to the nearest cluster, then the claimed equivalence between centroid-based clustering and the size-weighted centroid-separation objective fails on skewed data.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the combined intracluster/intercluster objective is not just a compromise but collapses exactly: minimizing Eq. (12) is equivalent to minimizing $-N N_+^2 N_-^2 \|\mu_+ - \mu_-\|_2^2$, i.e., maximizing the squared distance between the two cluster centroids weighted by the sizes of both clusters. Because the denominators $N_\pm$ are canceled by multiplying the combined objective by $N_+^2 N_-^2$, the resulting expression is quadratic in the binary cluster-assignment variables and can be written as a pairwise spin Hamiltonian, Eq. (A27), with no higher-order terms. This pairwise form is what makes the combined objective embeddable on D-Wave Advantage's Pegasus topology with zero chain breaks, while the other tested Hamiltonians—intra, intra*, and weighted max-cut—suffered chain breaks that destroyed solution quality. Empirically, the combined Hamiltonian produced Rand Index values higher than or equal to k-means on all four benchmark datasets while keeping comparable Silhouette Scores.

Load-bearing premise

The whole approach rests on the assumption that maximizing the cluster-size-weighted squared distance between the two centroids is a sensible clustering objective on real data, and that the algebraic clearing of denominators in Appendix A.3 preserves exactly which assignments are optimal; the paper tests this on only four small benchmark sets and does not analyze pathological cases such as singleton clusters.

Editorial extensions

If this is right

  • The combined objective's ground state is exactly a maximum of the size-weighted squared centroid distance, so tuning this objective means tuning a single interpretable quantity rather than two competing distance sums.
  • Because the combined Hamiltonian is purely quadratic, it can be targeted not only by quantum annealing but also by QAOA, VQE, or any quantum ground-state technique that handles Ising models, without extra slack qubits.
  • Constraints such as labeling, cardinality, and must-link/cannot-link can be appended as linear or quadratic penalty terms to the same objective, letting users enforce domain knowledge within a single QUBO.
  • The hierarchical k-clustering algorithm extends the binary Hamiltonian to $k > 2$ clusters by repeated binary splits, avoiding one-hot encodings and exclusivity penalties.
  • The intercluster-only objective requires higher-order terms and slack variables, so among the proposed Hamiltonians the combined form is the one best matched to current annealer hardware.

Reading between the lines

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

  • Not tested in the paper: on skewed data with a distant outlier, the size-weighted centroid-separation objective may favor isolating the outlier as a singleton cluster, because that maximizes $\|\mu_+-\mu_-\|^2$; this would be a failure mode for imbalanced real-world data.
  • The zero-chain-break finding may reflect the combined Hamiltonian's simple quadratic connectivity rather than a special property of clustering objectives; comparing it against a random quadratic Hamiltonian on the same graph would separate hardware embedding effects from clustering content.
  • The constrained-clustering curves show an initial drop in Rand Index when 10–30 percent of labels are revealed; a systematic sweep of the penalty weight $\lambda_p$ could determine whether that drop is an artifact of penalty calibration rather than a property of the objective.
  • A natural next step, consistent with the paper's own future-work list, is to use the centroid-separation value itself as a stopping criterion in the hierarchical splitting algorithm, thereby choosing $k$ from data instead of fixing it in advance.
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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 manuscript proposes a unified QUBO/Ising Hamiltonian framework for binary centroid-based clustering. It introduces distance functions for intracluster and intercluster objectives, constructs optimization problems for intra-only, inter-only, and combined cases, and derives explicit Pauli--Z Hamiltonians in Appendix A. The combined objective is shown to reduce to maximizing a weighted squared distance between cluster centroids. The authors also add penalty-term constructions for must-link, cannot-link, labeling, and cardinality constraints, and present a recursive algorithm for k-clustering. Experiments use exact enumeration, simulated annealing, and the D-Wave Advantage 6.4 quantum annealer on Iris, Wine, a synthetic overlapping Gaussian dataset, and a 0/1 MNIST subset, with claims of zero chain breaks for the intra-inter combined Hamiltonian.

Significance. The algebraic part of this work is useful: Appendix A provides explicit reductions of centroid-based objectives to Ising Hamiltonians, including the nontrivial simplification of the combined intra-inter objective to a pairwise Hamiltonian, and the derivation appears internally consistent. The constrained-clustering extensions are also valuable because they show how practical requirements can be folded into the same Hamiltonian framework. If the experiments were internally consistent, the zero-chain-break observation on a current quantum annealer would be a genuinely interesting data point for the quantum-annealing clustering community. However, the reported experimental tables contain a direct contradiction with the paper's own algebra, and this undermines the reliability of the empirical pipeline, including the D-Wave results, until it is resolved.

major comments (3)
  1. [Section III and Appendix A.1, vs. Tables II and III] Equation (A16) and the resulting Hamiltonian (A17) show that the objective labeled Intra* is exactly (1/2) sum_{i<j} ||x_i - x_j||^2 z_i z_j. This is a positive scalar multiple of the weighted MaxCut objective defined in Section III with a_ij = ||x_i - x_j||^2. Positive scaling preserves the full set of energy orderings and minimizers, so any optimizer of one objective is an optimizer of the other. Yet Table II reports exact-search Rand Index 0.858 for Intra* versus 0.847 for weighted MaxCut on Iris, with Silhouette Scores 0.422 versus 0.418, and Table III reports simulated-annealing Rand Index 0.939 versus 0.922. These differences are impossible as reported unless the two methods were evaluated on different random subsamples, implemented different objective functions, or used undocumented tie-breaking among degenerate ground states. The authors must reconcile this discrepancy, because the same experimental pipeline produced the central D-Wave claims.
  2. [Section V and Figures 2-3] The conclusion that 'at least one of our proposed Hamiltonians outperforms the weighted MaxCut' is not supported for Intra*, since Intra* is merely a positively scaled version of weighted MaxCut. Any apparent advantage of Intra* over weighted MaxCut in Figures 2 and 3 must therefore be an artifact of tie-breaking, sampling, or implementation details rather than evidence that centroid information improves clustering. The authors should remove or reinterpret this comparison after correcting the experimental pipeline.
  3. [Section IV.C, Table IV, and Figure 5] The central quantum-annealing claim is that the Intra-Inter combined Hamiltonian achieves zero chain breaks while Intra, Intra*, and weighted MaxCut produce near-random solutions on the same complete-graph problems. This is surprising because Intra* and weighted MaxCut are mathematically identical problems, and the combined Hamiltonian is also a pairwise Hamiltonian on the same complete graph. The paper does not report the chain strengths used, the embedding parameters returned by the clique sampler, the post-processing rule for broken chains, or the precise sampling protocol. Without these details, the qualitative difference in chain-break behavior cannot be reproduced or assessed; please provide these parameters and, ideally, the raw solver configuration for each Hamiltonian.
minor comments (4)
  1. [Appendix A.3, Eq. (A23)] Equation (A23) contains a typographical artifact in the factor written as '2N+1N-1'; this should read '2 N_+ N_-'.
  2. [Appendix B, Table IV] Several rows in Table IV report zero standard deviation for the Intra-Inter combined method; please state explicitly whether these are single-run values or averages over multiple runs, and how many quantum annealing repetitions were used for each entry.
  3. [Section IV.D] The constrained-clustering experiments depend on the penalty hyperparameter lambda_p, but the paper does not report the values used for Figures 6 and 7. Since the Inter method's variability is attributed to sensitivity to lambda_p, reporting the chosen values is necessary for reproducibility.
  4. [General] No code or data repository is provided. Given the role of random subsampling, tie-breaking, and hyperparameters in the reported results, a public artifact would materially strengthen the paper's reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Hamiltonian constructions are derived algebraically from explicit clustering objectives and evaluated on external benchmarks; the reported Intra* versus weighted-MaxCut inconsistency is a correctness concern, not a circular derivation.

full rationale

The derivation chain is self-contained. The paper defines the distance function l(mu,z,s) in Eq. (8), constructs intracluster, intercluster, and combined objectives in Eqs. (9)-(12), and then shows in Appendix A that after multiplying by the appropriate powers of N+ and N-, the combined objective reduces algebraically to the weighted centroid-separation expression in Eq. (13). This reduction is a direct expansion of the definitions and is not an assumed conclusion. The Hamiltonians in Eqs. (A12), (A17), and (A27) are obtained by the standard replacement of binary variables with Pauli Z operators, again an explicit mapping rather than a fitted or self-referential step. The performance claims are tested on Iris, Wine, a subset of MNIST, and a synthetic Gaussian dataset, with comparisons to k-means and weighted MaxCut; no parameter of the central unconstrained objectives is fitted to the reported Rand Index or Silhouette Score. There are no load-bearing self-citations and no imported uniqueness theorem; Ref. [35] is only used for a hierarchical k-clustering extension, and Ref. [38] supplies a standard pairwise penalty form. The one notable issue is an internal experimental inconsistency, not circularity: Appendix A.1 shows Intra* (Eq. (A16)) equals 1/2 times the weighted MaxCut objective (Eq. (4)), so exact optimizers of the two problems must return identical assignments, yet Tables II and III report different Rand Indices for them. That inconsistency concerns the experimental pipeline and implementation fidelity, not the logical derivation of the Hamiltonians from their stated objectives. The central result is therefore not circular; the numerical discrepancy should be treated as a reproducibility/correctness risk rather than a circularity finding.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its free parameters are limited to the constraint penalty strength lambda_p and design choices such as the scaling power. The main axioms are the standard QUBO mapping and the domain assumption that centroid-distance objectives are meaningful clustering criteria.

free parameters (1)
  • lambda_p = unspecified
    Hyperparameter for constraint penalties in Sec. IIID; chosen by user via grid or random search. It is not fitted to the test data and does not enter the central Hamiltonian derivations.
assumptions (4)
  • standard math QUBO/Ising mapping of combinatorial optimization problems
    Standard result (Lucas 2014), invoked in Eqs. (1) to (3).
  • domain assumption Squared Euclidean distance is an appropriate similarity measure for clustering
    Underlies all objective functions in Sec. III and the derived Hamiltonians.
  • domain assumption The ground state of the constructed Hamiltonian corresponds to the desired clustering
    The entire approach relies on this equivalence; it is tested empirically but not proven.
  • domain assumption D-Wave clique sampler can embed complete graphs up to size 175 with no chain breaks for the combined Hamiltonian
    Empirical claim in Sec. IVC; not guaranteed by the hardware and not independently verified.

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Cite this review

Pith. "Pith review of Hamiltonian formulations of centroid-based clustering." pith.science (2026). https://pith.science/paper/D4BOGEL3

@misc{pith2026250206542,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian formulations of centroid-based clustering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4BOGEL3}},
  note         = {Machine review of arXiv:2502.06542}
}
abstract

Clustering is a fundamental task in data science that aims to group data based on their similarities. However, defining similarity is often ambiguous, making it challenging to determine the most appropriate objective function for a given dataset. Traditional clustering methods, such as the $k$-means algorithm and weighted maximum $k$-cut, focus on specific objectives -- typically relying on average or pairwise characteristics of the data -- leading to performance that is highly data-dependent. Moreover, incorporating practical constraints into clustering objectives is not straightforward, and these problems are known to be NP-hard. In this study, we formulate the clustering problem as a search for the ground state of a Hamiltonian, providing greater flexibility in defining clustering objectives and incorporating constraints. This approach enables the application of various quantum simulation techniques, including both circuit-based quantum computation and quantum annealing, thereby opening a path toward quantum advantage in solving clustering problems. We propose various Hamiltonians to accommodate different clustering objectives, including the ability to combine multiple objectives and incorporate constraints. We evaluate the clustering performance through numerical simulations and implementations on the D-Wave quantum annealer. The results demonstrate the broad applicability of our approach to a variety of clustering problems on current quantum devices. Furthermore, we find that Hamiltonians designed for specific clustering objectives and constraints impose different requirements for qubit connectivity, indicating that certain clustering tasks are better suited to specific quantum hardware. Our experimental results highlight this by identifying the Hamiltonian that optimally utilizes the physical qubits available in the D-Wave System.

Figures

Figures reproduced from arXiv: 2502.06542 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of the intracluster distance and the intercluster [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The heatmap illustrates exact search results, showing the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The heatmap illustrates simulated annealing results, showing [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) A visualization of a complete graph with 5 vertices, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Chain break fraction for four different Hamiltonians using the D-Wave Advantage System 6.4. We conducted 200 samplings on a [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The plots display the Rand Index as a function of the percentage of known labels for the Iris (left) and Wine (right) datasets, under [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The top plots show the Rand Index across various methods with cardinality constraints applied to the Iris (left) and Wine (right) datasets. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Figure illustrates a 2D Gaussian-distributed synthetic dataset with overlapping clusters, showing Cluster 1 in blue and Cluster 2 in red. [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

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    (A1) and (A2): 𝑑∑︁ 𝑗=1  𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 2− 𝑁+𝑁2 − 𝑁2 +𝑁2− 𝑁∑︁ 𝑖 𝑥𝑖 𝑗 1+𝑧𝑖 2 !2 − 𝑁2 +𝑁− 𝑁2 +𝑁2− 𝑁∑︁ 𝑖 𝑥𝑖 𝑗 1−𝑧𝑖 2 !2

    Intracluster distance To capture the intracluster distance, we combine the following equations Eqs. (A1) and (A2): 𝑑∑︁ 𝑗=1  𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 2− 𝑁+𝑁2 − 𝑁2 +𝑁2− 𝑁∑︁ 𝑖 𝑥𝑖 𝑗 1+𝑧𝑖 2 !2 − 𝑁2 +𝑁− 𝑁2 +𝑁2− 𝑁∑︁ 𝑖 𝑥𝑖 𝑗 1−𝑧𝑖 2 !2 . (A5) Multiplying by𝑁2 +𝑁2 − to eliminate the deno...

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    (A3) and Eq

    Intercluster distance To capture the intercluster distance, we combine Eq. (A3) and Eq. (A4). After simplifying, we get: 𝑑∑︁ 𝑗=1  − 𝑁∑︁ 𝑖=1 𝑥𝑖2 𝑗+ 2𝑁 𝑁+𝑁− 𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 1+𝑧𝑖 2 𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 1−𝑧𝑖 2 − 𝑁+ 𝑁2− 𝑁∑︁ 𝑖=1 1−𝑧𝑖 2 !2 − 𝑁− 𝑁2 + 𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 1+𝑧𝑖 2 !2 . (A18) Mu...

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    Combining intra and intercluster distances To capture the intra and intercluster distances, we combine the following equations Eqs. (A1) + (A2) + (A3) + (A4): 𝑑∑︁ 𝑗=1 − 1 𝑁+ 𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 1+𝑧𝑖 2 !2 − 1 𝑁− 𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 1−𝑧𝑖 2 !2 + 2𝑁 𝑁+𝑁− 𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 1+𝑧𝑖 2 ! 𝑁∑︁ 𝑖=1 𝑥𝑖 𝑗 1−𝑧𝑖 ...

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Reviewed August 8, 2026 · model on record in the stance chip above.