Qubit Geometry and Conformal Mapping

In the past ten years, there has been a growing interest in the theory of quantum computation and quantum information.ð1Þ Due to the quantum parallelism, quantum computers have the potential to outperform the classical computers. The quantum computers are based on two-level quantum systems, which are called qubits.ð2Þ In this paper, we will investigate a representation of a single qubit operation on a complex plane. The idea that a state in a two-level quantum system can be described by a complex number in the complex plane is not new. The relationship between conformal mappings and spinor transforms is extensively studied by Rindler and Penrose.ð3Þ Urbantkeð4Þ considered a stereographic projection of two-level quantum system onto a complex plane, and their time evolution on the spheres. Recently, his work is extended by Mosser and Danoloff, ð5Þ who consider a S 7 Hopf fibration using quaternions instead of complex numbers for two qubits.ð7Þ The main purpose of our study is to