Bell Transform, Teleportation Operator and Teleportation-Based Quantum Computation

We introduce the concept of the Bell transform to represent known quantum gates in the literature, which are a unitary basis transformation from the product basis to the Bell states or the Greenberger-Horne-Zeilinger (GHZ) states. The algebraic structure of the four dimensional Bell transform has been studied systematically in this paper and we point out that it may be not a Clifford gate. The representative examples of the four dimensional Bell transform are verified as maximally entangling Clifford gates and some of them are recognized as parity-preserving gates or matchgates or Yang--Baxter gates. We define the teleportation operator in terms of the four dimensional Bell transform and apply it to the reformulation of the fault-tolerant construction of single-qubit gates and two-qubit gates in teleportation-based quantum computation. The algebraic structure of the higher dimensional Bell transform is also included and representative examples for it are verified as multi-qubit Clifford gates. Our research suggests that the Bell transform may play important roles in quantum information and computation as a new type of quantum transform.