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 nonstabilizer 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 ${\left[[n,1]\right]}_3$ qutrit stabilizer codes with a complete set of transversal Clifford gates for $n\le 23$, and all ${\left[[n,1]\right]}_3$ stabilizer codes with a transversal $H^2$ gate with $n\le 9$ 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 surpasses 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.

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• Quantum correlations, foundations & formalism
• Quantum error correction