Code whose only transversal gate is phase gate $ P $

Question

This is a follow-up to Example CSS codes and the properties "doubly even" and "self dual".

What is an example of a CSS code whose only single qubit transversal gate is phase gate $ P $?

The $ [\![7,1,3]\!] $ Steane code is a doubly even CSS code so it has $ P $ transversal but also it is self dual so the Hadamard gate $ H $ is transversal.

The $ [\![15,1,3]\!] $ quantum Reed-Muller code is a doubly even CSS code so it has $ P $ transversal but it is actually quadruply even so it has $ T=\sqrt{P} $ transversal also.

What is an example of a code that has $ P $ transversal but nothing else? (nothing else besides Pauli gates that is, since Pauli gates are transversal for every stabilizer code).

I know that doubly even CSS codes always have $ P $ transversal. So I guess I'm looking for some doubly even CSS code which is not quadruply even (so $ T $ not transversal) but also is not self-dual (so $ H $ not transversal).

Answer 1

TL;DR: Such codes may be obtained by concatenating the repetition code with a code whose only transversal $Z$ rotations are generated by the $P$ gate.

Preliminaries: Repetition code

Repetition code with the logical computational basis $|0_L\rangle=|00\rangle$ and $|1_L\rangle=|11\rangle$ is stabilized by the group generated by $ZZ$. Therefore, it is a CSS code. Moreover, $\overline X=XX$ and $\overline{R_Z(\theta)}=I\otimes R_Z(\theta)$ are transversal in this code for all $\theta\in[0,2\pi)$. In particular, $P=R_Z\left(\frac{\pi}{2}\right)$ is transversal. On the other hand, any logical gate that creates a superposition in the logical computational basis generates entanglement between physical qubits and hence cannot be transversal. Thus, Pauli operators and $Z$ rotations generate all transversal gates. In particular, Hadamard is not transversal.

CSS codes with transversal group $\langle P, X\rangle$

A quantum code whose single-qubit transversal group is generated by $X$ and $P$ may be obtained by first encoding into the $[\![2,1,1]\!]$ repetition code described above and then into a CSS code with transversal $P$ gate and no other transversal $Z$ rotations such as Steane $[\![7,1,3]\!]$ code. The stabilizer group of the resulting $[\![14,1,3]\!]$ code is generated by for example $$ \begin{array}{c|c|c} g_1&IIIXXXX,IIIIIII\\ g_2&IXXIIXX,IIIIIII\\ g_3&XIXIXIX,IIIIIII\\ g_4&IIIIIII,IIIXXXX\\ g_5&IIIIIII,IXXIIXX\\ g_6&IIIIIII,XIXIXIX\\ g_7&IIIZZZZ,IIIIIII\\ g_8&IZZIIZZ,IIIIIII\\ g_9&ZIZIZIZ,IIIIIII\\ g_{10}&IIIIIII,IIIZZZZ\\ g_{11}&IIIIIII,IZZIIZZ\\ g_{12}&IIIIIII,ZIZIZIZ\\ g_{13}&ZZZZZZZ,ZZZZZZZ \end{array} $$ Since the $X$ and $Z$ sectors of the stabilizer separate we see that the code is a CSS code. Moreover, the Hadamard entails generation of entanglement between the first seven qubits and the last seven qubits and thus cannot be implemented transversally. On the other hand, operators such as $I^{\otimes 7}\otimes P^{\dagger\otimes 7}$ induce the logical $P$ gate on the code subspace. The only other single-qubit transversal gates are the Pauli operators.