Enhanced fault-tolerant quantum computing in $d$-level systems

Error correcting codes protect quantum information and form the basis of fault tolerant quantum computing. Leading proposals for fault-tolerant quantum computation require codes with an exceedingly rare property, a transverse non-Clifford gate. Codes with the desired property are presented for $d$-level, qudit, systems with prime $d$. The codes use $n=d-1$ qudits and can detect upto $\sim d\mathrm{/}3$ errors. We quantify the performance of these codes for one approach to quantum computation, known as magic state distillation. Unlike prior work, we find performance is always enhanced by increasing $d$.