Quantum Computing in Technicolor: Topological Error Correction and the Color Code

Abstract

The manipulation of topologically-ordered phases of matter to encode and pro- cess quantum information forms the cornerstone of many approaches to fault-tolerant quantum computing. Thus, it is paramount that we build a comprehensive framework of the topological objects of these phases of matter and offer descriptions on different levels. On the macroscopic level, we study the excitations and their properties ab- stractly, allowing us to draw conclusions about which types of topological objects can be realised. Zooming in to the microscopic level, we can offer concrete realisations of said objects in lattice models. To describe the dynamics of these objects, we build a space-time picture. While the space-time of topologically ordered phases is isotropic on the macroscopic level, concrete microscopic realisations require us to treat the time direction distinctly. In this thesis, we apply the program laid out above to the color code phase. We chose the color code for our comprehensive analysis as it is relatively simple to study while offering an incredibly rich set of physical properties. Furthermore, it is one of the simplest phases to implement experimentally and offers practical advantages over comparable candidate models. We begin our exploration by introducing the theory of the excitations of topo- logically ordered phases – the anyons – as well as their properties. We then turn to the example of the color code, presenting its microscopic lattice model and iden- tifying the anyonic excitations in it explicitly. We build a comprehensive catalogue of the color codes topological objects, namely the boundaries and the corners that interface boundaries, the domain walls and the twist defects that terminate domain walls, and the sub-theories and interfaces thereof. We describe said objects on both, the macroscopic-abstract, as well as the microscopic-concrete level. Furthermore, we build a static 3-dimensional model, which when interpreted as a (2+1)-dimensional model describes the dynamics of said objects. We investigate how the same topologi- cal object can be used to perform different quantum information processing tasks as its orientation in space-time is changed. Finally, we showcase the usefulness of our framework and the found objects by applying them to optimise some pressing challenges in the field. Among other things, we construct codes with high encoding rates and ones tailored to specific noise models, we reduce the temporal overhead of performing fault-tolerant gates and find a powerful description of the novel Floquet codes.