Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all $[[n,1]]_3$ qutrit stabilizer codes with a complete set of transversal Clifford gates for $n\leq 23$, and all $[[n,1]]_3$ stabilizer codes with a transversal $H^2$ gate with $n \leq 9$ qudits. For $n=23$, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.

翻译：确定量子比特魔态蒸馏的最佳可达阈值直接关系到上下文性是否足以实现通用量子计算这一核心问题。本文证明，量子纠错码在魔态蒸馏中的性能可由其完全权重枚举器完全刻画。对于三量子比特奇异态——一种最大魔性的非稳定子态——其蒸馏码的性能则可由简单权重枚举器描述。基于此结论，我们得以对奇异态的高阈值魔态蒸馏方案展开广泛搜索。搜索范围涵盖所有具有完备横向Clifford门集的$[[n,1]]_3$三量子比特稳定子码（$n\leq 23$），以及所有具有横向$H^2$门的$[[n,1]]_3$稳定子码（$n \leq 9$）。在$n=23$时，我们发现了超过600个CSS码能够以三次噪声抑制实现三量子比特奇异态的蒸馏。尽管这些码均未超越11量子比特Golay码的阈值，但它们的存在表明：对于大规模编码，实现三量子比特奇异态蒸馏的能力具有一定普适性。