Anyon condensation and the color code

TL;DR

This paper develops a comprehensive theory of anyon condensation in topologically-ordered phases, applying it to the color code to classify topological objects, design fault-tolerant gates, and introduce a new Floquet code.

Contribution

It provides a constructive framework for understanding anyon condensation, explicitly applies it to the color code, and proposes new fault-tolerant logical gates and a Floquet code based on condensation.

Findings

Classified topological objects in the color code.

Designed novel fault-tolerant logic gates.

Introduced the Floquet color code as a dynamically driven code.

Abstract

The manipulation of topologically-ordered phases of matter to encode and process quantum information forms the cornerstone of many approaches to fault-tolerant quantum computing. Here we demonstrate that fault-tolerant logical operations in these approaches can be interpreted as instances of anyon condensation. We present a constructive theory for anyon condensation and, in tandem, illustrate our theory explicitly using the color-code model. We show that different condensation processes are associated with a general class of domain walls, which can exist in both space- and time-like directions. This class includes semi-transparent domain walls that condense certain subsets of anyons. We use our theory to classify topological objects and design novel fault-tolerant logic gates for the color code. As a final example, we also argue that dynamical `Floquet codes' can be viewed as a series of condensation operations. We propose a general construction for realising planar dynamically driven codes based on condensation operations on the color code. We use our construction to introduce a new Calderbank-Shor Steane-type Floquet code that we call the Floquet color code.