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What are the transversal gates of the perfect 5-qubit code ( [[5,1,3]] code) ?

Since this is a stabilizer code we have transversal Pauli gates.

Are there any others?

I am primarily interested in single qubit gates that are transversal for a single block of the code. But I would also be very curious to see any two qubit gates which are transversal for two blocks of the code (and higher for three blocks of the code etc... but I would imagine that is very hard to find).

I know that $ Z $ is transversal because $ |0_L> $ is a superposition of even parity bit strings and $ |1_L> $ is a superposition of odd parity bit strings.

However the phase gate $ S=\begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} $ is not transversal since $ |0_L> $ is not doubly even. See Transversal logical gate for Stabilizer (or at least Steane code)

Ian Gershon Teixeira
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This paper (page 4) lists Paulis + $HS$ gate.

unknown
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  • Very interesting! I would be very interested in some sort of conceptual reason why that true. Other than checking directly on logical basis states do you have any idea why that would be true? – Ian Gershon Teixeira Jun 29 '22 at 21:59
  • I'm afraid not. This paper https://arxiv.org/pdf/1603.03948.pdf (page 7) puts the HS gate in a family of 8 "octahedral gates"...all are transversal for the $[[5,1,3]]$ code. – unknown Jun 29 '22 at 22:13
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    Looked into it turns out transversal $ HS $ implements logical $ -HP $ which is interesting. If you want something that implements itself you can have $ \tilde{HP} $ to be the determinant 1 version of $ HS $ and then transversal $ \tilde{HP} $ implements logical $ \tilde{HP} $ . – Ian Gershon Teixeira Jul 06 '22 at 22:17
  • sorry that I kept switching between $ S $ and $ P $ in my notation I just mean the phase gate $ S=P=diag(1,i) $ – Ian Gershon Teixeira Jul 07 '22 at 20:57