Short answer: Color codes are a class of topological stabilizer codes with unusual transversality properties, introduced in two dimensions by Bombin and Martin-Delgado. Papers describe how they can implement some logic gates transversally, and how a "gauge" version can in principle support a universal gate set, with trade-offs. These are code-design results from theory papers, not hardware demonstrations.
What a color code is
A 2013 arXiv paper by Bombin, "Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes", describes color codes as topological stabilizer codes with remarkable transversality properties. It says the original color codes, by Bombin and Martin-Delgado, were defined in two dimensions with the aim of making the Clifford group of gates transversal. It adds that color codes can also be defined in higher dimensions on lattices called colexes.
A transversal gate, according to a 2014 paper that simplifies Bombin's presentation, is a code-space preserving unitary composed of separate unitaries applied to each physical qubit. That paper calls it the simplest form of fault-tolerant gate.
The limit on transversal gates
The 2014 paper cites a no-go theorem by Eastin and Knill: for any code that protects against arbitrary single-qubit errors, the set of transversal gates forms a finite group and therefore cannot be universal. It notes that approaches to circumvent this have been put forward, including gauge fixing, which Bombin applied to color codes in a d-dimensional lattice.
What the 2013 paper shows
The 2013 abstract says Bombin shows that the group of transversal gates of color codes is optimal and depends only on the spatial dimension, not the local geometry. It also introduces a generalized, subsystem version of color codes. In three dimensions, it says, these allow the transversal implementation of a universal set of gates by gauge fixing, while error-detecting measurements involve only 4 or 6 qubits.
The 2014 simplification
The 2014 paper says it gives a simplified yet rigorous and self-contained presentation of the ideas from Bombin's paper. It provides an explicit construction of a family of color codes in arbitrary dimensions and shows how to implement a generalized phase gate transversally. It describes implementing the Hadamard gate fault-tolerantly using code switching. In three dimensions, it says, this yields, together with the transversal CNOT, a fault-tolerant universal gate set without state distillation. That is the authors' theoretical claim about the construction.
Gauge color codes in two dimensions
A 2016 paper in Physical Review A presents a family of codes that support a universal set of transversal logic gates using only local operations on a two-dimensional array of physical qubits. It describes the construction as a subsystem version of color codes in which gauge fixing through local measurements dynamically determines which gates are transversal. The authors say the underlying code is not topological in structure, which is how the construction circumvents no-go constraints from the Bravyi-Konig and Pastawski-Yoshida theorems. They also say they provide strong evidence that the encoding has no error threshold in the conventional sense, though logical gates with error probability much lower than that of physical gates may still be possible.
What this does not tell you
This article summarises abstracts and introductions of three theory papers. It gives no resource estimates, and it does not say that any color code has been run on a quantum computer. Whether color codes beat surface codes for a given architecture is a separate question these sources do not settle.
FAQ
What is a color code?
A topological stabilizer code with unusual transversality properties, defined on lattices, per Bombin's 2013 paper.
What is a transversal gate?
A code-space preserving unitary composed of separate unitaries on each physical qubit.
Why not use transversal gates for everything?
The Eastin-Knill no-go theorem says transversal gates for such codes form a finite group and cannot be universal.
How do gauge color codes get around that?
By gauge fixing, which changes which gates are transversal, according to the 2013 and 2016 papers.
Do two-dimensional gauge color codes have a threshold?
The 2016 paper says it provides strong evidence they have no threshold in the conventional sense.
All field notes