Distillation with sublogarithmic overhead

TL;DR

This paper disproves a conjecture about the lower bound on noisy magic state distillation overhead, presenting quantum codes with sublogarithmic overhead for certain error rates.

Contribution

It introduces a family of quantum error correcting codes that enable magic state distillation with overhead less than logarithmic in error rate inverse.

Findings

Disproves the conjecture that distillation overhead is at least logarithmic in 1/ε.

Constructs quantum codes with transversal gates at the third level of Clifford hierarchy.

Achieves distillation overhead of O(log^γ(1/ε)) with γ<1 for large qubit systems.

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 $[[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]$ 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 $O(\log^\gamma (1/\epsilon))$ 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.