This paper is motivated by the problem of manipulating the state of a quantum bit in quantum computation. The problem of verifying the feasibility of a given operation in quantum computing is tackled as a the problem of verifying the reachability properties of a bilinear system whose state varies on the Lie group of special unitary matrices of dimensions two. The problem of studying how much time is needed for a given quantum operation (which amounts to a given state transfer) is considered. The contribution of the paper is as follows: 1) By application of previous theorems to the physical situation at hand it is shown that if two or three non parallel components of the input controlling electro-magnetic field can be varied for control the system can be driven to any desired final configuration (every arbitrary quantum logic operation) in arbitrary small time (no bound is placed on the magnitude of the controls). 2) It is shown that not every arbitrary state transfer can be obtained in arbitrary small time for the system with only one input, even if no restriction is placed on the magnitude of the input field. An exact characterization of the set of states reachable in arbitrary small time is given for this system. 3) It is proven that there exists a critical time T such that for every t>T every trasfer of state is possible (for the one input system) at time t. 4) Lower and upper bounds are derived for the critical time T obtaining so practical estimates for the speed of quantum logic operations.