Distillation with Sublogarithmic Overhead

It has been conjectured 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 $\epsilon$ is $\mathrm{\Omega }[\log (1/\epsilon )]$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $⟦\sum _{i=w+1}^m(\genfrac{}{}{0ex}{}{m}{i}),\sum _{i=0}^w(\genfrac{}{}{0ex}{}{m}{i}),\sum _{i=w+1}^{r+1}(\genfrac{}{}{0ex}{}{r+1}{i})⟧$ for any integers $m>2r$, $r>w\ge 0$, 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 $O\mathbf{(}{\log }^{\gamma }(1/\epsilon )\mathbf{)}$, where $\gamma =\log (n/k)/\log (d)<0.678$ for some $m$, $r$, $w$. The smallest code in our family for which $\gamma <1$ is on $\approx 2^{58}$ qubits.