Anyon condensation and the color 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.

55 pages, 57 figures, comments welcome; v2 - changes made in response to the peer-review process

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• Structured volume-law entanglement in an interacting, monitored Majorana spin liquid
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• Dynamical Logical Qubits in the Bacon-Shor Code
• XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime
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• Phase diagram of the three-dimensional subsystem toric code
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• Error Correction in Dynamical Codes
• The domain wall color code
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• Tailoring Dynamical Codes for Biased Noise: The X${}^3$Z${}^3$ Floquet Code
• A 3D lattice defect and efficient computations in topological MBQC
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