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

Question

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?

Answer 1

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.