# What are the transversal gates of the $[[5,1,3]]$ code?

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)

• IIRC it's only the Pauli gates. Jun 29, 2022 at 14:30

This paper (page 4) lists Paulis + $$HS$$ gate.
• I'm afraid not. This paper 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. Jun 29, 2022 at 22:13
• 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 $\tilde{HP}$ . Jul 6, 2022 at 22:17
• sorry that I kept switching between $S$ and $P$ in my notation I just mean the phase gate $S=P=diag(1,i)$ Jul 7, 2022 at 20:57