Low Overhead Qutrit Magic State Distillation

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 $k \leq 3m-2$ 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 $\epsilon$ is $O(\log^\gamma \epsilon^{-1})$, where the yield parameter $\gamma$ characterizes the overhead cost. For $k=3m-2$, our codes have $\gamma = \log_2 (2+\frac{6}{3 m-2})$, 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.