  Multiple quantum (MQ) NMR methods \cite{Baum} are applied to the analysis of
various problems of quantum information processing.
  It is shown that the two-spin/two-quantum Hamiltonian \cite{Baum} describing
MQ NMR dynamics is related to the flip-flop Hamiltonian of a one-dimensional
spin system in the approximation of the nearest neighbor interactions.
  As a result, it is possible to organize quantum state transfer along a linear
chain.
  MQ NMR experiments are performed on quasi-one-dimensional chains of
\BPChem{\^{19}F} nuclei in calcium fluorapatite \BPChem{Ca\_5(PO\_4)\_3F}.
Relaxation of the MQ NMR coherences is considered as the simplest model of
decoherence processes. A theory of the dipolar relaxation of the MQ NMR
coherences in one-dimensional systems is developed. A good agreement of the
theoretical predictions and the experimental data is obtained.
