  We bootstrap loop corrections to AdS${}_5$ supergravity amplitudes by
enforcing the consistency of the known classical results with the operator
product expansion of $\mathcal{N}=4$ super Yang-Mills theory. In particular
this yields much new information on the spectrum of double-trace operators
which can then be used, in combination with superconformal symmetry and
crossing symmetry, to obtain a prediction for the one-loop amplitude for four
graviton multiplets in AdS. This in turn yields further new results on
subleading $O(1/N^4)$ corrections to certain double-trace anomalous dimensions.
