In certain quantum models, e.g., interacting spin lattices (Kitaev's toric code), dynamics is expected to be noise-resilient: upon cyclic adiabatic deformation of the Hamiltonian, the dynamics depends only on the topology of the Hamiltonian’s trajectory in the parameter space, but not on the (potentially noisy) geometry of that trajectory. Such protected dynamics could be utilized in quantum computing, where precise quantum operations have to be performed in the presence of noise. The goal of the PhD work is to understand the most relevant theoretical models, and to bring them as close as possible to experimental realization.