Hybrid Quantum-Classical Learning for Multiclass Image Classification

Jishnu Mahmud; Muhammad Iqbal Hossain; Shuchismita Anwar; Sowmitra Das

REVIEW 2 major objections 5 minor 68 references

Hybrid Quantum-Classical Learning for Multiclass Image Classification

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A hybrid quantum-classical network that recycles qubits discarded at the first pooling layer beats both its own no-recycle baseline and two lightweight QCNN baselines across four-class MNIST, Fashion-MNIST, and OrganAMNIST.

desk verdict The paper's central claim—that recycled discarded-qubit measurements explain the 22–28 point accuracy jumps—doesn't hold up as stated, because the baseline differs in more than just the recycled branch; still, the idea is real and the architecture is new, so it deserves a serious referee with a matched-capacity ablation. read the letter →

arxiv 2508.18161 v1 pith:5VVVRMHZ submitted 2025-08-25 quant-ph cs.LG

Shuchismita Anwar , Sowmitra Das , Muhammad Iqbal Hossain , Jishnu Mahmud This is my paper

classification quant-phcs.LG

keywords quantumconvolutionalneuralnetworkshybridquantum-classicallearningdiscardedqubitreusepoolingmulticlassimageclassificationHadamardproductfusionNISQdevices

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 convolutional networks shrink their register by measuring and discarding qubits at pooling layers. The paper argues that those discarded qubits are not useless: because they were entangled with the surviving qubits, their measurements still carry class-relevant correlations. The proposed architecture measures both streams, rescales the probabilities to a symmetric interval, feeds each through a shallow classical head, and combines the two heads by element-wise multiplication before classification. Trained end-to-end, the recycled branch outperforms the no-recycle baseline by 22–28 percentage points on the three benchmarks while keeping the same quantum circuit and adding only 74 classical parameters.

What carries the argument

The load-bearing mechanism is the recycled branch: the qubits removed by the first pooling layer are measured at the end of the circuit, their probabilities are affinely rescaled from [0,1] to [-2,2] by p -> 4p - 2, and the result goes through a small dense head with a projection-expansion-contraction structure. That head's output is fused with the retained-qubit head by the element-wise Hadamard product z = y_ret ⊙ y_disc, a parameter-free operation that acts as a logical AND: a class fires only when both quantum streams agree. The fixed rescaling keeps activations in the high-gradient region of tanh and aligns their variance with the initialization criterion used in the paper. Back-propaga

What would settle it

Train the same quantum circuit with a matched-parameter baseline: keep the quantum part, loss, and optimizer exactly as in the proposed model, but feed the second head with fixed dummy inputs (zeros or scrambled measurements) instead of the discarded qubits' real probabilities, keeping all 74 extra parameters. If accuracy stays near the recycled model, the gain comes from classical capacity; if it drops back to the no-recycle baseline, the gain is specifically the discarded qubits' information.

Watch

Extended reading notes

Core claim

The paper's central claim is that information in qubits discarded by the first pooling layer can be recovered and used to make a quantum convolutional classifier substantially more accurate. On four-class splits of MNIST, Fashion-MNIST, and OrganAMNIST, measuring those discarded qubits, rescaling their probabilities to [-2,2], processing them through a small fully-connected head, and fusing that head with the retained-qubit head through a Hadamard product raises test accuracy from 68.51% to 93.55%, from 81.40% to 96.45%, and from 69.13% to 88.50%, respectively. The paper reports faster convergence and a narrower train-test gap, and its ablations indicate that recycling only the first pooling

Load-bearing premise

The empirical conclusion assumes the recycled and baseline models differ only in the presence of the recycled branch, yet the baseline is a single fully connected layer on retained qubits while the proposed model adds a second classical head, an expansion layer, and 74 extra parameters; if the baseline were given the same extra classical weights, part or all of the reported gain could disappear.

Editorial extensions

If this is right

Any QCNN that currently discards qubits after pooling can adopt this branch without changing circuit depth, qubit count, or noise budget.
The recycled architecture beats two published lightweight QCNN baselines on four-class MNIST and Fashion-MNIST splits while keeping a total of 262 trainable parameters.
Amplitude embeddings respond more strongly to qubit reuse than angle embeddings, suggesting the benefit depends on how information is encoded into the quantum state.
Only the first pooling stage's discarded qubits should be recycled; adding the second stage's discarded qubits gives negligible or unstable gains.
The gain is an optimization effect: faster convergence and a smaller generalization gap, not overfitting.

Reading between the lines

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

A capacity-matched ablation—extra classical weights fed with dummy inputs instead of real discarded-qubit measurements—would separate the contribution of quantum information from added classical capacity.
The same recycling idea should transfer to quantum-state discrimination tasks such as phase recognition or error-syndrome classification, where the input is already quantum and the entanglement structure is native.
All experiments use 8 qubits and four-class splits, so it remains to be seen whether the benefit persists at larger class counts or full ten-class datasets.
The scaling constant κ = 4 is justified by gradient and variance arguments; sweeping its value would test whether the exact choice matters or whether any symmetric rescaling works.

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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

The thing to know first: the strong claim in Section VI—that every gain traces directly to information from discarded qubits—is not supported by the experiments as described. The baseline is a single fully connected layer on retained qubits; the proposed model adds a tanh head, an expansion layer, a second head, and Hadamard fusion. That's roughly 74 extra parameters and a different nonlinearity structure. The stress-test note is right: any one of those changes could improve accuracy on MNIST, independent of whether the discarded measurements carry useful information. The paper even asserts 'the only difference is the presence or absence of the recycled branch,' which contradicts its own architecture description. That is a load-bearing flaw, not a nitpick.

But give credit where it's due. The core idea—reusing qubits that pooling discards, on the grounds that they remain entangled with retained ones—is a legitimate gap in the QCNN literature, and the specific recipe (first-pooling discarded qubits, affine rescale to [-2,2], two classical heads fused by Hadamard product) is not in the cited baselines. The motivation is clearly argued, and the paper is honest about some design choices, including why they reused only the first pooling's discarded qubits after seeing unstable results with both. That's transparent post-hoc selection, which is at least a step above hiding it.

Other soft spots are real but secondary. No error bars or multiple seeds are reported, so the 22–28 point gaps could in principle be noisy, though the consistency across three datasets makes that less worrying. No code or data is visible in the preprint despite the appendix promise. There are internal contradictions—three pooling stages vs two, 'three identical macro-blocks' vs the third without pooling—that would make reproduction genuinely hard. The kappa=4 choice is justified with variance arguments but is still a fixed constant without ablation.

Who gets value from this? People working on hybrid QCNN architectures, especially those interested in squeezing more out of small quantum registers. The paper does not address quantum advantage, noise, or scaling, and it doesn't claim to; its scope is modest. The central conclusion overreaches, but the underlying question is testable, and the flaw is fixable.

My recommendation: don't desk reject it. Send it to peer review with a request for a matched-capacity baseline (same number of classical weights and nonlinearity), error bars over multiple seeds, an ablation that zeros out the discarded branch while keeping the rest of the architecture, and code. If the authors can show that the gain survives those controls, this becomes a useful contribution. As it stands, the evidence doesn't support the causal claim, but the idea deserves a fair shot.

Referee Report

2 major / 5 minor

Summary. The paper proposes a hybrid quantum-classical QCNN that reuses measurements from qubits discarded after the first pooling layer. After an affine rescaling p -> 4p - 2, the retained and discarded measurements are processed by separate fully connected heads, and the two outputs are combined by an element-wise Hadamard product that serves as the classification logits. Experiments on four-class subsets of MNIST, Fashion-MNIST, and OrganAMNIST report large accuracy gains over a baseline without reuse (e.g., MNIST 68.51% to 93.55%). The central claim is that information from qubits otherwise discarded at the pooling stage is responsible for these gains.

Significance. If verified by controlled experiments, the core idea is valuable: it suggests a way to improve QCNN accuracy without adding quantum resources, by recycling measurements that current architectures throw away. The paper gives a concrete architectural proposal, provides explicit expressions for the circuit ansatz and classical heads, and reports head-to-head comparisons with prior lightweight quantum classifiers. The contribution would be stronger if the empirical comparison isolated the effect of the recycled branch. As it stands, the central claim is plausible but not established by the reported experiments.

major comments (2)

[§VI.A and §V.B, Eq. (8)-(9)] The paper states that 'the only difference is the presence or absence of the recycled branch' (§VI.A), but the baseline uses 'just a single fully connected layer tied to the retained qubits,' while the proposed model replaces that layer with a tanh-activated retained head (Eq. 8), adds a three-stage discarded head (§V.B.2), and fuses the two streams multiplicatively (Eq. 9). This comparison changes classical capacity (~74 additional parameters), introduces a nonlinearity on the retained stream, and changes the output function from affine logits to a Hadamard product. The 22–28 percentage point gains therefore cannot be attributed specifically to discarded-qubit information. A matched-capacity control is needed, e.g., a two-head Hadamard-fused model whose discarded inputs are replaced by zeros, random values, or duplicates of retained features, so that the only difference remains the actu
[§IV.C and §V.A.1] The measurement protocol for discarded qubits is internally inconsistent. Section IV.C says the two probability vectors are recorded at the circuit's end, with the discarded qubits 'still physically present at the circuit's end.' Section V.A.1, however, says 'Discarded qubits are measured immediately after the first pooling layer, precisely when they leave the quantum data path.' Mid-circuit measurement and end-of-circuit measurement are operationally different and could affect both the information content and noise behavior. The paper must specify which protocol was actually used in the experiments.

minor comments (5)

[§V.B.3 vs §II/Table I] Section V.B.3 says the Hadamard-fused vector z is treated directly as the logits, while Section II and Table I describe an additional final FC layer before the logits. Please clarify the actual output layer.
[§V.A.1] The scaling constant κ=4 is fixed and defended analytically, but no empirical ablation is provided. Since the scaling method is listed as a contribution, a small ablation over κ values (e.g., 2, 4, 6) would substantiate the claim.
[Tables II-III] Table III lacks the 'Qubits' column that appears in Table II, making it harder to compare configurations accurately. Also, the paper should state clearly that experiments use four-class subsets, not the full datasets.
[Throughout] Minor typographical issues: 'ater convolutions' in §IV.D should be 'later'; Eq. (8) introduces a tilde notation for y that is not used consistently later. No error bars, standard deviations, or multiple-seed results are reported for the main accuracy claims.
[§I] The introduction promises 'An appendix provides complete implementation details, code versions, hardware configurations, and reproducibility scripts,' but no appendix appears in this version. Either include the appendix or remove the promise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's claims are empirical comparisons, not derivations that reduce to their inputs.

full rationale

This manuscript does not contain a claimed derivation chain of the kind that can be circular. The central claim—reusing discarded-qubit measurements improves classification—is an empirical result backed by train/test evaluations on public benchmarks. There is no equation in which an output is defined in terms of the predicted quantity, no fitted parameter that is later renamed as a prediction, and no invocation of a uniqueness theorem or self-citation chain that forces a conclusion. The authors cite prior work only as background and baselines; none of the citations are self-citations, so no load-bearing weight is placed on the authors' own prior results. The paper does make post-hoc design choices (fixing κ=4, choosing the Hadamard product after preliminary trials, recycling only the first pooling discarded qubits after observing unstable gains from both sets), but these are model-selection decisions, not circular reductions. A separate validity concern is that the proposed model and the baseline differ not only in the recycled branch but also in classical head capacity and nonlinearity, so the reported gains may not be solely attributable to discarded-qubit information. That is a potential experimental confound, not a circularity in the sense of a derivation that feeds on its own output. Therefore no circular steps are identified and the score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central empirical claim rests on unverified modeling choices: that discarded-qubit measurements add non-redundant features, that the amplitude encoding of downsampled images is faithful, and that the fixed scaling and post-hoc pooling-stage selection are harmless. No new physical entities are introduced.

free parameters (2)

scaling constant kappa = 4 (fixed, hand-chosen)
Eq. (7) maps probabilities to [-2,2]; authors justify it via tanh gradient and variance heuristics but provide no ablation across kappa.
expansion factor k in discarded head = not reported
The deep feature expansion uses a dimension mk but the value of k is never stated, so the classical parameter count cannot be verified.

assumptions (4)

domain assumption Measurement outcomes of discarded qubits at the first pooling layer carry non-redundant information useful for classification.
This is the core mechanism claimed in Sections IV and V, but the ablation does not isolate it from added classical capacity.
domain assumption Preprocessed images can be faithfully represented by amplitude encoding on 8 qubits, i.e., 256 amplitudes.
The input dimensionality and downsampling procedure are not described, so the fidelity of this encoding is unknown.
ad hoc to paper The fixed affine rescaling with kappa=4 improves optimization without tuning.
The paper fixes kappa=4 and justifies it by heuristics; no experiments test other values.
standard math The 15-parameter SU(4) convolution ansatz is universal and can be compiled with nearest-neighbor CNOTs.
This follows from the cited Vatan-Williams decomposition and is standard in the QCNN literature.

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Cite this review

Pith. "Pith review of Hybrid Quantum-Classical Learning for Multiclass Image Classification." pith.science (2026). https://pith.science/paper/5VVVRMHZ

@misc{pith2026250818161,
  author       = {Pith},
  title        = {Pith review of: Hybrid Quantum-Classical Learning for Multiclass Image Classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5VVVRMHZ}},
  note         = {Machine review of arXiv:2508.18161}
}

read the original abstract

This study explores the challenge of improving multiclass image classification through quantum machine-learning techniques. It explores how the discarded qubit states of Noisy Intermediate-Scale Quantum (NISQ) quantum convolutional neural networks (QCNNs) can be leveraged alongside a classical classifier to improve classification performance. Current QCNNs discard qubit states after pooling; yet, unlike classical pooling, these qubits often remain entangled with the retained ones, meaning valuable correlated information is lost. We experiment with recycling this information and combining it with the conventional measurements from the retained qubits. Accordingly, we propose a hybrid quantum-classical architecture that couples a modified QCNN with fully connected classical layers. Two shallow fully connected (FC) heads separately process measurements from retained and discarded qubits, whose outputs are ensembled before a final classification layer. Joint optimisation with a classical cross-entropy loss allows both quantum and classical parameters to adapt coherently. The method outperforms comparable lightweight models on MNIST, Fashion-MNIST and OrganAMNIST. These results indicate that reusing discarded qubit information is a promising approach for future hybrid quantum-classical models and may extend to tasks beyond image classification.

Figures

Figures reproduced from arXiv: 2508.18161 by the authors.

**Figure 2.** FIG. 2: Quantum Convolution Layer Structure. This figure represents the quantum convolutional layers with the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Quantum Pooling Operation. This figure illustrates the application of a quantum pooling layer using a [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: Grouped accuracy comparison across datasets, class splits, and encoding types, contrasting configurations [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 6.** Figure 6: FIG. 6: Fashion-MNIST 1-2-8-9; training-loss [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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Reference graph

Works this paper leans on

68 extracted references · 50 canonical work pages

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The first pair of convolu- tions acts on the angle-encoded image and detects 5 FIG

Local feature extraction. The first pair of convolu- tions acts on the angle-encoded image and detects 5 FIG. 3: Quantum Pooling Operation. This figure illustrates the application of a quantum pooling layer using a 2-parameter ansatz with controlled quantum gates ( RZ & RX ). The pooling layer reduces qubit counts after the convolutional operations to man...

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The reduced register is processed by the next two convolutional layers, which integrate information across a larger effec- tive receptive field

Intermediate abstraction. The reduced register is processed by the next two convolutional layers, which integrate information across a larger effec- tive receptive field. A second pooling layer again halves the number of qubits, preserving only the qubits that carry the richest correlations

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The final convolutional pair operates on the remaining qubits to encode global patterns in the quantum state

Global representation. The final convolutional pair operates on the remaining qubits to encode global patterns in the quantum state. No further pooling is applied, leaving the surviving qubits available for measurement and subsequent classical processing. Where, n = initial qubits; d = feature dimension after pooling; c = number of classes. The joint quan...

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Proposed Scaling Method to Process Retained & Discarded Qubit States A pivotal part of the hybrid pipeline is a single affine rescaling that places quantum outputs in the numerical regime most amenable to classical optimisation. Without this step the raw measurement probabilities p ∈ [0, 1] would enter the classical layers highly skewed, hampering gradien...

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Outputs are centred at the origin, aligning with the most sensitive region of tanh and similar non-linearities

Normalization. Outputs are centred at the origin, aligning with the most sensitive region of tanh and similar non-linearities

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Positive and negative excursions are treated equally, preventing biased weight updates during back-propagation

Symmetry. Positive and negative excursions are treated equally, preventing biased weight updates during back-propagation

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The four-fold expan- sion of the raw range widens the gradient window, mitigating both vanishing and exploding tendencies while preserving the relative ordering of probabili- ties

Enhanced learning dynamics. The four-fold expan- sion of the raw range widens the gradient window, mitigating both vanishing and exploding tendencies while preserving the relative ordering of probabili- ties. 7 The affine map in Eq. (7) therefore constitutes a sim- ple yet essential bridge: it preserves the full informa- tional content of quantum measurem...

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The deriva- tive of tanh is largest for |x| ≲ 2

Preserving gradients under tanh. The deriva- tive of tanh is largest for |x| ≲ 2. Setting κ = 4 stretches the centred interval to z ∈ [−2, 2], en- suring that almost all activations lie within this high-sensitivity band. Smaller values of κ leave ac- tivations clustered near the origin, where tanh be- haves almost linearly, reducing expressive capacity. L...

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If p is approximately uniform, then Var[p] = 1/12

V ariance alignment with Glorot/Xavier ini- tialisation. If p is approximately uniform, then Var[p] = 1/12. Scaling by κ yields Var[z] = κ2/12. The Glorot criterion suggests maintaining variance close to one for stable forward and backward signal propagation, which implies κ ≈...
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We pass this vector through a single fully connected layer of width m matching the number of target classes followed by a tanh nonlinearity

Processing the Retained-Qubit Features The m measurement outcomes that survive the quantum-convolution and pooling stages form a compact, high-order feature vector. We pass this vector through a single fully connected layer of width m matching the number of target classes foll...
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8 We measure the discarded register, rescale the outcomes to [−2, 2], and obtain xdisc ∈ [−2, 2]m

Processing the Discarded-Qubit Features Although pooling removes half the qubits, their mea- surements remain informative via prior entanglement. 8 We measure the discarded register, rescale the outcomes to [−2, 2], and obtain xdisc ∈ [−2, 2]m. These values are then fed into a...
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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

The first pair of convolu- tions acts on the angle-encoded image and detects 5 FIG

Local feature extraction. The first pair of convolu- tions acts on the angle-encoded image and detects 5 FIG. 3: Quantum Pooling Operation. This figure illustrates the application of a quantum pooling layer using a 2-parameter ansatz with controlled quantum gates ( RZ & RX ). The pooling layer reduces qubit counts after the convolutional operations to man...

work page

[2] [2]

The reduced register is processed by the next two convolutional layers, which integrate information across a larger effec- tive receptive field

Intermediate abstraction. The reduced register is processed by the next two convolutional layers, which integrate information across a larger effec- tive receptive field. A second pooling layer again halves the number of qubits, preserving only the qubits that carry the richest correlations

work page

[3] [3]

The final convolutional pair operates on the remaining qubits to encode global patterns in the quantum state

Global representation. The final convolutional pair operates on the remaining qubits to encode global patterns in the quantum state. No further pooling is applied, leaving the surviving qubits available for measurement and subsequent classical processing. Where, n = initial qubits; d = feature dimension after pooling; c = number of classes. The joint quan...

work page

[4] [4]

Proposed Scaling Method to Process Retained & Discarded Qubit States A pivotal part of the hybrid pipeline is a single affine rescaling that places quantum outputs in the numerical regime most amenable to classical optimisation. Without this step the raw measurement probabilities p ∈ [0, 1] would enter the classical layers highly skewed, hampering gradien...

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[5] [5]

Outputs are centred at the origin, aligning with the most sensitive region of tanh and similar non-linearities

Normalization. Outputs are centred at the origin, aligning with the most sensitive region of tanh and similar non-linearities

work page

[6] [6]

Positive and negative excursions are treated equally, preventing biased weight updates during back-propagation

Symmetry. Positive and negative excursions are treated equally, preventing biased weight updates during back-propagation

work page

[7] [7]

The four-fold expan- sion of the raw range widens the gradient window, mitigating both vanishing and exploding tendencies while preserving the relative ordering of probabili- ties

Enhanced learning dynamics. The four-fold expan- sion of the raw range widens the gradient window, mitigating both vanishing and exploding tendencies while preserving the relative ordering of probabili- ties. 7 The affine map in Eq. (7) therefore constitutes a sim- ple yet essential bridge: it preserves the full informa- tional content of quantum measurem...

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[8] [8]

The deriva- tive of tanh is largest for |x| ≲ 2

Preserving gradients under tanh. The deriva- tive of tanh is largest for |x| ≲ 2. Setting κ = 4 stretches the centred interval to z ∈ [−2, 2], en- suring that almost all activations lie within this high-sensitivity band. Smaller values of κ leave ac- tivations clustered near the origin, where tanh be- haves almost linearly, reducing expressive capacity. L...

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