text: The ensemble is initialized to be the thermal equilibrium state (see quantum statistical mechanics). In mathematical parlance, this state is given by the density matrix:
  where H is the hamiltonian matrix of an individual molecule and
  where  is the Boltzmann constant and  the temperature. That the initial state in NMR quantum computing is in thermal equilibrium is one of the main differences compared to other quantum computing techniques, where they are initialized in a pure state. Nevertheless, suitable mixed states are capable of reflecting quantum dynamics which lead to Gershenfeld and Chuang to term them "pseudo-pure states."
  Operations are performed on the ensemble through radio frequency (RF) pulses applied perpendicular to a strong, static magnetic field, created by a very large magnet. See nuclear magnetic resonance.
  Consider applying a magnetic field along the z axis, fixing this as the principal quantization axis, on a liquid sample. The Hamiltonian for a single spin would be given by the Zeeman or chemical shift term:
  where  is the operator for the z component of the nuclear angular momentum, and  is the resonance frequency of the spin, which is proportional to the applied magnetic field.
  Considering the molecules in the liquid sample to contain two spin ½ nuclei, the system Hamiltonian will have two chemical shift terms and a dipole coupling term:
  Control of a spin system can be realized by means of selective RF pulses applied perpendicular to the quantization axis. In the case of a two spin system as described above, we can distinguish two types of pulses: “soft” or spin-selective pulses, whose frequency range encompasses one of the resonant frequencies only, and therefore affects only that spin; and “hard” or nonselective pulses whose frequency range is broad enough to contain both resonant frequencies and therefore these pulses couple to both spins. For detailed examples of the effects of pulses on such a spin system, the reader is referred to Section 2 of work by Cory et al.