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

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.

💡 Research Summary

The paper addresses the longstanding challenge of simultaneously handling data uncertainty and feature inter‑dependencies in image classification. Classical fuzzy neural networks (FNNs) excel at modeling uncertainty through membership functions but treat each feature independently, while deep learning architectures such as CNNs and Transformers capture hierarchical features and relationships at the cost of large memory and computational demands for high‑order joint distributions. The authors propose a novel quantum‑enhanced solution: the Dual‑Qubit Hierarchical Fuzzy Neural Network (DQ‑HFNN).

DQ‑HFNN extends the previously introduced quantum‑assisted hierarchical fuzzy neural network (QA‑HFNN), which encodes each feature into a single qubit, by pairing features (x_i, x_j) and encoding each pair onto two entangled qubits. The data‑encoding layer applies Ry rotations to map the raw pixel values onto the Bloch sphere, after which a trainable parameterized quantum circuit (PQC) containing single‑qubit rotations and CNOT entangling gates creates non‑classical correlations between the qubits. The joint measurement of Pauli‑Z operators yields a vector‑valued fuzzy membership f(x_i, x_j) that directly captures relational information.

The overall architecture comprises two parallel streams. The quantum fuzzy branch processes sampled feature pairs using the dual‑qubit circuit, aggregates the resulting fuzzy rules through a Σ‑layer, and produces a relational feature vector. Simultaneously, a conventional deep neural network (e.g., ResNet‑18) extracts high‑level semantic features from the full image. A fusion layer concatenates the quantum relational vector with the classical feature map, and a final classifier (softmax) predicts the image class.

A comprehensive ablation study explores seven circuit variants (A–G) differing in parameter count, symmetry, and entanglement strength, while keeping overall depth and total trainable parameters constant. Results demonstrate that circuits with entanglement consistently outperform their non‑entangled counterparts, confirming that the performance boost originates from relational modeling rather than mere expressivity gains.

Experimental evaluation on three benchmark datasets (CIFAR‑10, Fashion‑MNIST, Jaffe) shows that DQ‑HFNN achieves 2–4 % higher classification accuracy than QA‑HFNN and outperforms classical baselines (CNNs, Transformers, GNNs). Moreover, DQ‑HFNN requires roughly 30 % fewer parameters than QA‑HFNN for comparable performance and exhibits faster inference (≈ 45 % reduction in simulated runtime). Robustness tests under realistic quantum noise models (depolarizing and amplitude‑damping channels) reveal only marginal accuracy degradation up to 10 % gate error rates, indicating feasibility on noisy intermediate‑scale quantum (NISQ) hardware.

The authors acknowledge limitations: the current grid‑based pairing strategy is fixed and may not generalize to non‑image or irregular data; dynamic or graph‑based pairing could further enhance relational coverage. Additionally, circuit optimization for specific quantum hardware, error mitigation, and interpretability of the quantum‑fuzzy rules remain open research directions.

In summary, DQ‑HFNN introduces entanglement as a core mechanism for fuzzy relational learning, achieving superior accuracy, parameter efficiency, and noise resilience. This work represents a pioneering step toward practical quantum‑fuzzy hybrid models that combine uncertainty handling with relational reasoning, opening avenues for future AI systems that leverage quantum resources.