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

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

Blog Article

We present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates.One construction is depileve easy clean 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 $gamma ightarrow 1$.We also present a Reed-Muller based construction of these codes which obtains a worse $gamma$ but performs well at small sizes.Additionally, we present protocols based on checking the stabilizers of $CCZ$ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of.

Several examples, including a Reed-Muller code for $T$-to-Toffoli distillation, punctured Reed-Muller codes for $T$-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size.In particular, we find a $512$ T-gate ilootpaperie to $10$ Toffoli gate code with distance $8$ as well as triorthogonal codes with parameters $[[887,137,5]],[[912,112,6]],[[937,87,7]]$ with very low prefactors in front of the leading order error terms in those codes.