Magic-state distillation is a resource intensive subroutine for quantum computation. The ratio of noisy input states to output states with an error rate at most $\epsilon$ scales as $O\mathbf{(}{\log }^{\gamma }(1/\epsilon )\mathbf{)}$ [S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012)]. In a breakthrough paper, Hastings and Haah [Distillation with Sublogarithmic Overhead, Phys. Rev. Lett. 120, 050504 (2018)] showed that it is possible to construct distillation routines with a sublogarithmic overhead, achieving $\gamma \approx 0.6779$ and falsifying a conjecture that $\gamma$ is lower bounded by 1. They then ask whether $\gamma$ can be made arbitrarily close to 0. We answer this question in the affirmative for magic-state distillation routines using qudits of prime dimension ($d$ dimensional quantum systems for prime $d$).

• Received 7 January 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.070507