  According to the Gottesmann-Knill theorem the non-Gaussian states are
necessary component for a nontrivial quantum computation. We show two efficient
and deterministic methods of $\chi^{(3)}$ non-Gaussian state generation for a
cavity mode using a single trapped ion. Both require ion motional state
transfer to the cavity field. The methods are experimentally feasible. The
first is based on the well-known protocol for an ion finite motional
superposition state generation. It allows for an arbitrary good approximation
of $\chi^{(3)}$ non-Gaussian states. We give criteria based on the Wigner
function which quantify the error resulting from the approximation. The second
and novel method enables an exact non-Gaussian state generation using one laser
pulse only.
