Fast Navigation with Icosahedral Golden Gates

Abstract: An algorithm of Ross and Selinger for the factorization of diagonal elements of PU(2) to within distance $\varepsilon$ was adapted by Parzanchevski and Sarnak into an efficient probabilistic algorithm for any element of PU(2) using at most effective $3\log_p\frac{1}{\varepsilon<sup>{3}}$ factors from certain well-chosen sets associated to a number field and a prime $p$. The icosahedral super golden gates are one such set associated to $\mathbb{Q}(\sqrt{5})$. We leverage recent work of Carvalho Pinto, Petit, and Stier to reduce this bound to $\frac{7}{3}\log_{59}\frac{1}{\varepsilon<sup>3}$, and we implement the algorithm in Python. This represents an improvement by a multiplicative factor of $\log_259\approx5.9$ over the analogous result for the Clifford+$T$ gates. This is of interest because the icosahedral gates have shortest factorization lengths among all super golden gates.