NMR quantum computing with qubit systems represented by nuclear spins (I = 1/2) in small mols. in liqs. has led to the most successful exptl. quantum information processors so far. We use the quadrupolar spin-3/2 sodium nuclei of a NaNO3 single crystal as a virtual two-qubit system. The large quadrupolar coupling in comparison with the environmental interactions and the usage of strongly modulating pulses allow us to manipulate the system fast enough and at the same time keeping the decoherence reasonably slow. The exptl. challenge is to characterize the \"calcn.\" behavior of the quantum processor by process tomog. which is here adapted to the quadrupolar spin system. The results of a selection of quantum gates and algorithms are presented as well as a detailed anal. of exptl. results.