Sublogarithmic Distillation in all Prime Dimensions using Punctured Reed-Muller Codes
Tanay Saha, Shiroman Prakash
TL;DR
This work extends sublogarithmic magic state distillation to qudits of prime dimension $p$ by generalizing Hastings–Haah punctured Reed-Muller constructions to $p$-ary codes. It develops an analytically tractable puncturing scheme based on Manhattan weight, derives the distance and yield parameters, and shows the asymptotic yield $\gamma_0(p) \sim 1/\ln p$ as $p\to\infty$. It also demonstrates substantial practical improvements: block sizes needed for sublogarithmic overhead shrink dramatically with $p$, and randomized searches yield a $[[519,106,5]]_5$ code with $\gamma=0.99$. The results quantify a potential overhead reduction for fault-tolerant quantum computation using higher-dimensional qudits and open avenues for discovering better punctured 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 $ε$ is $O(\log^γ (ε^{-1}))$ where $γ$ is known as the yield parameter. Hastings and Haah derived a family of distillation protocols with sublogarithmic overhead (i.e., $γ< 1$) 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 $γ=0.99$.