  We give an approximation algorithm for MaxCut and provide guarantees on the
average fraction of edges cut on $d$-regular graphs of girth $\geq 2k$. For
every $d \geq 3$ and $k \geq 4$, our approximation guarantees are better than
those of all other classical and quantum algorithms known to the authors. Our
algorithm constructs an explicit vector solution to the standard semidefinite
relaxation of MaxCut and applies hyperplane rounding. It may be viewed as a
simplification of the previously best known technique, which approximates
Gaussian wave processes on the infinite $d$-regular tree.
