What is the definition of color codes?

Question

Is there a generally accepted definition of what a color code is?

I have found two definitions that I am not able to reconciliate with each other:

1. The error correction zoo defines color codes via homogeneous simplicial complexes as the triangulation of a single simplex and qubits are placed on simplices.

2. In An Introduction to Topological Quantum Codes in section 5, color codes are introduced as lattices that come with vertices and faces (at least, in the 2-D case) where qubits are on the vertices and the plaquettes denote X- and Z-stabilizers.

The examples that I have come accross so far (Steane, honeycomb, 4-8-8 lattice) all seem to fit the second definition, but not the first.

Are the two definitions equivalent? If yes, why? If not, what is the "correct" / generally accepted definition?

Answer 1

Ideally, using a definition one should be able to answer the question; is this circuit implementing the color code?

Given the two definitions in your question, it would not be clear how to argue if the circuits in these two papers are implementing the color code:

• Floquetifying the color code
• Engineering 3D Floquet codes by rewinding

For example, the lattice in Fig 23. of 2 does not satisfy either of the proposed definitions, but the color code is implemented on it.

This can be resolved by defining a topological code in terms of the corresponding anyon model. More specifically, any code whose excitations behave as color code anyons is a color code. Using this definition one can check that the circuits in the papers mentioned above implement the color code.

The anyon model of the 2D color code is given in section 3 of The boundaries and twist defects of the color code and their applications to topological quantum computation.

The three examples you've mentioned (Steane, honeycomb, 4-8-8 lattice) and the example Craig Gidney mentions (color code with Pauli boundaries) all satisfy this definition.

The honeycomb code is another example that shows why thinking of a code as an anyon model is useful. The behaviour of excitations in the honeycomb code is equivalent to that of excitations in the surface code, therefore the honeycomb code is a surface code.

One downside of this definition is that there is a lot of condensed matter physics jargon used in defining topological phases and anyon models.