In this paper, we study quantum generating functions (QGFs) as algorithmic primitives for quantum computation with a significant focus on machine learning. We develop a systematic formulation for encoding sequences and functions into quantum states, enabling efficient quantum algorithms for evaluation, transformation, and analysis.We demonstrate the utility of QGFs across three key areas of machine learning: i) quantum activation functions with efficient, explicit circuit implementations; ii) quantum kernel methods via generating function representations of feature maps; iii) quantum Kolmogorov-Arnold networks (KANs) for function approximation. For each area we provide proof-of-concept numerical simulations of the proposed circuits. We analyze circuit complexities and applications, offering a foundation for quantum machine learning. The quantum implementations of generating functions for structured function classes consist of simple quantum rotations. Therefore, they can provide efficient algorithmic primitives to design machine learning methods. At the same time, we are explicit about the scope of this efficiency: the simplest primitives are product-state circuits with classically computable angles, which are classically simulable, so a quantum advantage can only materialize when the constructions are embedded in coherent pipelines with entangling processing; we state these limitations alongside the corresponding results. Throughout the paper, complexity statements concern the core quantum computation; the costs of initial data loading and final classical readout are stated separately where relevant.
In this paper, we study quantum generating functions (QGFs) as algorithmic primitives for quantum computation with a significant focus on machine learning. We develop a systematic formulation for encoding sequences and functions into quantum states, enabling efficient quantum algorithms for evaluation, transformation, and analysis.We demonstrate the utility of QGFs across three key areas of machine learning: i) quantum activation functions with efficient, explicit circuit implementations; ii) quantum kernel methods via generating function representations of feature maps; iii) quantum Kolmogorov-Arnold networks (KANs) for function approximation. For each area we provide proof-of-concept numerical simulations of the proposed circuits. We analyze circuit complexities and applications, offering a foundation for quantum machine learning. The quantum implementations of generating functions for structured function classes consist of simple quantum rotations. Therefore, they can provide efficient algorithmic primitives to design machine learning methods. At the same time, we are explicit about the scope of this efficiency: the simplest primitives are product-state circuits with classically computable angles, which are classically simulable, so a quantum advantage can only materialize when the constructions are embedded in coherent pipelines with entangling processing; we state these limitations alongside the corresponding results. Throughout the paper, complexity statements concern the core quantum computation; the costs of initial data loading and final classical readout are stated separately where relevant.
The goal of generative machine learning is to model the probability distribution underlying a given data set. This probability distribution helps to characterize the generation process of the data samples. While classical generative machine learning is solely based on classical resources, generative quantum machine learning can also employ quantum resources - such as parameterized quantum channels and quantum operators - to learn and sample from the probability model of interest. Applications of generative (quantum) models are multifaceted. The trained model can generate new samples that are compatible with the given data and extend the data set. Additionally, learning a model for the generation process of a data set may provide interesting information about the corresponding properties. With the help of quantum resources, the respective generative models have access to functions that are difficult to evaluate with a classical computer and may improve the performance or lead to new insights. Furthermore, generative quantum machine learning can be applied to efficient, approximate loading of classical data into a quantum state which may help to avoid - potentially exponentially - expensive, exact quantum data loading. The aim of this doctoral thesis is to develop new generative quantum machine learning algorithms, demonstrate their feasibility, and analyze their performance. Additionally, we outline their potential application to efficient, approximate quantum data loading. More specifically, we introduce a quantum generative adversarial network and a quantum Boltzmann machine implementation, both of which can be realized with parameterized quantum circuits. These algorithms are compatible with first-generation quantum hardware and, thus, enable us to study proof of concept implementations not only with numerical quantum simulations but also real quantum hardware available today.
This paper explores a novel approach to feature extraction and representation learning within the context of quantum machine learning. The core claim is to enhance the performance and efficiency of machine learning algorithms by leveraging quantum computation for these foundational tasks. We introduce a methodology that utilizes quantum algorithms to transform raw data into representations optimized for subsequent machine learning processes. This represents a new direction in the burgeoning field of quantum machine learning, offering potential advantages over classical methods, particularly for complex datasets and computationally intensive tasks. The paper details the theoretical framework, outlines the key components of the proposed system, and discusses potential applications and future research directions. The primary focus is on the algorithmic design and mathematical foundations underpinning this approach, aiming to provide a solid basis for future development and experimentation.
Recent advances in Machine Learning have transformed numerous industrial sectors, yet classical paradigms face fundamental limitations: rapidly growing data volumes, rising computational costs, significant energy consumption, and the physical scaling limits of conventional hardware architectures. Quantum computing has emerged as a promising computational paradigm to address these challenges, giving rise to the field of Quantum Machine Learning (QML). In this thesis, the theoretical foundations of QML are investigated, with a focus on near-term and future practical applications. Three central challenges are addressed: the trainability of variational quantum circuits, their expressivity, and their resistance to efficient classical simulation. The trainability of Hamming-weight preserving variational quantum circuits is first studied, and theoretical guarantees are established that resolve an open conjecture on the absence of barren plateaus for this circuit family. Subspace-preserving QML algorithms are then introduced, including photonic circuits and quantum convolutional neural networks, and are designed to mimic classical ML subroutines while offering polynomial quantum advantage. Finally, variational quantum circuits are analyzed as quantum Fourier models, and a framework is derived to jointly characterize expressivity and trainability, from which conditions are obtained under which quantum models provably separate from their classical counterparts. These contributions are intended to advance the theoretical roadmap for harnessing near-term and future quantum technologies in real-world applications.
This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.
Abstract Quantum Machine Learning has emerged as a promising field at the intersection of Quantum Computing and Artificial Intelligence, offering the potential to achieve computational advantages over classical machine learning approaches. While classical ML has matured into a robust ecosystem with broad applications, QML remains nascent, constrained by hardware limitations, yet fueled by rapid theoretical and algorithmic developments. This paper presents a comprehensive comparative review of QML and classical ML, focusing on their foundations, complexity, and applications across domains. We analyze where QML models have shown advantages, where they lag behind, and try to provide possible explanations for these results. By synthesizing evidence across neural networks, kernel methods, generative models, representation learning, and clustering, we highlight patterns that indicate when quantum models tend to outperform or underperform classical baselines. We also review real-world applications of QML across diverse domains such as materials science, energy forecasting, healthcare and bioinformatics, transportation, and finance. Finally, we identify open challenges including state preparation, scalability, and benchmarking standards, and propose a roadmap for the development of QML into a field with demonstrable advantages.
Quantum Machine Learning (QML) represents an emerging interdisciplinary field that harnesses quantum computing principles to enhance machine learning algorithms and develop quantum-native learning paradigms. This convergence exploits fundamental quantum mechanical phenomena superposition, entanglement, and quantum interference—to potentially achieve exponential computational advantages over classical approaches for specific algorithmic tasks. The theoretical foundation of QML rests on quantum systems' ability to encode information in exponentially scaling state spaces, enabling complex data representation in logarithmically fewer qubits compared to classical bits. Key quantum advantage mechanisms include quantum parallelism through superposition states, entanglement based correlations for efficient encoding of data relationships, and quantum interference effects that amplify optimal solutions while suppressing suboptimal ones. Contemporary QML approaches encompass several algorithmic frameworks. Variational Quantum Algorithms (VQAs) represent the most promising near-term strategy, utilizing hybrid quantum-classical optimization with parameterized quantum circuits optimized through classical feedback loops. Quantum Neural Networks extend classical architectures into quantum domains using trainable quantum gates, while quantum kernel methods leverage quantum feature maps to project data into high-dimensional Hilbert spaces for enhanced classification and regression tasks.