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 $ε$ scales as $O\(\log^γ(1/ε\))$ (Bravyi and Haah, PRA 2012). In a breakthrough paper, Hastings and Haah (PRL 2018) showed that it is possible to construct distillation routines with sub-logarithmic overhead, achieving $γ\approx 0.6779$ 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 ($d$ dimensional quantum systems).