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REVIEW 5 major objections 5 minor 39 references

Performance Analysis of Convolutional Neural Network By Applying Unconstrained Binary Quadratic Programming

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Encoding a CNN loss as a QUBO matrix and annealing it yields 10-15% higher MNIST accuracy than back-propagation, at similar runtime.

desk verdict The QUBO encoding in Section III.C.2 is a sample/parameter category error, so the annealer is likely minimizing a different objective and the reported accuracy gains are not meaningful. read the letter →

arxiv 2506.00247 v1 pith:52LUWP2M submitted 2025-05-30 cs.LG cs.ET

Aasish Kumar Sharma , Sanjeeb Prashad Pandey , Julian M. Kunkel This is my paper
classification cs.LGcs.ET
keywords ConvolutionalneuralnetworksQuantumannealingQUBOHybridquantum-classicaloptimizationMNISTMeansquarederrorCross-entropyStochasticgradientdescent
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 establish that a hybrid CNN, which replaces the loss-minimization step of training with quantum annealing, can beat both back-propagation and simulated annealing on accuracy without sacrificing runtime. The method encodes the MSE or CE loss as a QUBO matrix, lets an annealer find a low-energy binary solution, and then uses SGD to update the network weights. On MNIST the authors report 10-15 percent higher accuracy than a standard BP-CNN baseline with similar execution times, with the largest gap (about 40 percent for MSE) appearing at learning rate 0.1 in 10-fold cross-validation. The evidence is limited to MNIST, two loss functions, and mostly an annealer simulator, with a single brief run on real quantum hardware.

What carries the argument

The load-bearing object is the QUBO matrix: an objective $y = x^{T}Qx$ over binary variables $x$, constructed from the CNN loss by the coefficient formulas in Section III.C.2. For MSE the bias and coupling terms are $a_i = (2/N)(f(x_i,\theta)-y_i)$ and $b_{ij} = (4/N^2)(f(x_i,\theta)-y_i)(f(x_j,\theta)-y_j)$; for CE they are $a_i = -(1/N)(y_i - f(x_i,\theta))$ and $b_{ij} = -(1/N^2)(y_i - f(x_i,\theta))(y_j - f(x_j,\theta))$. The annealer minimizes this QUBO to choose parameter adjustments, and SGD then updates the weights, replacing the Boltzmann-search step of the CNN-SA pipeline while keeping the classical training loop intact.

What would settle it

Take a fixed set of weights on MNIST, compute both the QUBO energy and the true MSE or CE loss for many candidate binary vectors, and check whether the vector with lowest QUBO energy also has the lowest true loss; if not, the annealer is minimizing a different objective and the reported accuracy gains do not come from the mechanism claimed. A second check is to rerun CNN-QA and CNN-BP from identical initializations and seeds: if the 10-15 percent gap disappears, the headline result is an artifact of comparison conditions.

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Extended reading notes

Core claim

The central claim is that inserting quantum annealing after loss computation improves CNN training: the MSE or CE loss is reformulated as a QUBO objective $y = x^{T}Qx$, minimized on an annealer, and the result feeds SGD weight updates. In the 10-fold cross-validation at learning rate 0.1, the CNN-QA model reports higher accuracy than CNN-BP and CNN-SA by roughly 40 percent for MSE and 6 percent for CE, summarized as a 10-15 percent overall improvement at execution times comparable to BP and far below SA. The authors interpret this as QA's binary search over parameter adjustments escaping poor local minima more effectively than gradient-based or thermal search.

Load-bearing premise

The QUBO matrix built from the paper's bias and coupling formulas is assumed to represent the CNN's MSE or CE loss, so that the annealer's minimum is the loss's minimum; this equivalence is asserted, not derived.

Editorial extensions

If this is right

  • In 10-fold cross-validation at learning rate 0.1, CNN-QA reports about 40 percent higher accuracy than CNN-BP and CNN-SA for MSE and about 6 percent higher for CE.
  • CNN-QA execution times are close to CNN-BP and roughly 2.8 times shorter than CNN-SA, so the accuracy gain is not bought with extra runtime.
  • Because only the loss function is annealed, the qubit count stays far below full quantum-gate CNN encodings, making the hybrid approach compatible with current annealer sizes.
  • The paper's scaling tests show CNN-QA keeps steady execution time as data size grows, unlike CNN-SA, suggesting the hybrid is the more scalable of the two non-BP optimizers.
  • The benefit is objective-dependent: the combinatorial nature of MSE maps more naturally to QUBO than logarithmic CE, so loss choice matters for real deployments.

Reading between the lines

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

  • The QUBO coefficient formulas are presented without a derivation showing that minimizing them is equivalent to minimizing the original MSE or CE loss; until that equivalence is checked, the accuracy gain cannot be confidently attributed to quantum annealing.
  • A useful control experiment would run the same QUBO-plus-SGD recipe with a classical QUBO solver; if a classical solver reproduces the accuracy gain, the quantum hardware is not the load-bearing part of the improvement.
  • The paper's in-text citation for the QUBO tutorial points to a reference that is not the tutorial named, and the same method is cited elsewhere in the text; readers tracing the derivation should resolve this citation inconsistency first.
  • The method should be tested on larger and more varied image datasets and with more loss functions; the authors tested only MNIST with MSE and CE, mostly on a simulator, so the 10-15 percent figure may not transfer.
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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

5 major / 5 minor

Summary. The paper proposes a hybrid CNN training method (CNN-QA) that converts MSE or cross-entropy losses into a QUBO matrix (Section III.C.2), optimizes it with D-Wave quantum annealing or its 'neal' simulator, and uses SGD for weight updates. Experiments on MNIST compare CNN-BP, CNN-SA, and CNN-QA under 10-fold cross-validation, 10-epoch, and data-size tests. The paper claims 10--15% accuracy improvement over the baseline with similar execution times, although Section IV.E.1 reports 40% (MSE) and 6% (CE) improvements.

Significance. If a valid QUBO encoding of CNN losses were optimized by quantum annealing with measured gains, this would be a useful contribution to hybrid quantum-classical training. The manuscript makes code available and reports a test on real D-Wave hardware, which is helpful. However, the central equivalence between the QUBO objective and the CNN loss is asserted rather than derived, the CNN-BP-MSE baseline operates at chance level, and the quantitative claims are mutually inconsistent. As presented, the results do not establish the proposed method's validity.

major comments (5)
  1. [III.C.2] Steps 4–5 define the QUBO bias and coupling coefficients a_i = 2(f(x_i,θ)-y_i)/N and b_ij = 4(f(x_i,θ)-y_i)(f(x_j,θ)-y_j)/N^2 (MSE), with analogous CE terms. These coefficients are functions of the solution variable θ, yet a QUBO matrix must contain constants over the binary variables. The paper gives no derivation connecting sample-indexed residuals to parameter-indexed binary variables, and no such connection exists for a nonlinear network for which f(x,θ) is not linear in the binary encoding of θ. For a single scalar parameter with f(x,θ)=θ, the true MSE is a quadratic in θ, but the proposed coefficients produce a different objective whose minimizer need not coincide with the MSE minimizer. The annealer therefore minimizes an objective that is not shown to be the stated loss, so the claimed accuracy gains cannot be attributed to the proposed QA optimization.
  2. [IV.E.1 / Figures 8-10] The CNN-BP-MSE baseline runs at approximately 10% accuracy on MNIST (e.g., validity accuracy 0.0986 and test accuracy 0.098 in Figure 8(b); 0.103/0.101 in Figure 9(a); 0.0951/0.0957 in Figure 10(b)). For a 10-class problem, 10% is chance level, indicating a broken or misconfigured baseline. Any relative improvement over such a baseline, including the reported 40% or 10–15%, is not a meaningful measure of the method's performance.
  3. [Abstract; IV.E.1; Figures 8-10] The reported improvements are mutually inconsistent: the Abstract claims 10–15%; Section IV.E.1 claims 'higher accuracy by 40% for MSE and by 6% for CE'; and the figure data imply roughly 410% relative accuracy gain for MSE (0.0986 to 0.5109) and about 3% relative gain for CE (0.8766 to 0.904). No error bars or variance estimates are provided for any of the curves, so the reader cannot distinguish signal from noise. The central empirical claim needs a single consistent, statistically supported number.
  4. [V.A] The conclusion's limitation paragraph states that 'the MSE function, aligning with the nature of combinatorial optimization, exhibited superior performance over the logarithmic CE objective function.' All reported data show the opposite: CE accuracies are 0.87–0.92 while MSE accuracies are 0.10–0.52 (e.g., Figures 8–10). This internal contradiction undermines the conclusions drawn about which loss function is compatible with the QA approach.
  5. [III.C.3; IV.C; IV.D] The experimental section omits essential details for reproducibility: the CNN architecture (number and layout of convolutional and dense layers), the D-Wave 'neal' simulator parameters (number of reads, annealing time, chain strength), the QUBO embedding strategy, and the exact way the QA solution is converted into parameter updates for SGD. Without these, the reader cannot reproduce or assess the validity of the reported accuracy and execution-time results.
minor comments (5)
  1. [III.C.1] The word 'insymmetric' should be 'in symmetric' (or 'in symmetric form') in the discussion of the Q matrix.
  2. [IV.B] 'hyprer-parameters' is a typo for 'hyper-parameters'.
  3. [IV.E.1] The bullet 'around 2.76 times compared to CNN-QA' is unclear; it should specify which method is the reference and which method's time is 2.76 times larger.
  4. [References / III.C.1] Reference [36] is listed as Nilsson's 'Introduction to machine learning', but the text in Section III.C.1 cites it as the Glover et al. QUBO tutorial; the intended reference is likely [20].
  5. [II.A.1 / Eq. (2)] Equation (2) is written for binary cross-entropy with a single output; the paper should explain how the CE loss is extended to the 10-class MNIST task.

Circularity Check

1 steps flagged · score 6.0 of 10

The QUBO encoding of the MSE/CE loss is self-referential: the proposed coefficients depend on the parameters the binary variables are supposed to represent, so the annealer minimizes a residual-dependent surrogate rather than the stated loss.

  1. self definitional [Section III.C.1 Eq. (8) and Section III.C.2 steps 2-5]
    "QUBO: minimize/maximize y = x^t Q x (8) where x is a vector of binary decision variables and Q is a square matrix of constants. ... Define Bias Terms (a_i): MSE bias terms are defined as: a_i = 2/N (f(x_i, theta)-y_i), and CE bias terms use: a_i = -1/N (y_i - f(x_i, theta))."

    Equation (8) requires Q to be a matrix of constants, but the coefficients supplied in steps 4-5 are functions of theta through the residuals f(x_i, theta)-y_i. Steps 2, 6, and 9 state that the binary variables represent the parameters theta, so the QUBO matrix is defined in terms of the same variables it is supposed to determine. No constant Q matrix exists, and the objective minimized by the annealer is not the MSE or CE of Eqs. (9)-(10). The claimed conversion of the loss into QUBO is therefore self-referential: the minimizer appears inside the coefficients of the objective.

full rationale

The empirical comparisons themselves are not circular: the MNIST validation and test accuracies are genuine external measurements rather than fitted predictions, so the accuracy figures have independent content. The circularity lies in the paper's derivation chain that QA minimizes the CNN loss. The paper's own equations define the QUBO coefficients as functions of the residuals f(x_i, theta)-y_i while simultaneously identifying the binary variables with the parameters theta, leaving no fixed Q matrix and no demonstrated equivalence with the MSE/CE objectives. The Note after step 13 and the Conclusion's limitation that CE is 'not inherently compatible with combinatorial optimization' acknowledge that the method's effectiveness is contingent on the objective and the hardware, but they do not repair the missing or self-referential derivation. The self-citations in Section II.B (references [32] and [33], the author's own ResearchGate and StackExchange posts) are used only as motivational community insights and are not load-bearing; the formal QUBO background is cited to the external Glover tutorial. Because the central methodological claim reduces to a residual-dependent surrogate rather than an independent encoding of the loss, while the reported accuracy numbers remain external empirical results, the circularity is partial rather than total.

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

The central claim depends on an unproven QUBO encoding of the loss and on several unspecified hyperparameters; no new physical entities are introduced.

free parameters (3)
  • QUBO bias coefficient a_i = a_i = 2/N (f(x_i,theta) - y_i) for MSE; a_i = -1/N (y_i - f(x_i,theta)) for CE
    Defined in Section III-C.2 step 4 with no derivation; the value changes per batch and determines the QA objective.
  • QUBO coupling coefficient b_ij = b_ij = 4/N^2 (f(x_i,theta)-y_i)(f(x_j,theta)-y_j) for MSE; b_ij = -1/N^2 (y_i - f(x_i,theta))(y_j - f(x_j,theta)) for CE
    Defined in Section III-C.2 step 5 with no derivation; controls pairwise qubit interactions.
  • Learning rate = 0.1 for main results; also 1 and 0.01 in cross-validation
    Hyperparameter chosen by the authors, not optimized.
assumptions (3)
  • ad hoc to paper The QUBO matrix built from the stated bias and coupling terms is equivalent to the CNN loss (MSE or CE) over the network parameters.
    Core premise of Section III-C.2; asserted without proof, no derivation connects the QUBO to the loss.
  • domain assumption Solving the QUBO via quantum annealing (or its classical simulation) yields a lower-cost solution that improves CNN training when combined with SGD.
    Assumed throughout Section III-C.3; no theoretical or empirical evidence isolates the QA contribution.
  • domain assumption Standard CNN training with backpropagation provides a valid baseline for comparison.
    The baseline BP-MSE model runs at ~10% accuracy, near random for 10 classes, so this assumption is questionable.

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Cite this review

Pith. "Pith review of Performance Analysis of Convolutional Neural Network By Applying Unconstrained Binary Quadratic Programming." pith.science (2026). https://pith.science/paper/52LUWP2M

@misc{pith2026250600247,
  author       = {Pith},
  title        = {Pith review of: Performance Analysis of Convolutional Neural Network By Applying Unconstrained Binary Quadratic Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/52LUWP2M}},
  note         = {Machine review of arXiv:2506.00247}
}
read the original abstract

Convolutional Neural Networks (CNNs) are pivotal in computer vision and Big Data analytics but demand significant computational resources when trained on large-scale datasets. Conventional training via back-propagation (BP) with losses like Mean Squared Error or Cross-Entropy often requires extensive iterations and may converge sub-optimally. Quantum computing offers a promising alternative by leveraging superposition, tunneling, and entanglement to search complex optimization landscapes more efficiently. In this work, we propose a hybrid optimization method that combines an Unconstrained Binary Quadratic Programming (UBQP) formulation with Stochastic Gradient Descent (SGD) to accelerate CNN training. Evaluated on the MNIST dataset, our approach achieves a 10--15\% accuracy improvement over a standard BP-CNN baseline while maintaining similar execution times. These results illustrate the potential of hybrid quantum-classical techniques in High-Performance Computing (HPC) environments for Big Data and Deep Learning. Fully realizing these benefits, however, requires a careful alignment of algorithmic structures with underlying quantum mechanisms.

Figures

Figures reproduced from arXiv: 2506.00247 by the authors.

Figure 2
Figure 2. Typical CNN Architecture An extension of ANNs, Deep Learning (DL), includes architectures of varying depth composed of input, hidden and output layers ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. The foundation of various ANN architectures, like [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. • SA, a classical method derived from annealing in physics, operates within the constraints of classical physics using [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Optimization Flowchart for CNN Using Simulated [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Diagrammatic representation of proposed methodology [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Flowchart of the SA Method B. Procedure for Analyzing CNN with SA (i) Model Preparation: The configuration and testing of the CNN model with SA optimization draw from methods outlined by Rere L.M. et al. [15]. (ii) Model Structure: The model comprises an input layer, c…
Figure 7
Figure 7. Figure 7: Flowchart of the QA Method necessitates the conversion of the loss function into a QUBO matrix through the natural QUBO formulation method for minimization matrices [20]. (iii) Execution of Quantum Annealer: The constructed QUBO matrix is then supplied to DWAVE’s quant…
Figure 10
Figure 10. Figure 10: Cumulative Data Increment Test Details possibly leveraging its inherent advantages in search space evaluations. • Overall, across both MSE and CE objectives, CNN-QA presented improved accuracy around 10-15% and faster execution times. 2) 10 Epochs Test: The 10-Epoch T…
Figure 9
Figure 9. Figure 9: Details of the 10 Epoch Test 10 15 20 25 30 35 40 45 50 Datasize 20 40 60 80 Validity Accuracy (%) Legends CNN-BP-MSE CNN-SA-MSE CNN-QA-MSE CNN-BP-CE CNN-SA-CE CNN-QA-CE 0.0 0.2 0.4 0.6 0.8 1.0 Datasize 0.0 0.2 0.4 0.6 0.8 Validity Accuracy (%) 0.0957 0.1014 0.1016 0.8…

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