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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?

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  • $\begingroup$ I'm no experimentalist, but this paper here: arxiv.org/pdf/1108.5738v1.pdf suggests that the color codes might sometimes have better thresholds, but maybe they are just similar. Also, on page 1 of the Blatt paper, they claim that they implement CSS, which is equivalent of the smallest 2d-color code. This is probably (?) not true for the surface code, so maybe that's why they didn't use the surface code. Also, I always thought that topological codes should be harder to implement on trapped ions due to the geometry, but maybe wrong there... $\endgroup$ – Martin Mar 12 '15 at 10:12
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    $\begingroup$ In addition to Martin's comment: After discussing with an expert in the field, he said that the trapped ion group created the CC (corrected both bit- and phase-flip), but was not above the CC-threshold, thus their correction mechanism didn't help. Whereas the superconducting qubit group implemented an 1-dimensional SC, only correcting bit-flip (not phase-flip, as 2-dim SC would allow), but they were above the SC-threshold, thus their correction actually improved their states. The SC has a better threshold than CC, thus will be more realistic in future. $\endgroup$ – Mario Krenn Mar 12 '15 at 18:06
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    $\begingroup$ I am not an expert on QEC, but it would make your question more precise if you asked about some figures of merit. I guess you might be asking about: the error threshold?, the decoding efficiency?, the gate overhead?, the weight of the stabilizer operators? Also, are you more interested in fault-tolerant quantum computing or on having a self-correcting quantum memory? Is your question about experimental implementations? $\endgroup$ – Juan Bermejo Vega Mar 27 '15 at 11:09
  • $\begingroup$ Your second question is very different from what the title says and also relates to this one $\endgroup$ – Juan Bermejo Vega Mar 27 '15 at 11:10
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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 has uses 7 qubits and was previously called Steane code. However the underlying geometry in which the qubits are laid out in the Blatt setup isa 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).

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