If quantum computational circuits are ever built, they will be relatively small and built from simple quantum gates. The power and limits of resource-bound quantum circuits is examined and compared to their classical counterparts.

In particular it is shown that;
- several of the most interesting quantum algorithms can be carried out using relatively small depth and width quantum circuits. This includes, for example, the efficient computation of the QFT.

- even constant-depth classical circuits are stronger than their classical counterparts. In particular such circuits can compute the parity function, a function which cannot be computed as efficiently classically.

- testing whether a constant depth quantum circuit has non-zero acceptance probability is a provably computationally difficult problem. It is hard for the polynomial time hierarchy.