# Is there a CSS code for which the controlled phase gate and all Clifford gates are transversal?

In a paper that I have recently read, a protocol was given that required a CSS code with certain properties for which all logical Clifford gates and standard measurements have a transversal implementation (the 7-qubit would be an example). Later in that same paper a procedure was required that required the controlled $$XP^{\dagger}$$ ($$C-XP^{\dagger}$$) gate to be transversal for that specific code.

Now my question is, is that even possible? As far as I know, the set {Clifford group + a single non-Clifford} gate is universal, and that no stabilizer code allows for transversal implementation of a universal set of gates without any further assumptions. The controlled phase gate $$C-P$$ is a non-Clifford gate and $$C-XP^{\dagger} = CNOT*(C-P^{\dagger})$$, so that $$C-XP^{\dagger}$$ also is a non-Clifford gate. Doesn't this make the requirement that the ($$C-XP^{\dagger}$$) gate be transversal impossible in the first place?

• just to be clear, when you're talking about C-P, you mean the gate which is diag(1,1,1,i)? – DaftWullie Oct 16 '20 at 15:52
• Clifford operations are certainly transversal on the [[7,1,3]] CSS code. – Condo Oct 16 '20 at 16:03
• @DaftWullie yes, or, equivalently, C-P^{\dagger} with diag(1,1,1,-i). – jgerrit Oct 16 '20 at 18:05
• You're right, this seems to be impossible by the Eastin-Knill theorem. $CP$ is in third level of the Clifford hierarchy. Perhaps they use a more general notion of transversality or some other relaxation. Could you link the paper? – Markus Heinrich Oct 17 '20 at 13:09
• Could you give us a link to the paper, just so we can check out the context of what they say a bit more carefully? – DaftWullie Oct 18 '20 at 15:32