  Let X be a noetherian scheme of finite Krull dimension, having 2 invertible
in its ring of regular functions, an ample family of line bundles, and a global
bound on the virtual mod-2 cohomological dimensions of its residue fields. We
prove that the comparison map from the hermitian K-theory of X to the homotopy
fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence
in general, and an integral equivalence when X has no formally real residue
field. We also show that the comparison map between the higher
Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an
isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum
conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of
complex algebraic varieties and rings of 2-integers in number fields, and hence
values of Dedekind zeta-functions.
