| Sumario: | <p>Quantum computing is a fascinating discipline, combining the laws of quantum physics with the practicalities of computing. Certain problems could potentially be solved faster on quantum computers, and researchers hope to use them to solve difficult problems in industries such as healthcare and finance. Quantum machine learning and quantum neural networks (QNNs) aim to solve some of these problems. Open questions remain for their utility, and this thesis aims to contribute to answers by exploring the foundation, optimisation, and application of quantum algorithms. We explore the foundation of QNNs by studying their bias and expressivity. We investigate whether a QNN can have a classical deep neural network's (DNN) bias, known as simplicity bias, which is hypothesised to contribute to the success of DNNs. We run numerical experiments and derive proofs to demonstrate how the type of QNN affects its bias and expressivity. Our results suggest a bias-expressivity tradeoff, highlighting the need for alternative frameworks to make QNNs effective general-purpose learning tools for classical data. We explore the optimisation of quantum circuits by creating a new quantum compiler with a novel technique: we automatically generate quantum circuits that are equivalent and design a tiling method to optimise larger circuits according to a customisable cost function. We generate novel equivalent circuits, compile custom quantum gates unlike previous compilers, and outperform existing compilers on certain circuits. We explore the application of quantum computers by constructing a quantum algorithm to solve a real-world problem in the mining industry. We provide a mapping for this problem onto a quantum circuit and run simulator and hardware experiments to find solutions to small-scale instances of the problem, which is the first attempt to apply such quantum algorithms to this mining problem. This thesis also presents a framework to identify definitions, key questions, and goals for quantum advantage.</p>
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