Short paths in PU(2)

Abstract

Parzanchevski--Sarnak \cite{PS} recently adapted an algorithm of Ross--Selinger \cite{RS16} for factorization of $\PU(2)$-diagonal elements to within distance $\eps$ into an efficient probabilistic algorithm for any $\PU(2)$-element, using at most $3\log_p(\nicefrac{1}{\eps^3})$ 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--Petit \cite{CPP} to improve this to $\frac{7}{3}\log_2(\nicefrac{1}{\eps^3})$, and implement the algorithm in Haskell.