Optimizing Quantum Embedding using Genetic Algorithm for QML Applications

Archisman Ghosh; Koustubh Phalak; Swaroop Ghosh

REVIEW 5 major objections 6 minor 15 references

Optimizing Quantum Embedding using Genetic Algorithm for QML Applications

T0 review · 5 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Genetic algorithm finds better quantum embeddings than random choice

desk verdict Solid empirical observation that embedding permutation matters, but the GA-vs-random claim needs repeated runs and error bars before it holds. read the letter →

arxiv 2412.00286 v1 pith:XIGAWVSE submitted 2024-11-29 quant-ph

Koustubh Phalak , Archisman Ghosh , Swaroop Ghosh This is my paper · ORCID

classification quant-ph

keywords quantumembeddingsgeneticalgorithmmachinelearningfeature-to-qubitmappingangleembeddingvariationalclassifierNISQdevicespermutationsearch

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

Quantum machine learning performance depends on which classical features are mapped to which qubits in the embedding circuit, not just on which embedding type is used. The authors frame the selection of a feature-to-qubit mapping as a discrete search problem and propose a genetic algorithm to solve it, instead of the usual continuous optimization of trainable embeddings. On binary classification tasks from MNIST and Tiny ImageNet, the GA finds embeddings with higher combined training and inference accuracy than randomly chosen mappings, by margins of 0.33–3.33 and 0.5–3.36 fitness points, while reducing runtime by up to 15% and 8.8%, respectively. The paper also reports that the GA scales with qubit count and outperforms three existing embedding methods, quantum embedding kernels, QAOA-based embedding, and trainable QRACs, by 1.003X, 1.03X, and 1.06X.

What carries the argument

The machinery is the angle-embedding circuit that pairs two features per qubit, one encoded in an RX gate and one in an RY gate, making a feature-to-qubit mapping a permutation of feature indices. The fitness score is the mean of training and inference accuracy after training the QNN for a fixed number of epochs. The genetic algorithm explores the permutation space with a population of 20, 5 generations, tournament selection of size 2, a crossover that takes a prefix from one parent and fills the remaining positions in the order they appear in the other parent, and a low-probability swap mutation (rate 0.001). The search is run under a fixed training configuration, and the best-scoring permutation is the proposed optimal embedding.

What would settle it

For a fixed binary-class pair such as MNIST classes 0 and 1 on 4 qubits, rerun the GA search multiple times with different random seeds and model initializations, or train the resulting embeddings for many more epochs. If the embeddings found by GA under the original fixed setup do not consistently outperform randomly sampled embeddings when the training configuration changes, the claim that GA finds a task-optimal embedding would be falsified.

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Extended reading notes

Core claim

The central discovery is that feature-to-qubit permutations are a meaningful axis of performance in quantum neural networks, and that a genetic algorithm can search this discrete space effectively. Evidence for the existence of an optimal mapping comes from a full sweep of all 720 permutations on 3 qubits for six MNIST binary-class pairs: under fixed data, model initialization, and hyperparameters, the combined accuracy score spans about 20 points, with the best embedding scoring 93.18 and the worst 74.75 on classes 2 and 6. The GA, using tournament selection, order-based crossover, and swap mutation over a population of 20 permutations for 5 generations, matches or exceeds random selection in both noiseless and noisy (FakeBrisbane) simulations; in the noisy case it often beats even the best randomly chosen embedding. On higher qubit counts (6–8 qubits) and on Tiny ImageNet, the advantage persists, with fitness-score gains up to 3.33–3.36 and runtime savings up to 8.8–15%.

Load-bearing premise

All fitness evaluations use one fixed training setup, 5 epochs, a learning rate of 3e-3, a fixed model initialization, and for noisy runs the FakeBrisbane noise model, so the best embedding found is assumed to be the best for the task under that single configuration.

Editorial extensions

If this is right

The feature-to-qubit mapping is a first-class design choice in quantum embedding, not a detail to be left to chance.
Because the GA only requires a fitness score from training, it can be applied to other embedding circuit families and to other classical datasets without changing the search procedure.
Under noisy simulation, the GA found embeddings that beat the best of 100 randomly chosen permutations, suggesting that evolutionary search is particularly useful when noise makes performance fluctuate.
The runtime of GA search is comparable to or lower than random selection at the same evaluation budget, so the search can be run as a standard part of QML model development on NISQ hardware.

Reading between the lines

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

One extension the paper leaves implicit is using the GA to search not only the feature order but also which embedding circuit template to use, since the same permutation representation extends to any circuit with a discrete parameter-assignment step.
The optimality claim rests on a single training configuration, so testing whether the GA-selected embedding remains optimal under different hyperparameters or seeds would tell whether the search is finding a property of the data or an artifact of the fixed training setup.
The reported gains over QEK, QAOA embedding, and QRAC are small in multiplicative terms, so the practical significance may lie less in raw accuracy and more in the ability to find a good embedding quickly without gradient-based training of the embedding itself.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper establishes one thing convincingly and one thing weakly. The strong part is the 720-permutation sweep on 3 qubits: embedding order moves the combined train/inference score by roughly 20 points under fixed hyperparameters. That is a concrete, reproducible observation and it justifies caring about the mapping problem. The weak part is the central claim that the GA beats random selection. The evidence for that is not there yet: each embedding is evaluated once, with one model initialization and 5 epochs; the GA is stochastic and no repeated runs are reported; and Table I shows several negative improvements (-1.7 to -3.8) that disappear from the abstract's "0.33-3.33" range. The stress-test note is on target: a single seed cannot separate the embedding effect from initialization noise, so the "optimal" permutation might be an artifact of that seed. The authors' own Section IV.E admits the search is backend-specific, which is honest but also means the claimed advantage is conditional on a particular noise model.

What's genuinely new: framing embedding selection as a discrete search and applying a GA to feature-to-qubit permutations is a natural extension of existing meta-heuristic kernel search, and the paper does the first full sweep for angle embedding that I know of. The SC trend in the histograms is a nice sanity check. The code is supposedly on GitHub, though I didn't verify it.

The comparison with QEK, QAOA, and QRAC is not controlled: different encoding methods, different feature counts, and QRAC numbers are lifted from the original paper. The 1.003X/1.03X/1.06X ratios are essentially meaningless as stated.

Bottom line: this is a workable engineering idea with an interesting observation, but the headline claim needs repeated runs over seeds, error bars, and a fairer baseline before I'd trust the GA advantage. It deserves a serious referee because the core observation is real and the flaws are fixable, not because the current evidence is convincing.

Referee Report

5 major / 6 minor

Summary. The paper proposes framing the selection of a quantum embedding (feature-to-qubit mapping) for QML as a discrete search problem, solved with a genetic algorithm (GA). Experiments on MNIST and Tiny ImageNet binary classification, using angle embedding and a strongly entangling PQC, compare GA-selected embeddings with randomly selected mappings, report runtime improvements, and compare against QEK, QAOA-embedding, and QRAC. The authors also present a full 720-permutation sweep on 3 qubits to argue that an optimal embedding exists.

Significance. The idea of using a genetic algorithm to search over embedding permutations is reasonable and addresses a real practical question in QML. The complete sweep of all 720 permutations on 3 qubits is a useful empirical observation, and the authors provide code on GitHub. However, the current evidence does not support the paper's main comparative claims: the GA-vs-random comparison is compromised by selection on the test metric, the noiseless sweep uses a single model initialization, and the abstract omits numerous negative results. With additional experiments and more careful statistical reporting, the approach could become a solid contribution.

major comments (5)

[III.A-B] The noiseless 720-permutation sweep uses a single fixed model initialization and only 5 training epochs. With such short training, final accuracies are strongly influenced by the random PQC initialization. The claim that the ~20-point spread "proves that there exists an optimal embedding configuration" is not supported without demonstrating that the ranking of permutations is stable across multiple initializations; otherwise the best permutation may be an artifact of one seed. Please repeat the sweep over several initializations and report rank correlations or the distribution of best/worst permutations.
[IV.B-C, Table I] The fitness function is the combined score, defined as the mean of training and inferencing accuracies. Using test-set accuracy in the fitness function means the GA is directly selecting on the test metric, while the random baseline performs no such selection. The comparison is therefore biased in favor of GA, and the reported fitness improvements are inflated. Use a separate validation set for selection and report test accuracy only after the final embedding is chosen.
[Abstract, Table I] The abstract states that GA outperforms random selection by 0.33-3.33 for MNIST, but Table I lists many negative improvements (GA best minus random best), e.g., -3.8 for (0,3,5) noiseless and -3.1 for (2,6,3) noisy. The quoted range is from the high-qubit MNIST subset only; the low-qubit noiseless results are mostly negative. Please report all results and make the abstract consistent with the full table.
[IV.D, Table III] The comparison with QEK, QAOA-embedding, and QRAC is not controlled: QEK and QRAC results are taken from the original papers without re-running under identical conditions, and no error bars are given for any method. The improvements (1.003X-1.06X) are small and may be within noise. Please either run all methods under the same setup or clearly label the comparison as indicative.
[IV.A, IV.C] The GA hyperparameters (population size 20, generations 5, crossover 0.8, retention 0.1, mutation 0.001) are fixed without sensitivity analysis, and no repeated GA runs are reported, so the reported fitness scores have no error bars. Additionally, the runtime advantage of GA over random selection is not explained given that both evaluate 100 permutations; please clarify how GA runtime can be lower and report variance across repetitions.

minor comments (6)

[III.B, IV.A, IV.C] There are several typos: "existance" should be "existence", "indiviuduals" should be "individuals", "Addtionally" should be "Additionally", and "varation" should be "variation".
[Table I] The header "mean best runtime best runtime" is ambiguous; please label columns clearly as RS and GA, and specify whether runtime refers to the best individual or the mean.
[II.D] The description of reference [3] as "quantum metric learning" does not match the cited paper (Lloyd et al., "Quantum embeddings for machine learning"); please correct the description.
[Algorithm 1] Variable names such as "f s" and "os" are not descriptive, and the "random(parents, 2)" notation is unclear; please use more explicit naming and explain the selection step.
[IV.D] In the sentence "where perform training for proposed GA and QAOA-embedding-based methods" there is a missing pronoun; should be "we perform training".
[Table III] The "1X" in the improvement column should be written as "1.0X" for consistency with the other ratios, and the decimal formatting should be uniform.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the GA search is evaluated against an external random baseline, and the exhaustive sweep is an independent empirical observation.

full rationale

The paper's central claims are empirical rather than derivational. The GA optimizes the same combined training/inference fitness score that is later used to report performance, but this is the algorithm's objective function, not a circular reduction: the random-selection baseline is an external control, and the reported improvements are measured quantities with concrete runtime and accuracy values. The 720-permutation sweep in Section III is an exhaustive empirical enumeration; the fact that a finite set of scores has a maximum is definitional, but the substantive finding is the observed spread in accuracy across permutations, which is not entailed by the definition. The paper relies on no load-bearing self-citations: references such as Schuld, Lloyd et al., and Thumwanit et al. are independent prior work, and no uniqueness theorem or ansatz is imported from the authors' own publications. The limitation stated in Section IV.E, that the GA result is valid only for the chosen backend, is an honest scope restriction rather than a circular step. Methodological concerns about fixed initialization, single-seed noiseless runs, and selection on the evaluation metric are robustness/fairness issues, not circularity, and do not raise the circularity score.

Assumptions & free parameters 8 free parameters · 4 assumptions · 0 invented entities

The central claims rest on a moderate number of hand-chosen hyperparameters (GA and training) and on domain assumptions about the embedding family, the noise model, and the transferability of the fixed-training-configuration evaluation. No new physical entities are introduced.

free parameters (8)

GA population size spop = 20
Chosen by hand in Algorithm 1; no sensitivity analysis provided.
GA generations g = 5
Chosen by hand; total fitness evaluations = spop * g = 100.
GA crossover rate cr = 0.8
Chosen by hand in Algorithm 1.
GA retention rate rr = 0.1
Chosen by hand; top 10% of population retained each generation.
GA mutation rate mr = 0.001
Chosen by hand in Algorithm 1.
Fitness function weights = 1/2 train + 1/2 inference
Combined score is defined as the mean of training and inference accuracies (Section III.A); this weighting is a design choice.
Training epochs and learning rate = 5 epochs, 3e-3 LR
Fixed for all fitness evaluations; the optimal embedding is defined relative to these training hyperparameters.
Features per qubit in angle embedding = 2 (RX and RY)
The search space is restricted to permutations into RX/RY pairs; changing this changes the problem.

assumptions (4)

domain assumption Angle embedding with two features per qubit (one RX, one RY) is a sufficient and representative embedding family for the search problem.
The entire search space is permutations over this fixed structure; other embedding methods (amplitude, basis) and other gate pairs are not considered, so optimality is only within this family.
domain assumption The best embedding under one fixed training configuration transfers to the QML task in general.
Fitness is evaluated with fixed initialization, 5 epochs, and a fixed learning rate; no retraining or re-initialization is tested (Section III.A, IV.B).
domain assumption FakeBrisbane simulated noise is representative of the target hardware.
All noisy results use FakeBrisbane with coupling 0-1-2; the authors acknowledge in Section IV.E that results are backend-specific.
domain assumption The genetic operators (tournament selection, order-preserving crossover, swap mutation) are effective for this permutation search space.
No analysis of the fitness landscape or operator performance is provided; the GA's effectiveness is assumed from generic combinatorial optimization practice.

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Cite this review

Pith. "Pith review of Optimizing Quantum Embedding using Genetic Algorithm for QML Applications." pith.science (2026). https://pith.science/paper/XIGAWVSE

@misc{pith2026241200286,
  author       = {Pith},
  title        = {Pith review of: Optimizing Quantum Embedding using Genetic Algorithm for QML Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XIGAWVSE}},
  note         = {Machine review of arXiv:2412.00286}
}

read the original abstract

Quantum Embeddings (QE) are essential for loading classical data into quantum systems for Quantum Machine Learning (QML). The performance of QML algorithms depends on the type of QE and how features are mapped to qubits. Traditionally, the optimal embedding is found through optimization, but we propose framing it as a search problem instead. In this work, we use a Genetic Algorithm (GA) to search for the best feature-to-qubit mapping. Experiments on the MNIST and Tiny ImageNet datasets show that GA outperforms random feature-to-qubit mappings, achieving 0.33-3.33 (MNIST) and 0.5-3.36 (Tiny ImageNet) higher fitness scores, with up to 15% (MNIST) and 8.8% (Tiny ImageNet) reduced runtime. The GA approach is scalable with both dataset size and qubit count. Compared to existing methods like Quantum Embedding Kernel (QEK), QAOA-based embedding, and QRAC, GA shows improvements of 1.003X, 1.03X, and 1.06X, respectively.

Figures

Figures reproduced from arXiv: 2412.00286 by the authors.

**Figure 1.** Example showing how permutation (f0, f1, f2, f3, f4, f5) is embedded on 3 qubits using angle embedding. A pair of two features is mapped on each qubit, such that the first feature is embedded using an RX gate, and the second feature is embedded using an RY gate. quantum feature spaces, enabling effective data representation in the quantum Hilbert Space. However, finding an optimal embedding pattern is essential due … view at source ↗

**Figure 2.** Plot representing the variation of training accuracy with respect to [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 4.** The figure represents a circuit diagram of a QNN. In state embedding, [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Various MNIST binary classes with varying SC values. The SC value increases from leftmost subplot ( [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: Histograms of sweeping all permutations of embeddings for various binary classes of MNIST dataset under noiseless conditions. The mean of [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 7.** Figure 7: Histograms of sweeping all permutations of embeddings for various binary classes of MNIST dataset under noisy conditions. Compared to noiseless [PITH_FULL_IMAGE:figures/full_fig_p004_7.png]

**Figure 8.** Figure 8: Noiseless and noisy plots for (0,1,4) MNIST case. Optimal fitness score found by GA is highlighted on red line along with value. the indiviuduals of the population are sorted in decreasing order, and the following steps are performed iteratively for g generations: ⃝1 t…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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work page 2018

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Maria Schuld. Supervised quantum machine learning models are kernel methods. arXiv preprint arXiv:2101.11020 , 2021

arXiv 2021

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Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran. Quantum embeddings for machine learning, 2020. 4GitHub repository link: https://github.com/KoustubhPhalak/ quantum-embed-genetic/

work page 2020

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Invited: Trainable discrete feature embeddings for quantum machine learning

Napat Thumwanit, Chayaphol Lortaraprasert, and Rudy Raymond. Invited: Trainable discrete feature embeddings for quantum machine learning. In 2021 58th ACM/IEEE Design Automation Conference (DAC), pages 1352–1355, 2021

work page 2021

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Simulating a perceptron on a quantum computer

Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. Simulating a perceptron on a quantum computer. Physics Letters A , 379(7):660–663, March 2015

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Stephanie Forrest. Genetic algorithms. ACM computing surveys (CSUR), 28(1):77–80, 1996

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Doriguello and Ashley Montanaro

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