Quantum Error Correction: Surface code vs. color code Recently, two groups working on quantum computers published results on quantum error correction.
The first was Rainer Blatt's group, who used trapped ions to perform a topologically encoded qubit using "color code": Quantum computations on a topologically encoded qubit (arxiv).
The other one was John Martinis' group, who used superconducting qubits and performed a simplified version of a "surface code": State preservation by repetitive error detection in a superconducting quantum circuit (arxiv).
Can anybody please explain to me, what is the difference between "color code" and "surface code"? What are the advantages and disadvantages of those? Why aren't the trapped-ion guys using the "surface code" or vice versa?
The Martinis group has shown recently that they are above the threshold for surface-code QEC arxiv. Has any ion group (maybe by Blatt) shown similar quality of single- and two-qubit gates?
Are there other groups who have the same level of control over their systems to perform multi-qubits quantum error-correction codes?
 A: The color code and surface code are very similar. They are stabilizer codes composed of qubits arranged in two dimensions, requiring only geometrically local stabilizer measurements.
From the theory point of view, the codes are very similar. In fact, with collaborators we have proven that the color code is equivalent to a surface code (paper) up to a geometrically local unitary (one which only makes nearby qubits interact). One can think by analogy of the  surface code* as a napkin with two rough and two smooth sides and the color code as folding this napkin along its diagonal. Because in the folded napkin, there are new things that are now close, it is possible to do more logical gates "transversally". This is good because it keeps errors from propagating and is relatively easy. However, the color code needs more qubits to interact in each stabilizer so ends up leading to a lower noise threshold.
So one can say that although very similar, each code has its advantages and disadvantages.
At this point, only very small versions of either of these codes are being demonstrated. The Rainer Blatt group demonstrated the smallest possible color-code which also uses 7 qubits (this instance is also referred to as the Steane code). However, the underlying geometry in which the qubits are laid out in the Blatt setup is a linear chain of ions, so I would say that this is not the natural setting to extend to larger and larger system sizes.
The superconducting qubit people (Martinis, IBM, DiCarlo, ...) on the other hand, are concentrating more on surface codes. While in principle, their architecture should allow them to go full fledge 2D, for now, they are having the classical logic come in from the sides, which is something that needs to change.
*There is actually an ambiguity as to what to call surface codes, but I will refer to the quantum double of Z2 with rough and smooth boundaries defined by Bravyi and Kitaev (paper).
