Transversal Gates in Nonadditive Quantum Codes

Transversal gates curb error propagation but are limited: any nontrivial single-qubit code admits only a finite subgroup of SU(2) transversally. We present a search framework that parametrizes logical subspaces on the Stiefel manifold and minimizes a composite loss enforcing the Knill-Laflamme conditions and a target transversal group. With it we find a new ((6,2,3)) code with a transversal Z(2pi/5) gate (C10), the smallest known distance-3 code with a non-Clifford transversal group, and several new ((7,2,3)) codes realizing the binary icosahedral group 2I. We further introduce the Subset-Sum-Linear-Programming (SS-LP) method for transversal diagonal groups, reducing the search to integer partitions under linear constraints and, in a constrained form, to binary-dihedral groups BD(2m). For n=7, SS-LP yields codes for all BD(2m) with 2m up to 36, including the first ((7,2,3)) codes with transversal T (BD16) and sqrt(T) (BD32), improving on the previous smallest examples ((11,2,3)) and ((19,2,3)). Extending SS-LP to ((8,2,3)) produces codes for larger 2m, including one with a transversal T^(1/4) (BD64). These results reveal a richer landscape of nonadditive codes and a tighter link between error correction and algebraic constraints on transversal groups.