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.
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.
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.
Bell states and quantum teleportation play crucial roles in quantum information and computation. However, a comprehensive theoretical study of both topics remains to be carried out. This work aims to investigate key algebraic properties of generalized Bell states and explore the topological features of quantum teleportation. First, the basis theorem and the basis group are introduced to show that the extension of a generalized Bell basis by a unitary matrix still forms an orthonormal basis. Then, a twist operator is defined to establish a connection between a generalized multi-qubit Bell state and a tensor product of two-qubit Bell states. In addition, the Temperley--Lieb algebra, the braid group relations, and the Yang--Baxter equation are employed to provide a topological description of generalized Bell states and quantum teleportation. The results demonstrate that our approach not only offers a clear illustration of relevant quantum information protocols but also reveals the topological nature of quantum entanglement and teleportation.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly from separable basis states important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
Quantum information processing has a wide range of uses, including the solution of complex algorithms, cryptography, dense coding, quantum computation, and quantum teleportation. Quantum entanglement, a phenomenon that is crucial to many different applications of quantum computation, is an essential resource for quantum computation, and the need for quantum circuits that generate entangled states is one of the ongoing needs for constructing quantum computers. In this study, we present a general approach to the design of quantum circuits that produce entangled states. The approach can be applied to the design of various entanglement circuits with any number of qubits (n-qubit systems), and it makes use of a set of CNOT and Hadamard gates, where the number of CNOT gates should always be (n-1) and each gate connects the two adjacent qubits. In this paper, a three-entangled teleportation scheme of a GHZ-like state (named after its inventor Greenberger-Horne-Zeilinger) through three particles as a quantum channel is presented. The probability of successful teleportation depends on the degree of entanglement of GHZ-like states. A 5-qubit quantum teleportation over a GHZ-like channel has also been used. And single-qubit gates are defined, which are Pauli gates. These gates are represented by arrays I, X, Y, and Z. The results in this paper are good and promising theoretical results in the field of quantum entanglement and quantum teleportation using the Mathematica program.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
ABSTRACT The quantum gate teleportation mechanism allows for the fault‐tolerant implementation of “Clifford hierarchies” of gates assuming, among other things, a fault‐tolerant implementation of the Pauli gates. We discuss how this method can be extended to assume the fault‐tolerant implementation of any orthogonal unitary basis of operators, in such a way that each of these bases can be used to implement its own hierarchy of gates. Noticing that all of these hierarchies are defined through the same algebraic construction, we study this construction from a group‐theoretic perspective. This allows us to deduce general results concerning all the hierarchies. We also discuss alternative hierarchies, study their properties, and compute some of their first few levels.