  We show that Aomoto's $q$-deformation of de Rham cohomology arises as a
natural cohomology theory for $\Lambda$-rings. Moreover, Scholze's $(q-1)$-adic
completion of $q$-de Rham cohomology depends only on the Adams operations at
each residue characteristic. This gives a fully functorial cohomology theory,
including a lift of the Cartier isomorphism, for smooth formal schemes in mixed
characteristic equipped with a suitable lift of Frobenius. If we attach
$p$-power roots of $q$, the resulting theory is independent even of these lifts
of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.
