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.
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.
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.
Quantum machine learning (QML) has attracted growing interest with the rapid parallel advances in large-scale classical machine learning and quantum technologies. Similar to classical machine learning, QML models also face challenges arising from the scarcity of labeled data, particularly as their scale and complexity increase. Here, we introduce self-supervised pretraining of quantum representations that reduces reliance on labeled data by learning invariances from unlabeled examples. We implement this paradigm on a programmable trapped-ion quantum computer, encoding images as quantum states. In situ contrastive pretraining on hardware yields a representation that, when fine-tuned, classifies image families with higher mean test accuracy and lower run-to-run variability than models trained from random initialization. Performance improvement is especially significant in regimes with limited labeled training data. We show that the learned invariances generalize beyond the pretraining image samples. Unlike prior work, our pipeline derives similarity from measured quantum overlaps and executes all training and classification stages on hardware. These results establish a label-efficient route to quantum representation learning, with direct relevance to quantum-native datasets and a clear path to larger classical inputs.
Machine learning is widely regarded as a promising application in which quantum computing could offer a computational advantage. However, general quantum machine learning models with provable guarantees on performance, trainability, and efficient inference remain limited. In this work, we introduce quantum Hamiltonian embeddings, a supervised learning framework in which a classical training dataset is encoded into a single effective, parameterized Hamiltonian. Training optimizes the parameters of this data-dependent Hamiltonian, whose approximate ground state defines the classification rule. Once the ground state has been approximated using a quantum algorithm, inference can be performed either classically using a classical representation of the learned state, or with a quantum representation of the state. We establish efficient methods for gradient estimation and prove favorable generalization guarantees. Furthermore, under standard cryptographic assumptions, we show that Hamiltonian embeddings can realize hypothesis functions beyond the reach of efficient classical algorithms. We then demonstrate the framework at scale on IBM Heron processors using a standard benchmark dataset. Together, these results establish Hamiltonian embeddings as a new framework for quantum machine learning, combining theoretical guarantees with a path for experimental investigations at scale on quantum hardware.
Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.
Quantum machine learning, as an extension of classical machine learning that harnesses quantum mechanics, facilitates effiient learning from data encoded in quantum states. Training a quantum neural network typically demands a substantial labeled training set for supervised learning. Human annotators, often experts, provide labels for samples through additional experiments, adding to the training cost. To mitigate this expense, there is a quest for methods that maintain model performance over fully labeled datasets while requiring fewer labeled samples in practice, thereby extending few-shot learning to the quantum realm. Quantum active learning estimates the uncertainty of quantum data to select the most informative samples from a pool for labeling. Consequently, a QML model is supposed to accumulate maximal knowledge as the training set comprises labeled samples selected via sampling strategies. Notably, the QML models trained within the QAL framework are not restricted to specific types, enabling performance enhancement from the model architecture's perspective towards few-shot learning. Recognizing symmetry as a fundamental concept in physics ubiquitous across various domains, we leverage the symmetry inherent in quantum states induced by the embedding of classical data for model design. We employ an equivariant QNN capable of generalizing from fewer data with geometric priors. We benchmark the performance of QAL on two classification problems, observing both positive and negative results. QAL effectively trains the model, achieving performance comparable to that on fully labeled datasets by labeling less than 7\% of the samples in the pool with unbiased sampling behavior. Furthermore, we elucidate the negative result of QAL being overtaken by random sampling baseline through miscellaneous numerical experiments. (character count limit, see the main text)