Towards low overhead magic state distillation

Magic state distillation is a resource intensive sub-routine for quantum computation. The ratio of noisy input states to output states with error rate at most ǫ scales as O(log γ (1/ǫ)) [1]. In a breakthrough paper, Hastings and Haah [2] showed that it is possible to construct distillation routines with sub-logarithmic overhead, achieving γ ≈ 0.6779 and falsifying a conjecture that γ is lower bounded by 1. They then ask whether γ can be made arbitrarily close to 0. We answer this question in the affirmative for magic state distillation routines using qudits of prime dimension (d dimensional quantum systems for prime d)