Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates

https://arxiv.org/abs/2408.09254 We construct quantum codes that support transversal gates over qudits of arbitrary prime power dimension (including ) 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 (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent as the block length , where and denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size . We construct our codes by combining two modular components, namely,