2026/07/17 by Gal Yehuda
Computer Science · Mathematics · #math.GR #cs.CC
We construct quantitative almost laws for SO(3). More precisely, there exist a constant c>0 and non-trivial words Wn∈ F2 such that, for every A,B∈ SO(3), ‖Wn(A,B)-I‖ ≤ exp (-c |Wn|δ), where δ=log2(x0)=0.879146… and x0>1 is the real root of x3=x2+x+1. This improves the exponent log2φ 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.