Distillation with Sublogarithmic Overhead
Matthew B. Hastings, Jeongwan Haah · Physical Review Letters · 2018
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 $\ensuremath{\epsilon}$ is $\mathrm{\ensuremath{\Omega}}[\mathrm{log}(1/\ensuremath{\epsilon})]$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $⟦\ensuremath{\sum}_{i=w+1}^{m}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=0}^{w}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=w+1}^{r+1}(\genfrac{}{}{0}{}{r+1}{i})⟧$ for any integers $m>2r$, $r>w\ensuremath{\ge}0$, by puncturing quantum Reed-Muller codes. When $m>\ensuremath{ u}r$, our code admits a transversal logical gate at the $\ensuremath{ u}$th level of Clifford hierarchy. In a distillation protocol for magic states at the level $\ensuremath{ u}=3$ ($T$ gate), the ratio of input to output magic states is $O\mathbf{(}{\mathrm{log}}^{\ensuremath{\gamma}}(1/\ensuremath{\epsilon})\mathbf{)}$, where $\ensuremath{\gamma}=\mathrm{log}(n/k)/\mathrm{log}(d)<0.678$ for some $m$, $r$, $w$. The smallest code in our family for which $\ensuremath{\gamma}<1$ is on $\ensuremath{\approx}{2}^{58}$ qubits.