Sfruttare la Sovraparametrizzazione dal Controllo Ottimale Quantistico al Machine Learning Quantistico

Quantum mechanics, the leading theory of modern physics, provides an unparalleled understanding of systems at atomic and subatomic levels. Unlike classical mechanics, which governs macroscopic objects, quantum mechanics is characterized by phenomena such as superposition and entanglement. These phenomena become especially relevant when simulating complex quantum systems, where classical simulation methods require exponentially increasing computational resources. To avoid this problem, the idea of developing quantum computers has caught on during the 20th century. Later in time, the field of Quantum Optimal Control (QOC) emerged to optimally guide the dynamics of quantum systems through external controls, while the combination of computation employing quantum computers and the emerging paradigm of artificial intelligence gave birth to the growing field of Quantum Machine Learning (QML), aiming at learning patterns from data and make autonomous decisions. This Ph.D. thesis is an attempt to unify these two seemingly distinct fields under a common theoretical framework. By presenting a detailed exploration of each field, we uncover the shared principles, particularly in optimization techniques and landscape, controllability, and the emerging concept of overparametrization. These shared features serve as the foundation for this work, offering new insights and applications across both domains. More specifically, the attention of this work is focused on overparametrization, a powerful tool enabling exponential convergence in optimization tasks that may also lead to undesired consequences. Here we leverage this technique on one side, and in parallel we propose novel techniques to prevent such undesired effects. In QOC, we apply overparametrization to tackle the longstanding challenge of obtaining deterministic entangling operations using weak photonic nonlinearities with high fidelity. By going beyond the quantum speed limit, which in the common language of QOC simply corresponds to access the overparametrized regime, we design a numerical protocol that optimally tunes nonlinear quantum interferometers, enabling the realization of deterministic entangling gates, such as the CNOT and Mølmer-Sørensen gates, using single-photon qubit encoding. This discovery is a significant advancement, pushing the boundaries of what can be achieved with weak photon-photon interactions in integrated quantum photonic circuits. In the context of QML, we address the potential pitfalls of overparametrization, particularly the risk of overfitting with Quantum Neural Networks (QNNs). To mitigate overfitting issues, we investigated the use of quantum dropout as a regularization technique, demonstrating its effectiveness in preventing overfitting while preserving the expressibility and entanglement associated to the QNN. By carefully controlling the dropout probability, we ensured that the model remained overparametrized without losing its ability to generalize. This study offers valuable guidelines for practitioners aiming to use overparametrized QNNs in machine learning tasks, presenting quantum dropout as a promising tool for enhancing model performance. Finally, we explore the diagnostic capabilities of the Quantum Neural Tangent Kernel (QNTK) as a tool for assessing the training and generalization abilities of overparametrized QNNs, without the need for extensive training. The QNTK provides an efficient means of predicting the behaviour of both shallows and overparametrized QNNs, particularly in scenarios in which overfitting hinders generalization or barren plateaus — a common issue in quantum optimization — make training unfeasible. We conclude that by leveraging the QNTK, researchers can bypass the computational difficulties associated with QNNs, significantly reducing the time and resources required for efficient training.

La meccanica quantistica, la principale teoria della fisica moderna, offre una comprensione senza precedenti dei sistemi ai livelli atomico e subatomico. A differenza della meccanica classica, che governa gli oggetti macroscopici, la meccanica quantistica è caratterizzata da fenomeni come la sovrapposizione e l'entanglement. Questi fenomeni diventano particolarmente rilevanti nella simulazione di sistemi quantistici complessi, dove i metodi di simulazione classica richiedono risorse computazionali in aumento esponenziale. Per evitare questo problema, l'idea di sviluppare computer quantistici ha preso piede nel corso del XX secolo. Successivamente, è emerso il campo del Controllo Ottimale Quantistico (QOC) per guidare ottimamente la dinamica dei sistemi quantistici attraverso controlli esterni, mentre la combinazione di calcolo con computer quantistici e il paradigma emergente dell'intelligenza artificiale ha dato vita al crescente campo del Machine Learning Quantistico (QML), mirato a imparare pattern dai dati e prendere decisioni autonome. Questa tesi di dottorato tenta di unificare questi due campi apparentemente distinti sotto un comune quadro teorico. Presentando un'esplorazione dettagliata di ciascun campo, scopriamo i principi condivisi, in particolare nelle tecniche di ottimizzazione, controllabilità e nel concetto emergente di sovraparametrizzazione. Queste caratteristiche comuni fungono da fondamento per questo lavoro, offrendo nuove intuizioni e applicazioni in entrambi i domini. In modo specifico, questo lavoro si concentra sulla sovraparametrizzazione, un potente strumento che consente una convergenza esponenziale nei compiti di ottimizzazione ma che può anche portare a conseguenze indesiderate. Qui sfruttiamo questa tecnica da un lato, proponendo contemporaneamente nuove tecniche per prevenire tali effetti indesiderati. Nel QOC, applichiamo la sovraparametrizzazione per affrontare la sfida di lunga data di ottenere operazioni entangled deterministiche