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 ancillas required to produce a magic state within a target error rate $\epsilon$ is $O\left[{ln}^{\gamma }\left({\epsilon }^{-1}\right)\right]$ 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 et al. and Krishna-Tillich, we generalize the Hastings-Haah construction to qudits of arbitrary prime dimension $p$. Although existing qubit-based constructions achieve sublogarithmic overhead, these advantages only emerge at block sizes that are prohibitively large for practical implementation. In contrast, using an analytically tractable puncturing scheme, we show that the block size required to achieve sublogarithmic overhead decreases drastically as $p$ increases for our qudit construction, with yield parameter approaching $\frac{1}{lnp}$ as $p\to \infty$. We also perform a small computational search for optimal puncture locations, which results in several interesting triorthogonal codes, including a ${\left[[519,106,5]\right]}_5$ code with $\gamma =0.99$. These results demonstrate that qudit-based distillation routines allow sublogarithmic overheads to be realized at much smaller and more physically viable block lengths.

Close

PrevNext

View figure in article

• Quantum error correction