This thesis investigates the development of Quantum Machine Learning (QML) methods for industrial applications, with a focus on bridging the gap between theoretical results and the constraints of current quantum hardware. Classical machine learning has transformed fields such as healthcare, finance, and logistics, yet it now faces several bottlenecks: the exponential growth of data, rising computational costs, concerns about privacy and security, and the massive energy consumption of large-scale models. Quantum computing, originally conceived for simulating physical systems, has emerged as a promising avenue for accelerating certain learning tasks. However, many quantum algorithms require fault-tolerant quantum computers, which remain out of reach. Variational quantum algorithms, better suited to today’s noisy devices, are considered the most realistic candidates, though they still suffer from training difficulties, a lack of theoretical tools to assess their usefulness, and the challenge of guaranteeing that their behavior cannot be efficiently simulated or approximated classically.
The first part of the thesis examines these issues through subspace-preserving quantum circuits, in particular those that preserve the Hamming weight of states. These circuits possess training-friendly properties, yet they are often classically simulable in polynomial time—illustrating the tension between trainability and genuine quantum advantage. This motivates a shift toward algorithms offering only polynomial advantages but equipped with convergence guarantees. Special attention is given to photonic architectures, which naturally preserve particle number and allow for high repetition rates. Their controllability is characterized, and new, albeit suboptimal, schemes are proposed to exploit polynomial-scale advantages that may have concrete industrial value. Building on these insights, the thesis introduces a family of subspace-preserving quantum algorithms that emulate key components of classical learning. These methods combine theoretical guarantees with favorable training behavior, offering a pragmatic pathway toward near-term industrial adoption of quantum learning, even in the absence of exponential speedups.
The second part presents a theoretical framework based on Fourier analysis of variational circuits. This approach jointly addresses expressivity and trainability and enables systematic comparisons between quantum circuits and classical methods. It identifies distinct convergence conditions and clarifies when surrogate models suffice to approximate quantum behavior. These results provide conceptual tools for designing circuits that are resistant to efficient classical simulation and for better understanding the scenarios in which a genuine quantum advantage may emerge.
By combining an analysis of trainability, the design of tailored architectures, and the development of new analytical tools, this thesis delineates the conditions under which quantum learning can achieve real industrial impact. It shows that while exponential advantages remain difficult to obtain with current devices, polynomial gains, particularly in high-throughput photonic architectures, may already offer a competitive edge. The results thus provide a theoretical and algorithmic foundation for designing quantum models that balance expressivity, trainability, and classical hardness, paving the way for concrete industrial applications of QML.