Why does code switching not allow for universal fault-tolerant quantum computation?

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?

• 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). May 3, 2022 at 6:43
• 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. May 3, 2022 at 14:18
• 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. May 3, 2022 at 14:24