- 2250 Elings Hall
We introduce a differential geometric framework for describing families of quantum error-correcting codes and for understanding quantum fault tolerance. This work unifies the notion of topological fault tolerance with fault tolerance in other kinds of quantum error-correcting codes. In particular, we use fibre bundles with a natural projectively flat connection to study the transformation of codewords under unitary fault-tolerant evolutions. We show that the fault-tolerant logical operations are given by the monodromy group for the bundles with projectively flat connection, which is always discrete. As a concrete realization of the general framework, we construct the bundles explicitly for two examples of fault-tolerant families of operations, the string operators in the toric code and the qudit transversal gates.