Dual-Qubit Hierarchical Fuzzy Neural Network for Image Classification: Enabling Relational Learning via Quantum Entanglement

TL;DR

The paper tackles uncertainty and relational learning in image classification by introducing the Dual-Qubit Hierarchical Fuzzy Neural Network (DQ-HFNN), which encodes feature pairs on entangled qubits to learn joint memberships $f_{oldsymbol{\theta}}(x_i, x_j)$. It combines a quantum fuzzy branch with a classical DNN branch, uses a hybrid pairing strategy to capture multi-scale relations, and fuses the two representations for final classification. Ablation studies show gains mainly arise from relational modeling enabled by entanglement rather than mere expressivity, with high parameter efficiency and robustness to simulated quantum noise. The approach demonstrates competitive accuracy on benchmarks and offers practical potential for NISQ devices, suggesting a viable path toward quantum-enhanced relational learning in vision tasks.

Abstract

Classical deep neural network models struggle to represent data uncertainty and capture dependencies between features simultaneously, especially under fuzzy or noisy conditions. Although a quantum-assisted hierarchical fuzzy neural network (QA-HFNN) was proposed to learn fuzzy membership for each feature, it cannot model dependencies between features due to its single-qubit encoding. To address this, this paper proposes a dual-qubit hierarchical fuzzy neural network (DQ-HFNN), encoding feature pairs onto a pair of entangled qubits, which extends the single-feature fuzzy model to a joint fuzzy representation. By introducing quantum entanglement, the dual-qubit circuit can encode non-classical correlations, enabling the model to directly learn relationship patterns between feature pairs. Experiments on benchmarks show that DQ-HFNN demonstrates higher classification accuracy than QA-HFNN, as well as classical deep learning baselines. Furthermore, ablation studies after controlling for circuit depth and parameter counts show that the performance gain mainly stems from the relational modeling capability enabled by entanglement rather than enhanced expressivity. The proposed DQ-HFNN model exhibits high parameter efficiency and fast inference speed. Experiments under noisy conditions suggest that it is robust against noise and has the potential to be implemented on noisy intermediate-scale quantum devices.

Figures (10)

• Figure 1: The overall architecture of the proposed dual-qubit hierarchical fuzzy neural Network (DQ-HFNN). The model consists of two parallel branches: a quantum fuzzy branch (top path) and a classical deep neural network (DNN) branch (bottom path). The quantum branch employs a pairing strategy to select feature pairs (e.g., pixels $p_i, p_j$), which are processed by class-specific dual-qubit circuits. The outputs are then aggregated by a fuzzy rule layer ($\Sigma^n$). Concurrently, the classical branch extracts high-level features using a standard DNN. Features from both branches are integrated in a fusion layer and passed to a final classifier for prediction.
• Figure 2: Quantum circuit designs for the trainable block $\boldsymbol{U}_q(\boldsymbol{\theta})$ used in the ablation study. The figure illustrates seven distinct dual-qubit architectures (A--G) with varying degrees of parameterization (P), symmetry (Sym/Asym), and entanglement (Ent). The yellow boxes represent the data encoding block ($\boldsymbol{U}_{\text{enc}}$), where input features $x_0$ and $x_1$ are encoded using $\boldsymbol{R}_y$ gates. The green boxes represent the trainable circuit blocks ($\boldsymbol{U}_q$), which consist of single-qubit rotation gates (e.g., $\boldsymbol{R}_x, \boldsymbol{R}_z$) and entangling CNOT gates. The final block ($\Pi$) denotes the joint measurement of Pauli-Z operators on both qubits.
• Figure 3: Illustration of the end-to-end workflow of the DQ-HFNN model on a sample image from the JAFFE lyons1998codinglyons2021excavating dataset. The model operates in two parallel streams. In the quantum stream (top), a grid-based pairing strategy is applied to the input image, sampling pixel pairs (e.g., $\textit{Pair}_1, \dots, \textit{Pair}_m$) to capture multi-scale spatial relationships. These pairs are then processed by the quantum fuzzy logic branch. In the classical stream (bottom), the full image is processed by a CNN to extract semantic features. Finally, the outputs from both streams are fused and classified.
• Figure 4: Training and validation curves for the DQ-HFNN model on the Dirty-MNIST dataset. The training loss decreases smoothly, with a scheduled drop at epoch 56 corresponding to learning rate decay. Both training and validation accuracies rise steadily and reach a stable plateau, demonstrating effective optimization.
• Figure 5: Multi-dimensional analysis of four distinct pairing strategies (a--d) on the JAFFE dataset. (a) Mutual information between features (Quantum, Classical, Combined) and labels. (b) Information redundancy between quantum and classical branches. Lower values indicate more complementary feature learning.(c) Information contribution of fixed vs. random pairs. (d) Fused feature quality measured by Linear Probe Accuracy. (e, f) Robustness of features and predictions against input noise (Gaussian noise with $\sigma = 0.05$). The robustness scores are computed by subtracting the raw distance/divergence metrics from their maximum values across all configurations, such that higher scores indicate better robustness. The best-performing strategy (c) (153 random pairs) demonstrates a favorable balance of information quality and prediction robustness.