  We develop a systematic unitarity method for loop-level AdS scattering
amplitudes, dual to non-planar CFT correlators, from both bulk and boundary
perspectives. We identify cut operators acting on bulk amplitudes that put
virtual lines on shell, and show how the conformal partial wave decomposition
of the amplitudes may be efficiently computed by gluing lower-loop amplitudes.
A central role is played by the double discontinuity of the amplitude, which
has a direct relation to these cuts. We then exhibit a precise, intuitive map
between the diagrammatic approach in the bulk using cutting and gluing, and the
algebraic, holographic unitarity method of arXiv:1612.03891 that constructs the
non-planar correlator from planar CFT data. Our analysis focuses mostly on
four-point, one-loop diagrams -- we compute cuts of the scalar bubble, triangle
and box, as well as some one-particle reducible diagrams -- in addition to the
five-point tree and four-point double-ladder. Analogies with S-matrix unitarity
methods are drawn throughout.
