GHZ transform (I): Bell transform and quantum teleportation

TL;DR

This paper explores the algebraic structure of the GHZ transform, focusing on the Bell transform, and applies it to reformulate quantum teleportation and fault-tolerant quantum gate construction, revealing new algebraic insights.

Contribution

It introduces the GHZ transform's algebraic structure, exemplifies it as multi-qubit Clifford gates, and applies the Bell transform to quantum teleportation and fault-tolerant quantum computation.

Findings

Bell transform as a basis for quantum circuit reformulation

Identification of the teleportation operator in Bell transform

Potential for algebraic structures in GHZ transform to advance quantum info

Abstract

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.