Qubit Geometry and Conformal Mapping
Chang Ho Kim,
and Soonchil Lee
Received January 14, 2002; accepted March 20, 2002
Identifying the Bloch sphere with the Riemann sphere(the extended complex plane),
we obtain relations between single qubit unitary operations and Mo
mations on the extended complex plane.
KEY WORDS: quantum computing; bloch sphere; qubit geometry; conformal
mapping; stereographic projection.
PACS: 03.67.-a, 03.67.Lx,03.67.Hk
In the past ten years, there has been a growing interest in the theory of
quantum computation and quantum information.
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.
In this paper, we will investi-
gate 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.
between conformal mappings and spinor transforms is extensively studied
by Rindler and Penrose.
considered a stereographic projec-
tion of two-level quantum system onto a complex plane, and their time
evolution on the spheres. Recently, his workis extended by Mosser and
who consider a S
Hopf ﬁbration using quaternions instead of
complex numbers for two qubits.
The main purpose of our study is to
1570-0755/02/0400-0129/0 # 2002 Plenum Publishing Corporation
Department of Chemistry, School of Molecular Science (BK 21), Korea Advanced Institute of
Science and Technology, Taejon 305-701, Korea.
Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701,
To whom correspondence should be addressed; E-mail: firstname.lastname@example.org
Quantum Information Processing, Vol. 1, Nos. 1/2, April 2002 (# 2002)