Curator's Take
This thesis tackles one of quantum computing's most fundamental challenges by exploring how geometric structures can help us understand and control quantum entanglement in spin systems, which form the backbone of many quantum computing architectures. The work's focus on the quantum brachistochrone problem - finding the fastest possible quantum evolution between states - has direct implications for optimizing quantum gate operations and reducing decoherence times in real quantum processors. By bridging classical geometric methods with quantum spin dynamics, this research provides new mathematical tools for analyzing how entanglement spreads and evolves in quantum systems, potentially leading to better strategies for quantum error correction and more efficient quantum algorithms. The geometric perspective on quantum states through the Fubini-Study metric offers fresh insights into why certain quantum operations are more robust than others, which could inform the design of next-generation quantum hardware.
— Mark Eatherly
Summary
This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.