  It is well-known that maximally entangled states such as the
Greenberger-Horne-Zeilinger (GHZ) states, with the Bell states as the simplest
examples, are widely exploited in quantum information and computation. We study
the application of such maximally entangled states from the viewpoint of the
GHZ transform, which is a unitary basis transformation from the product states
to the GHZ states. The algebraic structure of the GHZ transform is made clear
and representative examples for it are verified as multi-qubit Clifford gates.
In this paper, we focus on the Bell transform as the simplest example of the
GHZ transform and apply it to the reformulation of quantum circuit model of
teleportation and the reformulation of the fault-tolerant construction of
single-qubit gates and two-qubit gates in teleportation-based quantum
computation. We clearly show that there exists a natural algebraic structure
called the teleportation operator in terms of the Bell transform to catch
essential points of quantum teleportation, and hence we expect that there would
also exist interesting algebraic structures in terms of the GHZ transform to
play important roles in quantum information and computation.
