Elementary gates for quantum computation

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