No abstract

It is proposed that the state space of a quantum object with a complicated discrete spectrum can be used as a basis for multiqubit recording and processing of information in a quantum computer. As an example, nuclear spin 3/2 is considered. The possibilities of writing and reading two quantum bits of information, preparation of the initial state, implementation of the "rotation" and "controlled negation" operations, which are sufficient for constructing any algorithms, are demonstrated.We shall consider a nucleus with spin I = 3/2 and with an electric quadrupole moment in an axial symmetric crystalline electric field and a constant magnetic field parallel to the symmetry axis. We assume that the Zeeman energy is greater than the quadrupole energy, so that the nucleus possesses four nonequidistant spin energy levels E m with eigenfunctions χ m ( m = ±3/2, ±1/2), which are eigenfunctions of the z component I z of the nuclear spin I.Let a radio frequency (RF) field 2H a cosΩ a t, polarized along the y axis and resonant for a certain pair of energy levels ´Ω a = E m -E n , act on the spin. The interaction operator H a with the field in the interaction representation contains a timeindependent term + a, eff = ´γH a [P mn + P nm ] and rapidly oscillating (at frequencies which are sums of Ω a and (E k -E l )/´) terms. The role of these last terms in the evolution of the spin states is negligible. To simplify the notation here and below we use the projection operators P mn representation , where P mn is a 4x4 matrix with all matrix elements p kl equal to zero except p mn =1. Projection operators are very convenient because of their very simple properties: P kl P mn = δ lm P kn and P mn χ k = δ nk χ m . In addition, 1,2,3, and 4 will stand for the indices -3/2, -1/2, +1/2, +3/2, respectively.The interaction + a,eff leads to spin states evolution managed by propagator Uϕ a = [P mm + P nn ] Cos(ϕ a ) + [P kk + P ll ] + [P nm -P mn ] Sin(ϕ a ),

A procedure is described for storing a two-dimensional (2D) pattern consisting of 32×32=1024 bits in a spin state of a molecular system and then retrieving the stored information as a stack of nuclear magnetic resonance spectra. The system used is a nematic liquid crystal, the protons of which act as spin clusters with strong intramolecular interactions. The technique used is a programmable multifrequency irradiation with low amplitude. When it is applied to the liquid crystal, a large number of coherent long-lived H1 response signals can be excited, resulting in a spectrum showing many sharp peaks with controllable frequencies and amplitudes. The spectral resolution is enhanced by using a second weak pulse with a 90° phase shift, so that the 1024 bits of information can be retrieved as a set of well-resolved pseudo-2D spectra reproducing the input pattern.

A continuous, analog version of the Grover algorithm is realized using NMR. The system studied is 23Na in a liquid crystal medium. The presence of quadrupolar coupling makes the spin I=3/2 nucleus a 2-qubit system. Applying a specially designed pulse sequence, the time evolution of the spin density operator is described in an interaction representation which has no external time-dependent radio-frequency fields. This approach is used to implement one instance of the continuous Grover search for the transform of a uniform state to a target state, and the implementation provides a clear physical interpretation of the algorithm. The experimental results are in good agreement with the theory.Comment: 14 pages plus 2 figures, accepted for publication in Phys. Rev.