Variational Quantum Computing: Optimization and Geometry

Abstract

Quantum computing potentially offers unprecedented computational capabilities that transcend the limitations of classical computing paradigms. Despite its conceptual inception over three decades ago, recent years have witnessed remarkable progress in the realization of physical quantum computers, spurring a surge of research activity in the field. Although fault-tolerance devices remain unrealized, modern quantum hardware is getting less noisy, which allows us to investigate quantum algorithms that require only short depth circuits. One particular class of algorithms that falls into this category are variational quantum algorithms, which treat a quantum computer as a black box with tunable parameters that can be optimized via a classical optimization routine. This thesis delves into the realm of variational quantum algorithms and explores their optimization properties, trainability and geometric properties. Through a blend of numerical experiments, geometric insights, and mathematical analysis, it provides a comprehensive exploration of variational quantum algorithms paving the way for future advancements in variational quantum computing.