Sublogarithmic Distillation in all Prime Dimensions using Punctured Reed-Muller Codes

Abstract

Magic state distillation is a leading but costly approach to fault-tolerant quantum computation, and it is important to explore all possible ways of minimizing its overhead cost. The number of ancillae required to produce a magic state within a target error rate $\epsilon$ is $O(\log^{\gamma} (\epsilon^{-1}))$ where $\gamma$ is known as the yield parameter. Hastings and Haah derived a family of distillation protocols with sublogarithmic overhead (i.e., $\gamma < 1$) based on punctured Reed-Muller codes. Building on work by Campbell \textit{et al.} and Krishna-Tillich, which suggests that qudits of dimension $p>2$ can significantly reduce overhead, we generalize their construction to qudits of arbitrary prime dimension $p$. We find that, in an analytically tractable puncturing scheme, the number of qudits required to achieve sublogarithmic overhead decreases drastically as $p$ increases, and the asymptotic yield parameter approaches $\frac{1}{\ln p}$ as $p \to \infty$. We also perform a small computational search for optimal puncture locations, which results in several interesting triorthogonal codes, including a $[[519,106,5]]_5$ code with $\gamma=0.99$.