It remains unclear whether quantum machine learning (QML) has real advantages when dealing with practical and meaningful tasks. Encoding classical data into quantum states is one of the key steps in QML. Amplitude encoding has been widely used owing to its remarkable efficiency in encoding a number of $2^{n}$ classical data into $n$ qubits simultaneously. However, the theoretical impact of amplitude encoding on QML has not been thoroughly investigated. In this work we prove that under some broad and typical data assumptions, the average of encoded quantum states via amplitude encoding tends to concentrate towards a specific state. This concentration phenomenon severely constrains the capability of quantum classifiers as it leads to a loss barrier phenomenon, namely, the loss function has a lower bound that cannot be improved by any optimization algorithm. In addition, via numerical simulations, we reveal a counterintuitive phenomenon of amplitude encoding: as the amount of training data increases, the training error may increase rather than decrease, leading to reduced decrease in prediction accuracy on new data. Our results highlight the limitations of amplitude encoding in QML and indicate that more efforts should be devoted to finding more efficient encoding strategies to unlock the full potential of QML.
It remains unclear whether quantum machine learning (QML) has real advantages when dealing with practical and meaningful tasks. Encoding classical data into quantum states is one of the key steps in QML. Amplitude encoding has been widely used owing to its remarkable efficiency in encoding a number of 2n classical data into n qubits simultaneously. However, the theoretical impact of amplitude encoding on QML has not been thoroughly investigated. In this work we prove that under some broad and typical data assumptions, the average of encoded quantum states via amplitude encoding tends to concentrate towards a specific state. This concentration phenomenon severely constrains the capability of quantum classifiers as it leads to a loss barrier phenomenon, namely, the loss function has a lower bound that cannot be improved by any optimization algorithm. In addition, via numerical simulations, we reveal a counterintuitive phenomenon of amplitude encoding: as the amount of training data increases, the training error may increase rather than decrease, leading to reduced decrease in prediction accuracy on new data. Our results highlight the limitations of amplitude encoding in QML and indicate that more efforts should be devoted to finding more efficient encoding strategies to unlock the full potential of QML.
The evolution of quantum computers has encouraged research into how to handle tasks with significant computation demands in the past few years. Due to the unique advantages of quantum parallelism and entanglement, various types of quantum machine learning (QML) methods, especially variational quantum classifiers (VQCs), have attracted the attention of many researchers and have been developed and evaluated in numerous scenarios. Nevertheless, most of the research on VQCs is still in its early stages. For instance, as a consequence of the mathematical constraints imposed by the properties of quantum states, the majority of research has not fully taken into account the impact of data formats on the performance of VQCs. In this paper, considering a significant number of data in the real world exist in the form of complex numbers, i.e., phasor data in power systems and the result of Fourier transform on image processing, we develop two categories of data encoding methods, including coupling data encoding and splitting data encoding. This paper features the coupling data encoding method to encode complex-valued data in a way of amplitude encoding. By leveraging the property of quantum states living in a complex Hilbert space, the complex-valued data is embedded into the amplitude of quantum states to comprehensively characterize complex-valued information. Optimizers will be utilized to iteratively tune a parameterized ansatz, with the aim of minimizing the value of loss functions defined with respect to the specific classification task. In addition, distinct factors in VQCs have been explored in detail to investigate the performance of VQCs, including data encoding methods, loss functions, and optimizers. The experimental result shows that the proposed data encoding method outperforms other typical encoding methods on a given classification task. Moreover, different loss functions are tested, and the capability of finding the minimum value is evaluated for gradient-free and gradient-based optimizers, which provides valuable insights and guidelines for practical implementations.
Learning methodologies on quantum devices have shown that there are advantages in utilizing quantum properties. A requirement for using quantum computing in machine learning techniques is the data representation as quantum states. In Quantum Machine Learning, quantum state preparation is paramount to attain a functional pipeline in a model. One state preparation method, amplitude encoding, allows a dataset to be mapped or encoded more robustly and enhances the learning of quantum models. Albeit more densely represented, a dataset which has been prepared by amplitude encoding provides a more learnable input to a model. The two main advantages from using amplitude encoding are an increase in classification accuracy and reduced variability of learning epoch to epoch. In this paper, we compare the basic implementations of TensorFlow Quantum's Quantum Convolutional Neural Network and a hybrid quantum-classical network using angle encoding, with a third network of our design that utilizes amplitude encoding for enriched state preparation. Our results show there is a direct benefit in performing amplitude encoding before training a TensorFlow Quantum hybrid quantum-classical model. In the best case scenario, amplitude encoding made classifying the samples 8.9% more accurate.
Understanding the theoretical capabilities and limitations of quantum machine learning (QML) models to solve machine learning tasks is crucial to advancing both quantum software and hardware developments. Similarly to the classical setting, the performance of QML models can be significantly affected by the limited access to the underlying data set. Previous studies have focused on proving generalization error bounds for any QML models trained on a limited finite training set. We focus on the optimal QML models obtained by training them on a finite training set and establish a tight prediction error bound in terms of the number of trainable gates and the size of training sets. To achieve this, we derive covering number upper bounds and packing number lower bounds for the data re-uploading QML models and linear QML models, respectively, which may be of independent interest. We support our theoretical findings by numerically simulating the QML strategies for function approximation and quantum phase recognition.
When it comes to realizing immediate quantum advantages in practical applications like machine learning and optimization techniques, numerous methods have emerged as a frontrunner technique. Such algorithms often involve quantum circuits for data encoding and the training of Quantum Neural Networks (QNNs) to minimize target functions when applied to tasks involving classical data. Quantum Machine Learning (QML) is a promising area of quantum computing because it may effectively handle difficult learning tasks by taking advantage of the high dimensional Hilbert space to train better representations from limited data. Despite QML's rising popularity, few academic works have addressed the language's security features. There has been a lot of buzz about the promising new discipline of quantum machine learning. Training a parameterized quantum circuit is a common practice in contemporary QML for performing data analysis on classical or quantum datasets. To make accurate predictions on an independent test dataset, state-of-the-art QML techniques first variationally optimize a parameterized quantum circuit using the training dataset. Quantum modeling in QML has also been shown to offer an exponential benefit in sample complexity. In the realms of pattern recognition, machine learning, and data mining, feature dimensionality reduction as a crucial link in the process has become one hot and tough spot. Most academics have gravitated towards this topic because of the difficulty of its studies. The goal of this research is to learn how to implement low loss in the procedure of feature dimension reduction, preserve the integrity of the original data, locate the most effective mapping, and obtain the best possible low-dimensional data. Preliminary findings suggest that QML models may offer advantages over classical models for classical data analysis. In this paper, a thorough investigation into generalization performance in QML is performed. This paper presents a brief survey on Qubits Encoding using Quantum Machine Learning models that help researchers to identify the limitations of the existing models and to design new efficient models for better performance levels.
The encoding of classical data into quantum states constitutes the primary performance bottleneck in Quantum Machine Learning (qml) on Noisy Intermediate-Scale Quantum (nisq) devices. No existing framework jointly characterises resource cost, expressivity, and noise robustness, nor provides actionable selection guidelines for practitioners. This survey addresses that gap through a systematic review of 66 primary works (2017-2026) assembled via a PRISMA-adapted protocol across five academic databases. Four principal contributions are made. First, a three-axis cost-expressivity-robustness taxonomy classifies all major encoding families - basis, angle, dense-angle, amplitude, data re-uploading, and IQP - along independently measurable axes. Second, closed-form depth-fidelity bounds under nisq decoherence channels identify the critical gate-error rate p* ~ 10^-3 below which amplitude encoding is viable. Third, a unified treatment of Fourier expressivity, barren-plateau onset, and quantum kernel concentration as functions of the encoding circuit provides the first joint trainability analysis. Fourth, a five-regime decision framework maps (D, n, p, tau) - feature dimension, qubit budget, error rate, and task type - to a hardware-grounded encoding recommendation. The central finding is that for p >= 10^-3, shallow angle-based encodings consistently outperform amplitude encoding in practice, despite the latter's exponential qubit advantage.
As the limits of classical machine learning (ML) become increasingly evident—especially in terms of computational scalability and data processing—quantum computing emerges as a promising frontier capable of addressing these challenges. Quantum Machine Learning (QML) represents the intersection of quantum computing and traditional ML, aiming to leverage quantum phenomena such as superposition, entanglement, and quantum parallelism to enhance learning capabilities, accelerate training, and solve complex problems that are intractable for classical algorithms. This paper provides a comprehensive overview of the evolving field of QML, beginning with the theoretical foundations of quantum computing and its synergy with classical machine learning principles. We explore key QML algorithms, including quantum support vector machines (QSVM), variational quantum classifiers (VQC), quantum neural networks (QNN), and hybrid quantum-classical approaches that exemplify the practical integration of both paradigms. We further examine real-world applications of QML across domains such as drug discovery, finance, climate modeling, and optimization, where quantum enhancements have begun to show early but promising results. Despite its theoretical potential, QML faces substantial limitations, including quantum hardware instability, limited qubit scalability, noise sensitivity, and unresolved challenges in quantum data encoding and interpretability. In synthesizing recent advancements and persistent obstacles, this paper highlights the importance of continued interdisciplinary research, algorithmic refinement, and hardware evolution. We also outline future directions that could lead QML from experimental promise to practical utility, including developments in quantum error correction, benchmarking standards, and application-specific quantum architectures