Associated to each unitary fusion category is a quantum error correcting code. These codes are created by the Levin-Wen construction, closely related to the Turaev-Viro topological quantum field theory. The topological nature of these codes makes them inherently robust to local noise processes that are inevitable in realistic physical systems. Unfortunately, there is a tradeoff between the complexity of the experimental implementation and the power of the associated quantum code. In this work, we intend to attack this problem from both sides. In two dimensions, we will incorporate defects (bimodules and their functors) into the codes, and study the effect on the computational power. This approach side-steps the no-go theorems which forbid universal computing with simple codes. From the other side, in three dimensions we will design codes that maximize the benefit of more complicated codes. We intend to define codes based on the more complex two dimensional codes which are robust to errors in experiments, in addition to real environmental errors. Finally, we will use tensor networks to investigate fusion 2-categories, and their modules. By their nature, these techniques allow for computation of explicit data for these higher categorical structures. With these tensor networks in hand, they can be applied to the simulation of three dimensional topological phases.