Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates

With respect to the transversal gate group (an invariant of quantum codes), we demonstrate that non-additive codes can outperform stabilizer codes. We do this by constructing spin codes that correspond to permutation-invariant multiqubit codes that can implement generalized phase gates transversally. Of particular note, we construct permutation-invariant quantum codes that implement a transversal $T$ gate using fewer qubits and with a better minimum distance than is possible with the best known stabilizer codes.