Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates

Understand

We present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates.

• One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal $T$.
• We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency $\gamma\rightarrow 1$.
• We also present a Reed-Muller based construction of these codes which obtains a worse $\gamma$ but performs well at small sizes.