  We have developed methods for performing qudit quantum computation in the
Jaynes-Cummings model with the qudits residing in a finite subspace of
individual harmonic oscillator modes, resonantly coupled to a spin-1/2 system.
The first method determines analytical control sequences for the one- and
two-qudit gates necessary for universal quantum computation by breaking down
the desired unitary transformations into a series of state preparations
implemented with the Law-Eberly scheme. The second method replaces some of the
analytical pulse sequences with more rapid numerically optimized sequences. In
our third approach, we directly optimize the evolution of the system, without
making use of any analytic techniques. While limited to smaller dimensional
qudits, the third approach finds pulse sequences which carry out the desired
gates in a time which is much shorter than either of the other two approaches.
