New concepts are described for nuclear magnetic resonance (NMR) implementations of spin ensemble quantum computing in one and two dimensions. Similarities and differences between ensemble and pure state quantum computing are discussed by using a Liouville space formalism based on polarization and single transition operators. The introduction of an observer spin, that is coupled to the spins carrying the quantum bits, allows a mapping of the states of a quantum computer on a set of transitions between energy levels. This is spectroscopically favorable compared to a mapping on the energy levels themselves. Two complementary parallelization schemes for quantum computing are presented: one exploits the parallel processing feature inherent in multidimensional NMR, while the other employs mixed superposition states represented by operators in Liouville space. The spin swap operation, introduced in this paper, allows a convenient extension of quantum computing to spin systems where not all spin-spin couplings are resolved. The concepts are illustrated by implementations of logic operations and identities consisting of a sequence of basic logic gate operations.