Fibre bundle framework for quantum fault tolerance

We introduce a differential geometric framework for describing families of quantum error-correcting codes and for understanding quantum fault tolerance. In particular, we use fibre bundles and a natural projectively flat connection thereon to study the transformation of codewords under unitary fault-tolerant evolutions. We'll explain how the fault-tolerant logical operations are given by the monodromy group for the bundles with projectively flat connection, which is always discrete. We will discuss the construction of the said bundles for two examples of fault-tolerant families of operations, the string operators in the toric code and the qudit transversal gates. This framework unifies topological fault tolerance and fault tolerance based on transversal gates, and is expected to apply for all unitary quantum fault-tolerant protocols.