Distillation with sublogarithmic overhead

Abstract

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 $\epsilon$ is $\Omega (\log (1/\epsilon ))$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $[[{\sum }_{i=w+1}^m\left(\genfrac{}{}{0ex}{}{m}{i}\right),{\sum }_{i=0}^w\left(\genfrac{}{}{0ex}{}{m}{i}\right),{\sum }_{i=w+1}^{r+1}\left(\genfrac{}{}{0ex}{}{r+1}{i}\right)]]$ 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 $O({\log }^{\gamma }(1/\epsilon ))$ 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.

2 pages. v2: fixed typo in abstract

References in corpus (3)

• Magic state distillation with low overhead
• Multilevel distillation of magic states for quantum computing
• Fault-Tolerant Postselected Quantum Computation: Schemes

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