  We propose and experimentally demonstrate an efficient framework for the
quantum simulation of quantum channels in NMR. Our approach relies on the
suitable decomposition of non-unitary operators in a linear combination of $d$
unitary ones, which can be then experimentally implemented with the assistance
of a number of ancillary qubits that grows logarithmically in $d$. As a
proof-of-principle demonstration, we realize the quantum simulation of three
quantum channels for a single-qubit: phase damping (PD), amplitude damping
(AD), and depolarizing (DEP) channels. For these paradigmatic cases, we measure
key features, such as the fidelity of the initial state and the associated von
Neuman entropy for a qubit evolving through these channels. Our experiments are
carried out using nuclear spins in a liquid sample and NMR control techniques.
