19 - Topological codes

Summary

Introduction

What a good code is depends on the particular constraints of the problem at hand. In this chapter we address a constraint that is relevant to many physical settings: locality. In particular, we are interested in situations where geometrical locality is relevant. This typically means that the physical qubits composing the code are placed in some lattice and only interactions between nearby qubits are possible. In this case, it is desirable that syndrome extraction also be local, so that fault tolerance can possibly be achieved. Topological codes offer a natural solution to locality constraints, as they have stabilizer generators with local support.

In topological codes information is stored in global degrees of freedom, so larger lattices provide larger code distances. The nature of these global degrees of freedom is illustrated in Fig. 19.1, where several closed curves in a torus are compared. Consider curves a and b. They look the same if examined locally, as in the region marked with dotted lines. However, curve a is the boundary of a region, whereas curve b is not. In order to decide whether a curve is a boundary or not we need global information about it. This is, as we will see, a core idea in topological codes.