  We describe criteria for implementation of quantum computation in qudits. A
qudit is a d-dimensional system whose Hilbert space is spanned by states |0>,
|1>,... |d-1>. An important earlier work of Mathukrishnan and Stroud [1]
describes how to exactly simulate an arbitrary unitary on multiple qudits using
a 2d-1 parameter family of single qudit and two qudit gates. Their technique is
based on the spectral decomposition of unitaries. Here we generalize this
argument to show that exact universality follows given a discrete set of single
qudit Hamiltonians and one two-qudit Hamiltonian. The technique is related to
the QR-matrix decomposition of numerical linear algebra. We consider a generic
physical system in which the single qudit Hamiltonians are a small collection
of H_{jk}^x=\hbar\Omega (|k><j|+|j><k|) and H_{jk}^y =\hbar\Omega
(i|k><j|-i|j><k|). A coupling graph results taking nodes 0,... d-1 and edges
j<->k iff H_{jk}^{x,y} are allowed Hamiltonians. One qudit exact universality
follows iff this graph is connected, and complete universality results if the
two-qudit Hamiltonian H=-\hbar\Omega |d-1,d-1><d-1,d-1| is also allowed. We
discuss implementation in the eight dimensional ground electronic states of
^{87}Rb and construct an optimal gate sequence using Raman laser pulses.
