  We present a perturbative derivation of the T-system that is believed to
encode the exact spectrum of planar N=4 SYM. The T-system is understood as an
operator identity between some special line operators, the quantum transfer
matrices. By computing the quantum corrections in the process of fusion of
transfer matrices, we show that the T-system holds up to first order in a
semi-classical expansion. This derivation does not rely on any assumption. We
also discuss the extension of the proof to other theories, including models
describing string theory on various AdS spaces.
