  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 $\epsilon$ scales as $O(\log^{\gamma}(1/\epsilon))$ (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 $\gamma \approx 0.6779$ and falsifying a conjecture that $\gamma$ is
lower bounded by $1$. They then ask whether $\gamma$ 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).
