Short paths in PU(2)

Abstract

Parzanchevski and Sarnak recently adapted an algorithm of Ross and Selinger for factorization of PU(2)-diagonal elements to within distance $\epsilon$ into an efficient probabilistic algorithm for any PU(2)-element, using at most $3{\log }_p\frac{1}{{\epsilon }^3}$ factors from certain well-chosen sets. The Clifford+$T$ gates are one such set arising from $p=2$. In that setting, we leverage recent work of Carvalho Pinto and Petit to improve this to $\frac{7}{3}{\log }_2\frac{1}{{\epsilon }^3}$, and implement the algorithm in Haskell.