The authors study the algebraic K-theory and Grothendieck-Witt categories in the non-additive context, with a particular focus on classes of examples of ${\bf F}_1$-linear structure. The main results obtained are some analogues of some theorems of Quillen and Schlichting, which relate the K-theory or Grothendieck-Witt theory spaces of proto-exact categories by using the (hermitian) Quillen $Q$-construction and group completion. The first main result of this paper is a group completion theorem for uniquely split proto-exact categories. The second main result is a homotopy equivalence between certain homotopy fibers and a group completion. The third main result gives a weak homotopy equivalence of the Grothendieck-Witt space ${\cal G} {\cal W}({\cal A})$ with a disjoint sum over elements of $W_0({\cal A})$, where $\cal A$ is a uniquely split proto-exact category with duality.