Transversal Gates in Nonadditive Quantum Codes

Transversal gates play a crucial role in suppressing error propagation in fault-tolerant quantum computation, yet they are intrinsically constrained: any nontrivial code encoding a single logical qubit admits only a finite subgroup of $\mathrm{SU}(2)$ as its transversal operations. We introduce a systematic framework for searching codes with specified transversal groups by parametrizing their logical subspaces on the Stiefel manifold and minimizing a composite loss that enforces both the Knill-Laflamme conditions and a target transversal-group structure. Applying this method, we uncover a new $((6,2,3))$ code admitting a transversal $Z\bigl(\tfrac{2\pi}{5}\bigr)$ gate (transversal group $\mathrm{C}_{10}$), the smallest known distance $3$ code supporting non-Clifford transversal gates, as well as several new $((7,2,3))$ codes realizing the binary icosahedral group $2I$. We further propose the \emph{Subset-Sum-Linear-Programming} (SS-LP) construction for codes with transversal \emph{diagonal} gates, which dramatically shrinks the search space by reducing to integer partitions subject to linear constraints. In a more constrained form, the method also applies directly to the binary-dihedral groups $\mathrm{BD}_{2m}$. Specializing to $n=7$, the SS-LP method yields codes for all $\mathrm{BD}_{2m}$ with $2m\le 36$, including the first $((7,2,3))$ examples supporting transversal $T$ gate ($\mathrm{BD}_{16}$) and $\sqrt{T}$ gate ($\mathrm{BD}_{32}$), improving on the previous smallest examples $((11,2,3))$ and $((19,2,3))$. Extending the SS-LP approach to $((8,2,3))$, we construct new codes for $2m>36$, including one supporting a transversal $T^{1/4}$ gate ($\mathrm{BD}_{64}$). These results reveal a far richer landscape of nonadditive codes than previously recognized and underscore a deeper connection between quantum error correction and the algebraic constraints on transversal gate groups.