Abstract In a recent paper, we defined a type of weighted unitary design called a twisted unitary 1-group and showed that such a design automatically induced error-detecting quantum codes. We also showed that twisted unitary 1-groups correspond to irreducible products of characters thereby reducing the problem of code-finding to a computation in the character theory of finite groups. Using a combination of GAP computations and results from the mathematics literature on irreducible products of characters, we identify many new non-trivial quantum codes with unusual transversal gates. Transversal gates are of significant interest to the quantum information community for their central role in fault tolerant quantum computing. Most unitary $$\text {t}$$ t -designs have never been realized as the transversal gate group of a quantum code. We, for the first time, find nontrivial quantum codes realizing nearly every finite group which is a unitary 2-design or better as the transversal gate group of some error-detecting quantum code.

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.