Meet-in-the-Middle Compilation of the $[\mathrm{CNOT}]$ Class with Fibonacci Anyons

Abstract

Compilation of two-qubit Fibonacci anyon gates is complicated by leakage ($\ell$) from the four-dimensional computational sector to the fifth state of the fusion space. We construct a GPU pipeline to find low-leakage representatives of the $[\text{CNOT}]$ class on different subsets of the braid group. We exhaustively explore the full positive five-generator alphabet up to $L=21$ with minimal leakage $\ell\sim10^{-3}$ and error $d_{[\text{CNOT}]}\sim 10^{-8}$ on it. To reach larger lengths, the search space is successively narrowed to three-generator, boundary-$σ_3$, and palindromic braids, and by considering only the Garside left normal form. These optimizations allow us to reach $L=72$. For $L=69$, a braid with $\ell\sim10^{-10}$ and $d_{[\text{CNOT}]}\sim 10^{-10}$ is found. In addition to directly enumerating optimal braids, we proved that $\ell\in\mathbb Z[φ]$, and any nonzero leakage obeys an exponential lower bound in the braid length, and obtained the limit laws $F_{3,\infty}(\ell)=\ell^2$ and $F_{5,\infty}(\ell)=\ell^4$ for our sets. Using asymptotics for the distribution function $F$, we showed that for sufficiently long braids, the asymptotics of exhaustive search are better than the asymptotics of random sampling for finding low-leakage braids.