Final Activity Report Summary - QUANTUM COMPUTATION (Robust Quantum Computation with Geometric Phases) Quantum computation with geometric phase is thought to hold the promise to realise robust quantum computations against errors. In such schemes, gate operations depend on topological features of geometric phases and not on the way the loops are actually performed, and are therefore largely insensitive to local inaccuracies and fluctuations. Though conceptually fascinating, many details remain unclear regarding, in particular, the physical foundation of geometric phases in noisy environments. Therefore, the objective of this project was to achieve a deeper theoretical understanding and practical realisation of fast robust quantum computation with topologic gates. Using state of the art Nuclear magnetic resonance (NMR) technologies, we experimentally investigated the properties of geometric phases in noisy environments and accumulated significant experimental evidence about the robustness of topologically-stabilised quantum computations. More specifically, we achieved the following milestones: 1. we chose a suitable sample and utilised the gradient-diffusion techniques to generate variable noise strengths; 2. an NMR interferometer with controllable noise-rate was set up to measure the effect of noise on geometric phases in an open system; 3. we designed a specific path for the evolution of one qubit with different noise power in the abovementioned interferometer and measured the corresponding geometric phases; 4. we implemented asymmetric phase-covariant cloning and experimentally realised an optimal quantum cloning machine for two qubits which did not require ancilla qubits. In addition, we experimentally demonstrated complete measurement of quantum states with a single observable; 5. we experimentally investigated a quantum mechanical phase factor that reflected the topology of the SO(3) group and observed a topological phase in the maximally entangled state of a pair of qubits via the nuclear magnetic resonance interferometer; 6. we experimentally demonstrated a unified framework for two existing definitions, namely Uhlmann and Sjoqvist definitions, of geometric phase in a mixed-state scenario within a single, common, formalism based on simple interferometry. The relevant paper was submitted to Physical Review Letters. Taking into account all the novel scientific results that were obtained in this project, we anticipated that they would have a strong direct impact on both fundamental research and technological progress in this extremely challenging area.