Understanding transversal gates for the 7 qubit steane code

Question

How can one derive the complete list of transversal operators for the 7-qubit Steane code? I can derive the Clifford operators that are transversal, but I do not understand an easy way to check for non-Clifford transversal operators, and some idea on how to do this in general would be helpful. If there is a reference that would help.

Answer 1

By Eastin-Knill theorem, no quantum error correcting code capable of detecting arbitrary errors implements a universal gate set transversally (see this paper). Clifford group together with any non-Clifford operator is universal (see this paper). Therefore, if such a code implements all Clifford operators transversally then no operator outside the Clifford group can have a transversal implementation in the code.

Now, since the 7-qubit Steane code implements all Clifford operators transversally (see this answer, the introduction in this paper or section 10.6.2 of Nielsen & Chuang), we conclude that no operator outside the Clifford group admits a transversal implementation in Steane code.

Thus, your list of transversal operators is already complete.