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REVIEW 4 major objections 7 minor 50 references

Learning Complex Word Embeddings in Classical and Quantum Spaces

T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper shows that complex-valued word embeddings, including quantum states prepared by parameterised circuits, can match classical Skip-gram on similarity benchmarks via a two-stage pipeline that scales to a 426k vocabulary.

desk verdict Useful, honest scaling pipeline for quantum word embeddings, but the headline claim of parity with classical is a few points overstated. read the letter →

arxiv 2412.13745 v1 pith:LZ2CS4DJ submitted 2024-12-18 cs.CL

Carys Harvey , Stephen Clark , Douglas Brown , Konstantinos Meichanetzidis This is my paper · ORCID
classification cs.CL
keywords complexwordembeddingsquantumnaturallanguageprocessingSkip-gramwithnegativesamplingparameterisedcircuitssimilaritybenchmarksfidelityoverlaptwo-stageembeddingpipelineword2vec
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

The paper tries to show that word embeddings can live in complex vector spaces, and ultimately as quantum states prepared by parameterised quantum circuits, without losing the quality of classical real-valued embeddings. It develops a complex-valued version of Skip-gram with negative sampling, replacing real vectors with complex vectors and the inner product with a scaled fidelity measure, then evaluates on WordSim353, MEN, RG-65, and SCWS. The key result is a two-stage pipeline: train complex embeddings with an efficient C implementation on a 3.8B-word corpus with a 426k vocabulary, then fit a separate PQC to each word. The fitted quantum embeddings reproduce the classical complex embeddings' scores almost exactly, so the paper claims high-quality quantum word embeddings at scale.

What carries the argument

The two-stage pipeline is the load-bearing mechanism. Stage one is a complex-valued Skip-gram with negative sampling, implemented by modifying the C word2vec code to store real and imaginary parts separately and to feed the scaled fidelity $F_D(v_f,v_c) = D(2|\langle v_f|v_c\rangle|^2 - 1)$ into a sigmoid loss; for the PyTorch models the overlap $|\langle v_f|v_c\rangle|^2$ is also used directly as a probability in the cross-entropy loss. Stage two fits, for each word, a parameterised quantum circuit—a unitary built from single-qubit Y-rotations and controlled X-rotations, with three layers of Ansatz 5 or Ansatz 14—to the normalised complex vector from stage one, using the overlap as the fitting loss. The fidelity overlap is the object that connects classical complex geometry to quantum states: normalised complex vectors are quantum states, and the squared absolute inner product is the Born-rule probability between pure states.

What would settle it

Compute the final per-word fidelity loss for all 426,507 fitted PQCs and inspect the tail; if a non-negligible number of words, or a distinct semantic class of words, have loss far above zero, the claim that the fitted PQCs reproduce the complex embeddings across the vocabulary fails. A simpler check is to hold out a random set of words from the fitting stage and see whether their PQC embeddings still match the target vectors.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that parameterised quantum circuits can serve as word embeddings without a quality penalty, provided they are fitted in a second stage to complex embeddings produced by an efficient classical Skip-gram variant. Directly training PQCs as part of the Skip-gram objective degrades performance, especially when both focal and context words are circuits, but the two-stage procedure—first learn arbitrary complex vectors from the corpus, then fit a PQC to each normalised vector—yields PQC embeddings whose WordSim353, MEN, and RG-65 correlations match the complex vectors exactly, and SCWS within 0.2 points. The paper attributes this to the expressivity of the ansatz: three layers of Ansatz 5 or Ansatz 14 fit the arbitrary complex states with the loss 'effectively going to zero.' Because the fitting stage only touches each vocabulary item and not each token, the route scales with vocabulary size rather than corpus size.

Load-bearing premise

The claim that the whole 426k-vocabulary PQC set is high quality rests on the assumption that three layers of the chosen ansatz can fit every word's complex embedding to effectively zero loss; the paper reports that loss 'effectively went to zero' but verifies it mainly through identical WordSim353 scores rather than a per-word fitting-loss distribution.

Editorial extensions

If this is right

  • If the central claim holds, a 426k-word vocabulary of PQC word embeddings is available for quantum natural language processing models at inference time, without needing a quantum device during training.
  • Complex-valued Skip-gram with the fidelity overlap can be trained as efficiently as classical word2vec on multi-billion-word corpora, since the overhead is a constant factor and gradients are computed explicitly.
  • Direct PQC training is not the route to good embeddings; the two-stage fitting procedure is, so future work on quantum embeddings should treat the circuit as a compression of a classically learned complex vector.
  • On the additional similarity datasets, the fitted PQC matches the complex embedding scores on three of four datasets and stays within 0.2 points on the fourth, so the result is not specific to WordSim353.

Reading between the lines

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

  • If the per-word fit is truly lossless, the quantum circuits inherit the full information of the complex embeddings, making the PQC set a drop-in replacement for complex vectors in any downstream compositional model, not just similarity benchmarks.
  • The closeness of the 64-dimension complex model to the 100-dimension real baseline, and the lack of gain from 64 to 128 complex dimensions, suggests the fidelity geometry rather than raw dimension is what drives performance; a controlled test with matched parameter counts across more dimensions would clarify this.
  • A natural extension is to use the same two-stage idea for GloVe-style objectives, where the overlap would predict log co-occurrence counts; success there would broaden the method beyond Skip-gram.
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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

4 major / 7 minor

Summary. This paper introduces complex-valued extensions of the Skip-gram word embedding model, replacing real-valued vectors with complex vectors and the inner product with a fidelity-based overlap. The authors describe a PyTorch implementation for small corpora and a modified C implementation of word2vec for a 3.8B-word corpus with a vocabulary of over 400k. They then train parameterized quantum circuits (PQCs) in two ways: directly, with the PQC producing focal or both focal and context embeddings, and through a two-stage procedure that fits a PQC to each already-trained complex embedding. The models are evaluated on WordSim353, MEN, RG-65, and SCWS. The main empirical findings are that the arbitrary complex embeddings are competitive with the classical Skip-gram baseline and that the two-stage fitted PQCs essentially reproduce the evaluation scores of the complex embeddings. The paper argues that this provides a scalable route to producing quantum word embeddings for large vocabularies.

Significance. If the results are reproducible, the two-stage pipeline is a meaningful practical contribution: it avoids corpus-scale PQC training, scales with vocabulary size rather than corpus size, and yields PQC-parameterized states for hundreds of thousands of words that can be used in QNLP inference or compositional models. The paper also provides a useful empirical comparison of circuit ansatze and layer counts for fitting arbitrary complex vectors. The C implementation that enables training on a 3.8B-word corpus is a nontrivial engineering contribution, and the authors are transparent about the limitations of direct PQC training. However, the absence of code or data release and the lack of per-word fitting evidence currently temper the significance of the claims.

major comments (4)
  1. [Section 3.4] The claim that the fitted PQCs reproduce the arbitrary complex embeddings 'perfectly,' with the loss going to 'effectively zero,' is not backed by per-word evidence. WordSim353 contains 353 word pairs and Spearman correlation is an aggregate rank measure, so identical WordSim353 scores do not rule out a heavy tail of poorly fit words; Table 7 itself shows a 0.2-point gap on SCWS (65.7 vs 65.9), so the fit is not exactly lossless for every word. Please report the distribution of the final fitting loss or fidelity over the full 426,507-word vocabulary (or a random sample), including the fraction of words below a stated fidelity threshold. This is load-bearing for the abstract's claim that the quantum embeddings 'perform as well' for the whole vocabulary.
  2. [Section 3.1 and Section 3.5] WordSim353 is used both for hyperparameter selection (e.g., the D scaling factor, ansatz, number of layers, learning rate) and as the main evaluation set; Section 3.5 further states that 'WordSim353 is used as a validation set to choose the best-performing model' before reporting the additional datasets. No held-out split or significance tests are provided, and several of the headline differences are small relative to the reported standard deviations (e.g., Table 2: 64.6 vs 63.0 at dimension 64, with SDs of 0.33 and 0.50). Please add paired significance tests (e.g., bootstrap over runs) or a properly separated validation set, and report the scores of all models without selection on the test set.
  3. [Sections 3.2-3.4] The paper introduces a custom C implementation, a newly created 3.8B-word corpus, and per-word PQC fitting code, but no code, embeddings, or fitted parameters are released. This prevents independent verification of both the large-scale training results and the central per-word fitting claim. At minimum, please release the trained complex embeddings and the fitted PQC parameters for the evaluation vocabulary, along with a script that recomputes the evaluation scores, or provide a clear statement about any restrictions.
  4. [Abstract and Section 3.4] The two-stage PQC result is partly by construction: once a PQC is fitted to a complex embedding, its evaluation score is, by design, nearly identical to that of the complex embedding. The abstract's wording could be read as an independent demonstration that quantum embeddings match classical ones, which is not what the experiment shows. The paper should explicitly state that the value of the two-stage method is the scalable production of PQC-prepared states that inherit the semantic quality of the complex embeddings, and that the direct PQC-training results (Tables 3, 4, 6) are the appropriate tests of PQC-based learning.
minor comments (7)
  1. [Section 2.1] The phrase 'to the predict the context word' should be 'to predict the context word'.
  2. [Table 4 header] 'Anzatze' should be 'Ansätze'.
  3. [Section 3.4] 'in the worse case' should be 'in the worst case'.
  4. [Section 3.5] 'WordsSim353' should be 'WordSim353'.
  5. [Table 6] The WordSim353 score for the 1-way PQC model (64.0) does not match the corresponding dimension-64 entry in Table 3 (63.5); please clarify whether these are the same best-run weights or different selections.
  6. [Section 2.1 vs Section 3.1] Section 2.1 says 'a value of around D = 3 works well in practice,' but Section 3.1 states D = 3.5 for all experiments using loss (4); please reconcile this or comment on the sensitivity to D.
  7. [Table 7] The abstract's phrase 'comparable numbers of parameters' should be quantified: a 100-dimensional classical embedding has 100 real parameters per word, a 64-dimensional complex embedding has 128, and a 3-layer A5 PQC on 6 qubits has (log_2^2(64)+3 log_2(64))*3 = 162 parameters per word.

Circularity Check

1 steps flagged · score 4.0 of 10

The two-stage PQC result inherits its WordSim score from the fitted complex vectors by construction, so the claim that quantum embeddings 'perform as well' is not an independent prediction.

  1. fitted input called prediction [Section 3.4, Table 5 discussion; echoed in the Abstract]
    "The PQC for each word was 3 layers of A5 and, since the loss effectively went to zero, the performance of the PQCs on WordSim353 was identical to the arbitrary complex embeddings on which the PQCs were trained."

    The PQC embeddings are produced by minimizing a fitting loss against the stage-1 complex embeddings; with the loss at 'effectively zero', the PQC's WordSim353 score is identical to the target's score by construction. The abstract's 'quantum word embeddings from the two-stage process perform as well as the classical Skip-gram embeddings' therefore reduces to the stage-1 complex model's already-reported score, not to any independent quality of the PQC representation. The only genuinely new content is that the ansatz can approximate the target vectors, but that is verified by the fitting loss, not by the downstream evaluation.

full rationale

The paper is largely self-contained and its pipeline is clearly described: a complex Skip-gram model is trained on a corpus, and then PQCs are fitted to those complex vectors. The ansatz choice and expressivity discussion rely on external prior work (Sim et al. [SJAG19]), not on a self-citation chain. No uniqueness theorem is imported from the authors, and no known result is merely renamed. The central circularity is limited to the two-stage PQC evaluation: because the fitting loss is reported as effectively zero, the 'quantum embedding' scores on WordSim353, MEN, RG-65, and SCWS are, up to small fitting error, the same as the stage-1 complex embedding scores by construction. The paper is transparent about this, but the abstract's framing that the two-stage quantum embeddings 'perform as well' presents an inherited result as a finding about the PQC model. There is also a mild selection effect in that WordSim353 is used as a validation set for choosing hyperparameters and the best run, but that is an overfitting concern rather than a definitional circularity. Overall, the non-redundant contribution—scalable fitting of PQCs to complex embeddings—does not itself reduce to the evaluation, so the circularity score is moderate rather than severe.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the empirical validity of the fidelity-based loss and the expressivity of the PQC ansatze, plus standard NLP evaluation assumptions. The main hand-tuned free parameter is D; other hyperparameters are conventional.

free parameters (3)
  • D scaling factor = 3.5
    Introduced in Eq. (3) and (6) to map the fidelity overlap into the sigmoid range; the text says 'around D = 3 works well in practice' and D = 3.5 is used for all experiments, so it is a hand-tuned constant.
  • word frequency cutoff = 17 (small), 100 (large)
    Applied to the corpora in Section 3.1 before training; changes the vocabulary size and is chosen by the authors without a stated criterion.
  • PQC layers and ansatz = 3 layers of A5 or A14
    Selected based on performance on WordSim353; the number of layers and ansatz choice determine the number of trainable parameters and are not derived from first principles.
assumptions (4)
  • domain assumption The distributional hypothesis: words appearing in similar contexts should have similar embeddings.
    Underlies the Skip-gram objective in Section 2.1.
  • domain assumption The fidelity measure |<v|w>|^2, scaled by D, is a suitable objective for learning semantic similarity in complex spaces.
    Introduced in Section 2.1 as the replacement for the inner product; no derivation linking this measure to semantic quality is given.
  • domain assumption The chosen PQCs (3 layers of A5 or A14) are expressive enough to fit arbitrary complex embedding vectors.
    Stated in Section 3.4 where the fitting loss 'effectively went to zero'; this is verified only indirectly through WordSim353 performance, not through a formal expressivity guarantee for these circuits.
  • domain assumption Human similarity and relatedness ratings in WordSim353, MEN, RG-65, and SCWS are a valid evaluation of embedding quality.
    Used in Section 3.1 as the evaluation protocol; the assumption that Spearman correlation on these datasets measures embedding quality is standard but not argued.

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Pith. "Pith review of Learning Complex Word Embeddings in Classical and Quantum Spaces." pith.science (2026). https://pith.science/paper/LZ2CS4DJ

@misc{pith2026241213745,
  author       = {Pith},
  title        = {Pith review of: Learning Complex Word Embeddings in Classical and Quantum Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LZ2CS4DJ}},
  note         = {Machine review of arXiv:2412.13745}
}
read the original abstract

We present a variety of methods for training complex-valued word embeddings, based on the classical Skip-gram model, with a straightforward adaptation simply replacing the real-valued vectors with arbitrary vectors of complex numbers. In a more "physically-inspired" approach, the vectors are produced by parameterised quantum circuits (PQCs), which are unitary transformations resulting in normalised vectors which have a probabilistic interpretation. We develop a complex-valued version of the highly optimised C code version of Skip-gram, which allows us to easily produce complex embeddings trained on a 3.8B-word corpus for a vocabulary size of over 400k, for which we are then able to train a separate PQC for each word. We evaluate the complex embeddings on a set of standard similarity and relatedness datasets, for some models obtaining results competitive with the classical baseline. We find that, while training the PQCs directly tends to harm performance, the quantum word embeddings from the two-stage process perform as well as the classical Skip-gram embeddings with comparable numbers of parameters. This enables a highly scalable route to learning embeddings in complex spaces which scales with the size of the vocabulary rather than the size of the training corpus. In summary, we demonstrate how to produce a large set of high-quality word embeddings for use in complex-valued and quantum-inspired NLP models, and for exploring potential advantage in quantum NLP models.

Figures

Figures reproduced from arXiv: 2412.13745 by the authors.

Figure 1
Figure 1. Ansatz 14 taken from [SJAG19]. 2.2 Parameterising Quantum Spaces Section 2.1 described arbitrary complex embeddings in which the real values of the classical word2vec em￾beddings are simply replaced with complex numbers. In this section we provide a more “physically-inspired” parameterisation in which the complex vectors are produced by parameterised quantum circuits (PQCs). In a PQC, the parameters are not the comp… view at source ↗
Figure 2
Figure 2. Original Skip-gram (left), unitary implementation (right, top) and quantum circuit (right, bottom). [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Arbitrary encoding circuit for calculating the overlap between two quantum states. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Ansatz 5 from [SJAG19]. Model Embedding Dim Anzatze Correlation PQC (focal, context) Skip-gram w/sigmoid 64 A5 53.4 0.10 PQC (focal, context) Skip-gram w/sigmoid 64 A14 57.8 0.41 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: WordSim353 scores for a variety of ansatze from Sim et al. with 6 qubits and L = no. of layers. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.