Quantum Machine Learning: continuous-variable approach with application to Neural Networks

The work presented in this thesis is focused on quantum computing and in particular on the interplay between quantum devices and machine learning. This interaction gives birth to a relatively recent area of study, called Quantum Machine Learning (QML), which currently represents a hot research field. This work was carried out in collaboration with an important company with home in Turin, DATA Reply S.r.l., whose main focuses are Big Data, Artificial Intelligence, Machine Learning and Quantum Computing. We start by introducing the fundamental concepts of quantum mechanics, presenting the topic in a formal way from a mathematical point of view while keeping things as simple as possible, as long as they allow us to understand quantum computing basics. After an overview on motivations, possible employments of quantum devices and an analysis of risks and benefits of their development, we proceed to explain quantum computing main concepts, such as qubits and quantum gates. The introduced formalism is the so-called Discrete Variable (DV) formalism, which is the most prominent choice in literature for describing quantum systems. However, as extensively explained during the course of this work, this approach presents several drawbacks: one of the main goals of this thesis is to introduce the alternative formalism of Continuous Variables (CV) and investigate how it can contribute to improve this research area. Currently, one of the preferred tools for quantum machine learning algorithms are the Variational - or Parametrized - circuits: they allow to train the quantum devices in the same way as a classical neural network and allow to address ML problems, both supervised and unsupervised. After a detailed description of these tools, we proceed to introduce PennyLane, a Python library for QML which is very well suited for dealing with variational circuits. In the core part of this work, we concentrate on some QML examples, in particular we analyze some applications in which using the CV formalism turns out to be the most convenient choice. Integration, more specifically Monte Carlo integration, is one of those fields which the CV formalism suits best; in a similar way, Gaussian process regression in its continuous variables version provides a great advantage, namely an exponential speedup, as long as some hypothesis are satisfied. We only provide a brief overview of these applications, since we opted to leave more room to experiments involving variational classifiers and quantum neural networks. Three use cases are analyzed, all involving the CV formalism. The first one is a variational classifier, employed in a supervised setting for the classification of a simple dataset. The second one is a quantum neural network, used to solve a problem of function fitting. The results and the chosen hyperparameters are presented, together with a formal explanation of the theoretical concepts behind them. The last use case, the most complex and detailed one, focuses on a time series forecasting task, carried out with a quantum neural network, and also presents a comparison with the classical approach.