Improved Almost laws for $SO(3)$

We construct quantitative almost laws for $SO(3)$. More precisely, there exist a constant $c>0$ and non-trivial words $W_n\in F_2$ such that, for every $A,B\in SO(3)$, $$\mathrm{\parallel }{W}_n(A,B)-I\mathrm{\parallel }\le \exp \text{\hspace{0.17em}⁣}(-c\mathrm{\mid }{W}_n{\mathrm{\mid }}^{\delta }),$$ where $\delta ={\log }_2(x_0)=0.879146\dots$ and $x_0>1$ is the real root of $x^3=x^2+x+1$. This improves the exponent ${\log }_2\phi$ obtained from Elkasapy’s lower-central-series construction. As an application, we show how this result improves the word-length threshold in Kuperberg’s Solovay–Kitaev algorithm for single-qubit gates.