Gate Operations of Qubit and Conformal Mapping

A two-level quantum system is a quantum bit(qubit).Without the consideration of quantum coherence, qubit can be expressed on Riemann complex sphere,i.e. Bloch sphere representation. Identifying the Bloch sphere representation of qubit with the extended complex plane by means of stereographic projection and considering the gate operations of single qubit,we obtain equivalance relation between unitary operations and one special kind of conformal mappings.The investigation shows that there is close relation between gate operations of qubit and conformal mappings.