Quantum Codes with Transversal $CCZ$ Gates and Sublinear $Z$-Stabilizers

Abstract: We construct quantum CSS codes with transversal (CCZ) gates whose (Z)-stabilizers admit sublinear-weight generating sets. We build on the algebraic puncturing framework of Guruswami and Golowich \cite{GG24}, which turns classical codes with the required Schur-product and distance conditions into CSS codes with transversal (CCZ). However, applying the framework directly to the algebraic expander codes of \cite{KT26} runs into their small dual distance, and therefore produces only sublinear quantum dimension. Our main technical step is a refined puncturing theorem in which the global dual-distance assumption is replaced by a condition only on the selected puncturing set. Applying this theorem to algebraic expander codes gives explicit growing-alphabet CSS codes with parameters ([[N,Θ(N),Ω(N^{1/m})]]), for every fixed (m\geq 3), and with transversal (CCZ) gates. Moreover, the (Z)-stabilizer space has an explicit generating set of weight (O(N^{1/m})). We also reduce the alphabet to a fixed prime field using a projective-multiplicity version of multiplication-friendly codes. The resulting fixed-prime-field CSS code triples, of length (n), still have transversal (CCZ) gates. Their dimension is near-linear, their distance is (n^{1/m}) up to polylogarithmic factors, and the (Z)-stabilizer locality remains sublinear, again up to polylogarithmic losses.