Small Codes for Magic State Distillation

TL;DR

This paper investigates small stabilizer codes for magic state distillation, revealing some codes can distill states effectively even without transversality, including states near the undistillable boundary.

Contribution

It identifies and analyzes small stabilizer codes with interesting distillation properties, expanding understanding beyond transversality requirements.

Findings

Some codes distill states near the undistillable boundary.

Certain codes distill towards previously unconsidered non-stabilizer states.

Many codes exhibit inefficient but notable distillation behavior.

Abstract

Magic state distillation is a critical component in leading proposals for fault-tolerant quantum computation. Relatively little is known, however, about how to construct a magic state distillation routine or, more specifically, which stabilizer codes are suitable for the task. While transversality of a non-Clifford gate within a code often leads to efficient distillation routines, it appears to not be a necessary condition. Here we have examined a number of small stabilizer codes and highlight a handful of which displaying interesting, albeit inefficient, distillation behaviour. Many of these distill noisy states right up to the boundary of the known undististillable region, while some distill toward non-stabilizer states that have not previously been considered.