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arXiv:2508.02054

Enhancement of Quantum Semi-Supervised Learning via Improved Laplacian and Poisson Methods

Hamed Gholipour, Farid Bozorgnia, Hamzeh Mohammadigheymasi, Kailash Hambarde, Javier Mancilla, Hugo Pedro Martins Carriço Proença, João C. Neves, Moharram Challenger

Published Aug 4, 2025• 0 citations
arXivHTMLPDF

Abstract

This paper develops a hybrid quantum approach for graph-based semi-supervised learning to enhance performance in scenarios where labeled data is scarce. We introduce two enhanced quantum models, the Improved Laplacian Quantum Semi-Supervised Learning (ILQSSL) and the Improved Poisson Quantum Semi-Supervised Learning (IPQSSL), that incorporate advanced label propagation strategies within variational quantum circuits. These models utilize QR decomposition to embed graph structure directly into quantum states, thereby enabling more effective learning in low-label settings. We validate our methods across four benchmark datasets like Iris, Wine, Heart Disease, and German Credit Card -- and show that both ILQSSL and IPQSSL consistently outperform leading classical semi-supervised learning algorithms, particularly under limited supervision. Beyond standard performance metrics, we examine the effect of circuit depth and qubit count on learning quality by analyzing entanglement entropy and Randomized Benchmarking (RB). Our results suggest that while some level of entanglement improves the model's ability to generalize, increased circuit complexity may introduce noise that undermines performance on current quantum hardware. Overall, the study highlights the potential of quantum-enhanced models for semi-supervised learning, offering practical insights into how quantum circuits can be designed to balance expressivity and stability. These findings support the role of quantum machine learning in advancing data-efficient classification, especially in applications constrained by label availability and hardware limitations.

DOI
10.48550/arxiv.2508.02054

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This paper develops a hybrid quantum approach for graph-based semi-supervised learning to enhance performance in scenarios where labeled data is scarce. We introduce two enhanced quantum models, the Improved Laplacian Quantum Semi-Supervised Learning (ILQSSL) and the Improved Poisson Quantum Semi-Supervised Learning (IPQSSL), that incorporate advanced label propagation strategies within variational quantum circuits. These models utilize QR decomposition to embed graph structure directly into quantum states, thereby enabling more effective learning in low-label settings. We validate our methods across four benchmark datasets like Iris, Wine, Heart Disease, and German Credit Card -- and show that both ILQSSL and IPQSSL consistently outperform leading classical semi-supervised learning algorithms, particularly under limited supervision. Beyond standard performance metrics, we examine the effect of circuit depth and qubit count on learning quality by analyzing entanglement entropy and Randomized Benchmarking (RB). Our results suggest that while some level of entanglement improves the model's ability to generalize, increased circuit complexity may introduce noise that undermines performance on current quantum hardware. Overall, the study highlights the potential of quantum-enhanced models for semi-supervised learning, offering practical insights into how quantum circuits can be designed to balance expressivity and stability. These findings support the role of quantum machine learning in advancing data-efficient classification, especially in applications constrained by label availability and hardware limitations.
Laplacian learning method is a well-established technique in classical graph-based semi-supervised learning, but its potential in the quantum domain remains largely unexplored. This study investigates the performance of the Laplacian-based Quantum Semi-Supervised Learning (QSSL) method across four benchmark datasets -- Iris, Wine, Breast Cancer Wisconsin, and Heart Disease. Further analysis explores the impact of increasing Qubit counts, revealing that adding more Qubits to a quantum system doesn't always improve performance. The effectiveness of additional Qubits depends on the quantum algorithm and how well it matches the dataset. Additionally, we examine the effects of varying entangling layers on entanglement entropy and test accuracy. The performance of Laplacian learning is highly dependent on the number of entangling layers, with optimal configurations varying across different datasets. Typically, moderate levels of entanglement offer the best balance between model complexity and generalization capabilities. These observations highlight the crucial need for precise hyperparameter tuning tailored to each dataset to achieve optimal performance in Laplacian learning methods.
Abstract The Laplacian learning method has proven effective in classical graph-based semi-supervised learning, yet its quantum counterpart remains underexplored. This study systematically evaluates the Laplacian-based quantum semi-supervised learning (QSSL) approach across four benchmark datasets—Iris, Wine, Breast Cancer Wisconsin, and Heart Disease. By experimenting with varying qubit counts and entangling layers, we demonstrate that increased quantum resources do not necessarily lead to improved performance. Our findings reveal that the effectiveness of the method is highly sensitive to dataset characteristics, as well as the number of entangling layers. Optimal configurations, generally featuring moderate entanglement, strike a balance between model complexity and generalization. These results emphasize the importance of dataset-specific hyperparameter tuning in quantum semi-supervised learning frameworks.
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Label propagation is an essential semi-supervised learning method based on graphs, which has a broad spectrum of applications in pattern recognition and data mining. This paper proposes a quantum semi-supervised classifier based on label propagation. Considering the difficulty of graph construction, we develop a variational quantum label propagation (VQLP) method. In this method, a locally parameterized quantum circuit is created to reduce the parameters required in the optimization. Furthermore, we design a quantum semi-supervised binary classifier based on hybrid Bell and Z bases measurement, which has a shallower circuit depth and is more suitable for implementation on near-term quantum devices. We demonstrate the performance of the quantum semi-supervised classifier on the Iris data set, and the simulation results show that the quantum semi-supervised classifier has higher classification accuracy than the swap test classifier. This work opens a new path to quantum machine learning based on graphs.
Semi-supervised learning (SSL) has emerged as a critical paradigm for leveraging abundant unlabeled data alongside scarce labeled samples, yet classical approaches often struggle in sparse label regimes where the ratio of labeled to unlabeled data is extremely low. Meanwhile, quantum machine learning (QML) has demonstrated that quantum feature spaces can induce rich kernel structures with provable representational advantages, but these methods typically require quantum hardware that remains inaccessible for practical deployment. In this work, we propose a quantum-inspired semi-supervised learning framework that harnesses the geometric properties of quantum Hilbert space embeddings while remaining entirely classical and hardware-independent. Our approach, which we term Quantum-Inspired SemiSupervised Learning (QISSL), constructs feature maps that mimic the action of quantum encoding circuits via classically simulable kernel functions derived from randomized trigonometric expansions. We provide rigorous theoretical analysis establishing: (1) a bounded approximation guarantee between our classical embeddings and their quantum counterparts; (2) a generalization bound for QISSL under low-label regimes via Rademacher complexity analysis; and (3) a convergence proof for the alternating optimization procedure that jointly trains the embedding and classifier modules. Extensive experiments on benchmark datasets demonstrate that QISSL achieves state-of-the-art performance in extreme low-label settings (0.1%–1% labeled data), outperforming strong baselines including FixMatch, MixMatch, and FlexMatch by margins of 2–7% in accuracy. Our results suggest that the inductive biases of quantum feature spaces—when properly classically approximated—offer a powerful and practical tool for semi-supervised learning without quantum hardware.
This paper introduces an innovative theoretical framework for quantum-inspired data embeddings, grounded in foundational concepts of quantum mechanics such as superposition and entanglement. This approach aims to advance semi-supervised learning in contexts characterized by limited labeled data by enabling more intricate and expressive embeddings that capture the underlying structure of the data effectively. Grounded in foundational quantum mechanics concepts such as superposition and entanglement, this approach redefines data representation by enabling more intricate and expressive embeddings. Emulating quantum superposition encodes each data point as a probabilistic amalgamation of multiple feature states, facilitating a richer, multidimensional representation of underlying structures and patterns. Additionally, quantum-inspired entanglement mechanisms are harnessed to model intricate dependencies between labeled and unlabeled data, promoting enhanced knowledge transfer and structural inference within the learning paradigm. In contrast to conventional quantum machine learning methodologies that often rely on quantum hardware, this framework is fully realizable within classical computational architectures, thus bypassing the practical limitations of quantum hardware. The versatility of this model is illustrated through its application to critical domains such as medical diagnosis, resource-constrained natural language processing, and financial forecasting—areas where data scarcity impedes the efficacy of traditional models. Experimental evaluations reveal that quantum-inspired embeddings substantially outperform standard approaches, enhancing model resilience and generalization in high-dimensional, low-sample scenarios. This research marks a significant stride in integrating quantum theoretical principles with classical machine learning, broadening the scope of data representation and semi-supervised learning while circumventing the technological barriers of quantum computing infrastructure.