  We say that collection of $n$-qudit gates is universal if there exists
$N_0\geq n$ such that for every $N\geq N_0$ every $N$-qudit unitary operation
can be approximated with arbitrary precision by a circuit built from gates of
the collection. Our main result is an upper bound on the smallest $N_0$ with
the above property. The bound is roughly $d^8 n$, where $d$ is the number of
levels of the base system (the '$d$' in the term qu$d$it.) The proof is based
on a recent result on invariants of (finite) linear groups.
