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In this paper, the authors briefly mention that one proposed method to bypass the Eastin-Knill theorem is to perform code-switching. That is, given codes $C_1$ and $C_2$ which permit a complementary set of transversal gates, one can encode their data using code $C_1$ and perform some logical operations there, and when one wants to implement a gate which is not transversal in $C_1$ but is transversal in $C_2$, switch to $C_2$ and implement the desired logical gate there. See here for details about how to implement code switching between the 5 and 7-qubit codes.

In principle, at least to me, this seems like it should work but they claim that "such schemes do not yield a set of universal operations". Does it not work in general, or is it that no one has yet found a pair of codes that do this?

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  • $\begingroup$ The two papers that you link to are very different methods for achieving a similar result. You seem to be conflating them. I believe that the second method does also work between the 7-qubit Steane code and 15-qubit Reed-Muller code to offer universal QC. (Indeed, it should work very well because they have so many stabilizers in common). $\endgroup$
    – DaftWullie
    May 3, 2022 at 6:43
  • $\begingroup$ I'm not talking at all about code concatenation, which is done in the first paper. I just noticed that in that paper, the authors claim that this other technique (I've heard it called code switching) does not yield a set universal operations. Can you point to a resource which shows that code switching between the 7 and 15-qubit codes gives universal QC? I don't doubt you but I'm trying to learn more about this. $\endgroup$
    – SescoMath
    May 3, 2022 at 14:18
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    $\begingroup$ I had one of my PhD students looking at this about 6 months ago. I've asked him if he can post something. But the key point is that once you accept that the switch can be done, you get the transversal gates for both codes. As the first paper you mention states, the 7 and 15 qubits codes between them fulfil universality. $\endgroup$
    – DaftWullie
    May 3, 2022 at 14:24

1 Answer 1

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One reference is here. The authors switch between the 7-qubit Steane code and 15-qubit Reed-Muller code. Clifford group operators can be performed in the Steane code, and the T-gate in the 15-qubit code.

In general if there are two codes which between them contain a set of gates which is universal for quantum computation, then a scheme which switches between these codes would provide access to both sets of gates, and hence provide a universal gate set. One issue is ensure that this can be done in such a way as to preserve an appropriate code distance throughout the code switching process. In the case of these two codes, the 15-qubit code, the 7-qubit code and all the intermediate codes in the switching process can be viewed as gauge-fixings of a slightly larger code with distance 3. This ensures that the distance 3 of the code is maintained throughout the process.

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