One- and two-dimensional ensemble quantum computing in spin Liouville space

New concepts are described for nuclear magnetic resonance (NMR) implementat ions of spin ensemble quantum computing in one and two dimensions. Similari ties and differences between ensemble and pure state quantum computing are discussed by using a Liouville space formalism based on polarization and si ngle transition operators. The introduction of an observer spin, that is co upled to the spins carrying the quantum bits, allows a mapping of the state s 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 mu ltidimensional NMR, while the other employs mixed superposition states repr esented by operators in Liouville space. The spin swap operation, introduce d in this paper, allows a convenient extension of quantum computing to spin systems where not all spin-spin couplings are resolved. The concepts are i llustrated by implementations of logic operations and identities consisting of a sequence of basic logic gate operations. (C) 1998 American Institute of Physics. [S0021-9606(98)01348-8].