[Paper Review] Anyon Condensation and the Color Code

Markus S. Kesselring, Julio C. Magdalena de la Fuente+4 more|arXiv (Cornell University)|Nov 30, 2022

Topological Materials and Phenomena108 references15 citations

TL;DR

This paper establishes anyon condensation as a unifying framework for fault-tolerant quantum computation in topological codes, demonstrating that logical operations in the color code—such as state preparation, measurement, and gate implementation—can be understood as condensation processes. It introduces novel structures like semi-punctures and semi-transparent domain walls, and constructs a new Floquet color code via periodic condensation, enabling universal, low-overhead quantum computation with enhanced fault tolerance.

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.

Motivation & Objective

• To unify diverse fault-tolerant operations in topological quantum codes under the framework of anyon condensation.
• To develop a constructive theory of anyon condensation in stabilizer codes, particularly the color code, enabling microscopic realization of abstract condensation processes.
• To classify and engineer new topological defects—such as semi-punctures and semi-transparent domain walls—via condensation.
• To demonstrate that dynamical Floquet codes can be understood as sequences of condensation operations, leading to a new class of topological codes.

Proposed method

• Constructs a general theory of anyon condensation in Abelian anyon models, distinguishing maximal, partial, and trivial condensation.
• Applies the theory to the color code by identifying condensation processes that realize logical qubit encoding, initialization, and measurement.
• Introduces semi-punctures by condensing only a single anyon type (a boson) in a region, generalizing standard punctures.
• Uses domain walls in (2+1)D space-time to describe temporal and spatial logical operations, including time-like domain walls for state preparation and measurement.
• Develops a microscopic lattice construction for condensation, enabling explicit realization of condensation in stabilizer codes.
• Proposes the Floquet color code as a dynamically driven code where periodic condensation operations implement logical gates, using detection cells and syndrome tracking.

Experimental results

Research questions

• RQ1 How can anyon condensation be used to describe and construct fault-tolerant logical operations in topological stabilizer codes like the color code?
• RQ2 What new types of topological defects—such as semi-punctures and semi-transparent domain walls—emerge from partial condensation, and how do they enable new computation schemes?
• RQ3 Can dynamical, periodically driven codes (Floquet codes) be understood as sequences of condensation operations, and if so, how can they be systematically constructed?
• RQ4 How do temporal and spatial domain walls interplay to ensure fault tolerance in logical operations such as initialization and measurement?
• RQ5 What is the role of Lagrangian subgroups in condensation processes, and how do they determine the resulting topological phase and logical operations?

Key findings

• The paper introduces semi-punctures—regions where only a single anyon type (a boson) is condensed—enabling new code deformation schemes and generalizing standard puncture-based encoding.
• It demonstrates that logical state preparation and measurement in the color code can be realized via partial condensation of anyons, specifically by condensing a Lagrangian subgroup to project onto a logical state.
• The theory identifies semi-transparent domain walls that condense only a subset of anyons, allowing for controlled, non-maximal phase transitions and enabling new types of code deformation.
• A new Floquet color code is constructed by applying periodic condensation operations, which supports universal, fault-tolerant quantum computation with reduced resource overhead.
• Numerical simulations show that the Floquet color code outperforms the honeycomb code in error correction under similar noise models, indicating improved logical error suppression.
• The framework unifies various known protocols—including lattice surgery, code deformation, and measurement-based quantum computation—under the single principle of anyon condensation.