Quantum neural networks (QNNs) have been a promising framework in pursuing near-term quantum advantage in various fields, where many applications can be viewed as learning a quantum state that encodes useful data. As a quantum analog of probability distribution learning, quantum state learning is theoretically and practically essential in quantum machine learning. In this paper, we develop a no-go theorem for learning an unknown quantum state with QNNs even starting from a high-fidelity initial state. We prove that when the loss value is lower than a critical threshold, the probability of avoiding local minima vanishes exponentially with the qubit count, while only grows polynomially with the circuit depth. The curvature of local minima is concentrated to the quantum Fisher information times a loss-dependent constant, which characterizes the sensibility of the output state with respect to parameters in QNNs. These results hold for any circuit structures, initialization strategies, and work for both fixed ansatzes and adaptive methods. Extensive numerical simulations are performed to validate our theoretical results. Our findings place generic limits on good initial guesses and adaptive methods for improving the learnability and scalability of QNNs, and deepen the understanding of prior information's role in QNNs.

Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.

Effective and accurate diagnosis of diseases such as cancer, diabetes, and heart failure is crucial for timely medical intervention and improving patient survival rates. Machine learning has revolutionized diagnostic methods in recent years by developing classification models that detect diseases based on selected features. However, these classification tasks are often highly imbalanced, limiting the performance of classical models. Quantum models offer a promising alternative, exploiting their ability to express complex patterns by operating in a higher-dimensional computational space through superposition and entanglement. These unique properties make quantum models potentially more effective in addressing the challenges of imbalanced datasets. This work evaluates the potential of quantum classifiers in healthcare, focusing on Quantum Neural Networks (QNNs) and Quantum Support Vector Machines (QSVMs), comparing them with popular classical models. The study is based on three well-known healthcare datasets -- Prostate Cancer, Heart Failure, and Diabetes. The results indicate that QSVMs outperform QNNs across all datasets due to their susceptibility to overfitting. Furthermore, quantum models prove the ability to overcome classical models in scenarios with high dataset imbalance. Although preliminary, these findings highlight the potential of quantum models in healthcare classification tasks and lead the way for further research in this domain.

Artificial intelligence in dynamic, real-world environments requires the capacity for continual learning. However, standard deep learning suffers from a fundamental issue: loss of plasticity, in which networks gradually lose their ability to learn from new data. Here we show that quantum learning models naturally overcome this limitation, preserving plasticity over long timescales. We demonstrate this advantage systematically across a broad spectrum of tasks from multiple learning paradigms, including supervised learning and reinforcement learning, and diverse data modalities, from classical high-dimensional images to quantum-native datasets. Although classical models exhibit performance degradation correlated with unbounded weight and gradient growth, quantum neural networks maintain consistent learning capabilities regardless of the data or task. We identify the origin of the advantage as the intrinsic physical constraints of quantum models. Unlike classical networks where unbounded weight growth leads to landscape ruggedness or saturation, the unitary constraints confine the optimization to a compact manifold. Our results suggest that the utility of quantum computing in machine learning extends beyond potential speedups, offering a robust pathway for building adaptive artificial intelligence and lifelong learners.

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical neural networks, ensuring that every continuous function can be arbitrarily well approximated uniformly on a compact set of a Euclidean space, some recent works have established analogous results for QNNs, ranging from single-qubit to multi-qubit QNNs, and even hybrid classical-quantum models. In this paper, we study the approximation capabilities of QNNs for periodic functions with respect to the supremum norm. We use the Jackson inequality to approximate a given function by implementing its approximating trigonometric polynomial via a suitable QNN. In particular, we see that by restricting to the class of periodic functions, one can achieve a quadratic reduction of the number of parameters, producing better approximation results than in the literature. Moreover, the smoother the function, the fewer parameters are needed to construct a QNN to approximate the function.

The expressive power of a QNN model can be characterized by the function classes that it can approximate.

In this paper, we make progress in filling this gap by considering the common data encoding strategies based on parameterized quantum circuits.

Pneumonia poses a significant global health challenge, demanding accurate and timely diagnosis. While deep learning, particularly Convolutional Neural Networks (CNNs), has shown promise in medical image analysis for pneumonia detection, CNNs often suffer from high computational costs, limitations in feature representation, and challenges in generalizing from smaller datasets. To address these limitations, we explore the application of Quanvolutional Neural Networks (QNNs), leveraging quantum computing for enhanced feature extraction. This paper introduces a novel hybrid quantum-classical model for pneumonia detection using the PneumoniaMNIST dataset. Our approach utilizes a quanvolutional layer with a parameterized quantum circuit (PQC) to process 2x2 image patches, employing rotational Y-gates for data encoding and entangling layers to generate non-classical feature representations. These quantum-extracted features are then fed into a classical neural network for classification. Experimental results demonstrate that the proposed QNN achieves a higher validation accuracy of 83.33 percent compared to a comparable classical CNN which achieves 73.33 percent. This enhanced convergence and sample efficiency highlight the potential of QNNs for medical image analysis, particularly in scenarios with limited labeled data. This research lays the foundation for integrating quantum computing into deep-learning-driven medical diagnostic systems, offering a computationally efficient alternative to traditional approaches.

In the ever-evolving landscape of astrophysics and machine learning, understanding the internal kinematics of galaxies remains a formidable challenge. Traditional techniques for modeling galaxy dynamics have offered valuable insights but are often limited by their inability to capture complex, non-linear relationships in high-dimensional data. Recent advances in quantum computing and explainable artificial intelligence (XAI) provide new avenues for addressing these challenges, paving the way for more sophisticated and interpretable models in astrophysical research [19] [20] [21]. Galaxy velocity dispersion is a critical parameter that underpins our understanding of the mass distribution, dynamical state, and evolutionary history of galaxies. By analyzing detailed stellar population and kinematic properties--such as morphological classification, effective radius, and gradients in stellar age and metallicity, the prediction of velocity dispersion becomes central to characterizing the intricate interplay between a galaxy's structure and its dynamic behavior. The MaNGA dataset, with its rich set of 11 features, offers a robust platform for exploring these phenomena and highlights the technical demands of achieving accurate predictions in this domain [1].