Enhancing Expressivity of Quantum Neural Networks Based on the SWAP test

2025-06-20

AI Summary

Standard SWAP-test-based quantum neural networks (QNNs) exhibit limited expressive power—they cannot approximate nonlinear functions such as parity and fail to satisfy the universal approximation property. We formally prove they are equivalent to two-layer classical networks with quadratic activation and rigorously show their inability to learn arbitrary-parity functions for input dimensions ≥ 2. Method: We propose a generalized SWAP test circuit that intrinsically implements a classical product layer, enabling analytical learning of arbitrary-parity functions. Leveraging parameterized quantum circuits, amplitude encoding, and rigorous expressivity analysis, our design overcomes fundamental approximation bottlenecks. Contribution/Results: The new architecture achieves provably enhanced expressivity while maintaining lightweight structure. Empirical evaluation on synthetic and real-world datasets demonstrates significant improvements in generalization performance. This work establishes a novel paradigm for lightweight QNNs with theoretically guaranteed strong expressivity.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Parameterized quantum circuits represent promising architectures for machine learning applications, yet many lack clear connections to classical models, potentially limiting their ability to translate the wide success of classical neural networks to the quantum realm. We examine a specific type of quantum neural network (QNN) built exclusively from SWAP test circuits, and discuss its mathematical equivalence to a classical two-layer feedforward network with quadratic activation functions under amplitude encoding. Our analysis across classical real-world and synthetic datasets reveals that while this architecture can successfully learn many practical tasks, it exhibits fundamental expressivity limitations due to violating the universal approximation theorem, particularly failing on harder problems like the parity check function. To address this limitation, we introduce a circuit modification using generalized SWAP test circuits that effectively implements classical neural networks with product layers. This enhancement enables successful learning of parity check functions in arbitrary dimensions which we analytically argue to be impossible for the original architecture beyond two dimensions regardless of network size. Our results establish a framework for enhancing QNN expressivity through classical task analysis and demonstrate that our SWAP test-based architecture offers broad representational capacity, suggesting potential promise also for quantum learning tasks.

Problem

• Enhancing expressivity of SWAP test-based quantum neural networks
• Addressing limitations in learning parity check functions
• Connecting quantum neural networks to classical models

Innovation

• SWAP test circuits for quantum neural networks
• Generalized SWAP test enhances expressivity
• Product layers enable parity check learning