Distillation with sublogarithmic overhead

It has been conjectured [1] that for any distillation protocol for magic states for the $T$ gate, the number of noisy input magic states required per output magic state at output error rate $ϵ$ is $\mathrm{\Omega }(\log (1\mathrm{/}ϵ))$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $[[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]$foranyintegers$ m > 2r$,$r > w \ge 0$, by puncturing quantum Reed-Muller codes. When $m>\nu r$, our code admits a transversal logical gate at the $\nu$-th level of Clifford hierarchy. In a distillation protocol for magic states at the level $\nu =3$ ($T$-gate), the ratio of input to output magic states is $O({\log }^{\gamma }(1\mathrm{/}ϵ))$ where $\gamma = \log(n/k)/\log(d)< 0.678$forsome$m,r,w$\mathrm{.}Thesmallestcodeinourfamilyforwhich$\gamma < 1$ison$\approx 2^{58}$ qubits.