In this mini-course, we will cover mathematical topics in topological phases of matter from a quantum information perspective. The course consists of four parts in which the following topics will be discussed. All lectures will be on a blackboard.

1. We will first discuss about general quantum error correction, where one is given with a subspace of a large Hilbert space, and is concerned with operations within the subspace. To achieve the error correcting capability, it is necessary that the operations can only be performed with non-local operators. These will appear routinely in the study of topological order, when the subspace is identified with the ground state subspace. Error correction criteria will be connected to the entanglement structure of the ground state subspace.

2. We will next examine simple examples, toric code and double semion models, which can be viewed as error correcting codes on a torus. Lessons from the first quarter will be applied and reviewed. We will then discuss a (Wegner) duality by which these models are mapped to a trivial phase and a Z2-symmetry protected topological phase, respectively.

3. In the third quarter, we will discuss local unitary transformations and invariants of topological phases (mainly, the topological entanglement entropy and the topological S-matrix through so-called twist product). The local unitary transformations will lead us to a tensor network description (MERA in this case) for the ground states of topologically ordered systems.

4. Finally, we will take a journey to a relatively new class of phase of matter in three spatial dimensions, so-called fracton topological order. It is a kind of quantum error correcting code, but previously existing quantities or viewpoints on topological order do not capture the system properly, and necessitates modifications. The example of the cubic code model will be primarily discussed.