A Search for High-Threshold Qutrit Magic State Distillation Routines

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.