We show that a set of gates that consists of all one-bit quantum gates [U (2)] and the two-bit exclusive-O R gate [that maps Boolean values (x, y) to (x, x⊕ y)] is universal in the sense that all unitary operations on arbitrarily many bits n [U (2 n)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U (2) transformation to one input bit if and only if the logical a n d of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.