Layer-wise QUBO-Based Training of CNN Classifiers for Quantum Annealing

Mostafa Atallah; Rebekah Herrman

quant-phMar 3, 2026

Layer-wise QUBO-Based Training of CNN Classifiers for Quantum Annealing

Authors: Mostafa Atallah, Rebekah Herrman

Organizations: Department of Industrial and Systems Engineering, University of Tennessee - Knoxville, Knoxville, 37996, TN, USA. · Department of Physics, Faculty of Science, Cairo University, Giza, 12613, Egypt.

Abstract

Variational quantum circuits for image classification suffer from barren plateaus, while quantum kernel methods scale quadratically with dataset size. We propose an iterative framework based on Quadratic Unconstrained Binary Optimization (QUBO) for training the classifier head of convolutional neural networks (CNNs) via quantum annealing, entirely avoiding gradient-based circuit optimization. Following the Extreme Learning Machine paradigm, convolutional filters are randomly initialized and frozen, and only the fully connected layer is optimized. At each iteration, a convex quadratic surrogate derived from the feature Gram matrix replaces the non-quadratic cross-entropy loss, yielding an iteration-stable curvature proxy. A per-output decomposition splits the $C$ -class problem into $C$ independent QUBOs, each with $(d+1)K$ binary variables, where $d$ is the feature dimension and $K$ is the bit precision, so that problem size depends on the image resolution and bit precision, not on the number of training samples. We evaluate the method on six image-classification benchmarks (sklearn digits, MNIST, Fashion-MNIST, CIFAR-10, EMNIST, KMNIST). A precision study shows that accuracy improves monotonically with bit resolution, with 10 bits representing a practical minimum for effective optimization; the 15-bit formulation remains within the qubit and coupler limits of current D-Wave Advantage hardware. The 20-bit formulation matches or exceeds classical stochastic gradient descent on MNIST, Fashion-MNIST, and EMNIST, while remaining competitive on CIFAR-10 and KMNIST. All experiments use simulated annealing, establishing a baseline for direct deployment on quantum annealing hardware.

Figures & tables

Figure 1: Training pipeline matching Algorithm 1 . Green blocks: one-time initialization (frozen CNN features and Gram matrix). Gray blocks: iterative phases 1 and 3. Cyan blocks: Phase 2 inner loop over

C

classes. Outer loop repeats

T

iterations.

Figure 2: CNN architecture used in all experiments. Convolutional layers are randomly initialized and frozen; only the fully connected (FC) classifier head is trained via iterative QUBO solves.

Parameter	Value
Dataset	sklearn digits (8 $\times$ 8 images)
Training samples	1000
Test samples	540
Number of classes	10
CNN architecture	1 conv layer ( $3\times 3$ kernel), $2\times 2$ pooling, FC output
CNN filters	2

Table 1: Experimental Settings ( sklearn Digits Dataset)

$K$ (bits)	Logical Vars $n=(d{+}1)K$	Quadratic Terms $\frac{n(n-1)}{2}$
5	95	4,465
10	190	17,955
15	285	40,470
20	380	72,010

Table 2: Per-Output QUBO Size by Bit Precision for all datasets (

d=18

, one of

C=10

independent QUBOs)

Method	Train Acc	Test Acc	Time (s)	Final Loss
Classical FC (SGD)	82.2%	79.8%	22.7	0.872
QUBO 5-bit	33.9%	33%	225.6	2.704
QUBO 10-bit	77.7%	77.4%	767.3	0.758
QUBO 15-bit	83.9%	80%	2605.4	0.677
QUBO 20-bit	84.9%	81.5%	9712.6	0.671

Table 3: Layer-wise Gram QUBO Results on sklearn Digits (1000 iterations)

Panel of Figure 3 — Figure 3: Convergence curves for Classical FC (SGD) and QUBO methods with different bit precisions over 1000 iterations.

Figure 4: Sample predictions from Classical FC (SGD) and QUBO models with different bit precisions. Each row shows predictions for one test digit, with columns showing probability distributions from each model.

True Digit	Classical	5-bit	10-bit	15-bit	20-bit	Result
0	91.6% (T)	83.4% (T)	98.1% (T)	96% (T)	96.7% (T)	All correct
1	29.8% (T)	1.8% (F)	34.6% (F)	41.6% (T)	55.3% (T)	Mixed
2	61.6% (T)	43.8% (T)	58.3% (T)	79.7% (T)	78.9% (T)	All correct
3	66.3% (T)	80.4% (T)	49.9% (T)	86.2% (T)	77.7% (T)	All correct
4	68.3% (T)	98.1% (T)	84.2% (T)	80.3% (T)	85.3% (T)	All correct
5	48% (T)	44.2% (F)	66.2% (T)	67.3% (T)	64.2% (T)	Mixed

Table 4: Sample Prediction Analysis: Probability of Correct Class. T = correct prediction (True), F = incorrect prediction (False).

Parameter	Value
Datasets	MNIST, Fashion-MNIST, CIFAR-10, EMNIST, KMNIST
Image resolution	8 $\times$ 8 grayscale
Training samples	1000
Test samples	500
Number of classes	10
Random seeds	5 (42, 43, 44, 45, 46)

Table 5: Benchmark Experimental Settings

Metric	Formula	Range	Chance Level ( $C{=}10$ )
Accuracy	$\frac{\text{TP}+\text{TN}}{\text{TP}+\text{TN}+\text{FP}+\text{FN}}$	$[0,1]$	10%
Precision	$\frac{\text{TP}}{\text{TP}+\text{FP}}$	$[0,1]$	10%
Recall	$\frac{\text{TP}}{\text{TP}+\text{FN}}$	$[0,1]$	10%
F1 Score	$\frac{2\cdot\text{Prec}\cdot\text{Rec}}{\text{Prec}+\text{Rec}}$	$[0,1]$	10%
Cohen’s $\kappa$	$\frac{p_{o}-p_{e}}{1-p_{e}}$	$[-1,1]$	0%
MCC	$\frac{\text{TP}\cdot\text{TN}-\text{FP}\cdot\text{FN}}{\sqrt{(\text{TP}+\text{FP})(\text{TP}+\text{FN})(\text{TN}+\text{FP})(\text{TN}+\text{FN})}}$	$[-1,1]$	0%

Table 6: Evaluation Metrics Summary. For all metrics, higher values indicate better performance. The Chance Level column shows the expected value for a random classifier with

C=10

equiprobable classes.

Dataset	Method	Train Acc	Test Acc	Precision	Recall	F1	Kappa	MCC
MNIST	Classical	$81.6\pm 0.5$	$78.2\pm 1.9$	$78.7\pm 1.5$	$78.1\pm 1.7$	$77.8\pm 1.8$	$75.8\pm 1.8$	$75.9\pm 1.8$
	QUBO 5b	$37.2\pm 3.9$	$36.5\pm 4.2$	$35.9\pm 3.9$	$36.4\pm 3.7$	$35.5\pm 3.6$	$29.5\pm 4.1$	$29.6\pm 4.1$
	QUBO 10b	$76.5\pm 1.8$	$74.4\pm 2.9$	$74.7\pm 2.4$	$74.2\pm 2.8$	$74\pm 2.7$	$71.6\pm 2.9$	$71.6\pm 2.9$
	QUBO 15b	$83.7\pm 0.5$	$80.1\pm 1.9$	$80.3\pm 1.7$	$80\pm 1.7$	$79.8\pm 1.9$	$77.9\pm 1.8$	$78\pm 1.8$
	QUBO 20b	$\mathbf{84\pm 1}$	$\mathbf{81.3\pm 1.4}$	$\mathbf{81.6\pm 1.2}$	$\mathbf{81.1\pm 1.2}$	$\mathbf{80.9\pm 1.3}$	$\mathbf{79.2\pm 1.4}$	$\mathbf{79.3\pm 1.4}$
Fashion	Classical	$54.5\pm 0.9$	$52.1\pm 1.1$	$52.6\pm 2.3$	$51.8\pm 0.4$	$50\pm 1$	$46.7\pm 1$	$47.1\pm 1$

Table 7: Consolidated Benchmark Results (%, mean

\pm

std over 5 seeds). Best per dataset in bold.

Dataset	Method	0	1	2	3	4	5	6	7	8	9
MNIST	Classical	98 $\pm$ 1	71 $\pm$ 7	91 $\pm$ 4	80 $\pm$ 6	88 $\pm$ 3	65 $\pm$ 7	88 $\pm$ 1	73 $\pm$ 5	49 $\pm$ 6	78 $\pm$ 5
	QUBO 5b	51 $\pm$ 26	22 $\pm$ 11	31 $\pm$ 20	37 $\pm$ 14	38 $\pm$ 10	33 $\pm$ 11	48 $\pm$ 16	47 $\pm$ 13	22 $\pm$ 14	35 $\pm$ 20
	QUBO 10b	95 $\pm$ 5	61 $\pm$ 3	75 $\pm$ 12	75 $\pm$ 3	85 $\pm$ 3	68 $\pm$ 5	90 $\pm$ 2	71 $\pm$ 5	50 $\pm$ 7	73 $\pm$ 9
	QUBO 15b	98 $\pm$ 1	77 $\pm$ 7	92 $\pm$ 4	83 $\pm$ 3	87 $\pm$ 5	70 $\pm$ 6	88 $\pm$ 2	73 $\pm$ 5	57 $\pm$ 8	74 $\pm$ 7
	QUBO 20b	98 $\pm$ 2	78 $\pm$ 7	92 $\pm$ 3	84 $\pm$ 4	88 $\pm$ 3	73 $\pm$ 4	90 $\pm$ 2	76 $\pm$ 5	53 $\pm$ 8	78 $\pm$ 4
Fashion	Classical	64 $\pm$ 6	71 $\pm$ 12	36 $\pm$ 12	49 $\pm$ 15	27 $\pm$ 15	70 $\pm$ 10	20 $\pm$ 8	64 $\pm$ 20	53 $\pm$ 11	64 $\pm$ 12

Table 8: Per-Class Recall Across All Datasets (%, mean

\pm

std over 5 seeds). All datasets have

C=10

classes: MNIST/Fashion/CIFAR-10 use digits/items/objects 0–9; EMNIST uses letters A–J; KMNIST uses 10 hiragana characters.

Dataset	5-bit	10-bit	15-bit	20-bit
MNIST	$160\pm 5$	$448\pm 48$	$1,063\pm 127$	$2,449\pm 229$
Fashion	$157\pm 9$	$464\pm 50$	$1,128\pm 127$	$2,433\pm 241$
CIFAR-10	$157\pm 8$	$466\pm 49$	$1,118\pm 119$	$2,351\pm 192$
EMNIST	$157\pm 7$	$464\pm 49$	$1,125\pm 121$	$2,370\pm 199$
KMNIST	$148\pm 17$	$438\pm 55$	$1,014\pm 122$	$2,424\pm 234$

Table 9: Execution Time by Dataset and Bit Precision (seconds, mean

\pm

std over 5 seeds)

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Beyond Barren Plateaus: A Scalable Quantum Convolutional Architecture for High-Fidelity Image Classification

Radhakrishnan Delhibabu

School of Computing Science and Engineering (SCOPE), Vellore Institute of Technology, Vellore, 632014, India.

Quantum Convolutional Neural Networks Quantum Circuit Trainability Barren Plateaus Image Classification

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While Quantum Convolutional Neural Networks (QCNNs) offer a theoretical paradigm for quantum machine learning, their practical implementation is severely bottlenecked by barren plateaus -- the exponential vanishing of gradients -- and poor empirical accuracy compared to classical counterparts. In this work, we propose a novel QCNN architecture utilizing localized cost functions and a hardware-efficient tensor-network initialization strategy to provably mitigate barren plateaus. We evaluate our scalable QCNN on the MNIST dataset, demonstrating a significant performance leap. By resolving the gradient vanishing issue, our optimized QCNN achieves a classification accuracy of 98.7%, a substantial improvement over the baseline QCNN accuracy of 52.32% found in unmitigated models. Furthermore, we provide empirical evidence of a parameter-efficiency advantage, requiring

\mathcal{O}(\log N)

fewer trainable parameters than equivalent classical CNNs to achieve

>95\%

convergence. This work bridges the gap between theoretical quantum utility and practical application, providing a scalable framework for quantum computer vision tasks without succumbing to loss landscape concentration.

Abstract

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