  From a geometric approach, we derive the minimum number of applications
needed for an arbitrary Controlled-Unitary gate to construct a universal
quantum circuit. A new analytic construction procedure is presented and shown
to be either optimal or close to optimal. This result can be extended to
improve the efficiency of universal quantum circuit construction from any
entangling gate. Specifically, for both the Controlled-NOT and Double-CNOT
gates, we develop simple analytic ways to construct universal quantum circuits
with three applications, which is the least possible.
