Low Overhead Qutrit Magic State Distillation

Abstract

We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of $[[9m-k,k,2]{]}_3$ triorthogonal qutrit error-correcting codes for any positive integers $m$ and $k$ with $k\le 3m-2$ that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate $ϵ$ is $O({\log }^{\gamma }{ϵ}^{-1})$, where the yield parameter $\gamma$ characterizes the overhead cost. For $k=3m-2$, our codes have $\gamma ={\log }_2(2+\frac{6}{3m-2})$, which tends to $1$ as $m\to \infty$. Moreover, the $[[20,7,2]{]}_3$ qutrit code that arises from our construction when $m=3$ already has a yield parameter of $1.51$ which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.