Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

We construct quantum codes that support transversal $CCZ$ gates over qudits of arbitrary prime power dimension q (including $q=2$) such that the code dimension and distance grow linearly in the block length. The only previously known construction with such linear dimension and distance required a growing alphabet size q (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent γ=log(n/k)/log(d)→ 0 as the block length n→∞, where k and d denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size q. We construct our codes by combining two modular components, namely, (i) a transformation from classical codes satisfying certain properties to quantum codes supporting transversal $CCZ$ gates, and (ii) a concatenation scheme for reducing the alphabet size of codes supporting transversal $CCZ$ gates. For this scheme we introduce a quantum analogue of multiplication-friendly codes, which provide a way to express multiplication over a field in terms of a subfield. We obtain our asymptotically good construction by instantiating (i) with algebraic-geometric codes, and applying a constant number of iterations of (ii). We also give an alternative construction with nearly asymptotically good parameters (k,d=n/2O(log^*n)) by instantiating (i) with Reed-Solomon codes and then performing a superconstant number of iterations of (ii).