Teleportation-based quantum computation, extended Temperley–Lieb diagrammatical approach and Yang–Baxter equation

Abstract

This paper focuses on the study of topological features in teleportation-based quantum computation as well as aims at presenting a detailed review on teleportaiton-based quantum computation (Gottesman and Chuang, Nature 402, 390, 1999). In the extended Temperley-Lieb diagrammatical approach, we clearly show that such topological features bring about the fault-tolerant construction of both universal quantum gates and four-partite entangled states more intuitive and simpler. Furthermore, we describe the Yang-Baxter gate by its extended Temperley-Lieb configuration, and then study teleportation-based quantum circuit models using the Yang-Baxter gate. Moreover, we discuss the relationship between the extended Temperley-Lieb diagrammatical approach and the Yang-Baxter gate approach. With these research results, we propose a worthwhile subject, the extended Temperley-Lieb diagrammatical approach, for physicists in quantum information and quantum computation.

Quantum teleportation and Birman–Murakami–Wenzl algebra

“…with the symbol T denoting the matrix transpose. Note that the property that a single-qubit gate flows on the configuration plays a crucial role in the extended Temperley-Lieb diagrammatical approach to quantum teleportation [20,21]. In quantum teleportation [3,4,5,6], Alice has an unknown qubit |α to be transmitted to Bob and meanwhile shares the Bell state |Ψ with Bob, namely, Alice and Bob prepare the state |α ⊗ |Ψ .…”

“…Here the Yang-Baxter gate B is the Bell transform (38), see (36) and (44), and especially the inverse of the Yang-Baxter gate B is related to the Yang-Baxter gate B by the local action of single-qubit gates, see (37) and (46). Hence we define the teleportation operator as the tensor product in terms of the identity operator and the Yang-Baxter gate B, namely (B ⊗ 1 1 2 )(1 1 2 ⊗ B) for simplicity (instead of (B −1 ⊗ 1 1 2 )(1 1 2 ⊗ B) orginally introduced in [21,27]…”

“…where single-qubit gates K i,j,k,l = (−1) j·l+i·j+k X j+k+1 Z i+l+1 are the Pauli gates with phase factors. Furthermore, the teleportation operator can be defined in the other way, namely (1 1 2 ⊗ B 0 )(B 0 ⊗ 1 1 2 ) (refer to [21]), and it is related to the teleportation equation…”

“…The detailed strategy of fault-tolerantly constructing single-qubit gates and two-qubit gates in teleportation-based quantum computation has been presented in [21], and hence we will make a brief sketch on the relevant results in this paper. In quantum circuit model [1], single-qubit gates with an entangling two-qubit gate form a universal quantum gate set, namely all quantum gates can be generated as a tensor product of gates from this gate set.…”

“…It is well-known that how to construct an interesting braid representation (2) (or an interesting solution of the Yang-Baxter equation [10]) is always a meaningful and challenge problem in the research frontier, refer to [9,10,11,12,13,21,34], and the same is true for the construction of an interesting representation of the BMW algebra [16], refer to [9,17,18,19,22,23]. In the previous research [21], a method of constructing the Yang-Baxter gate in the extended Temperley-Lieb diagrammatical approach has been proposed, and hence in this section we want to exploit the extended Temperley-Lieb diagrammatical approach to construct interesting representations of the BMW algebra different from the representation using the Temperley-Lieb projector E (8) and the Yang-Baxter gate B (9). In view of the research result in the last section that the tangle relations (4) of the BMW algebra have a natural interpretation of quantum teleportation in the extended Temperley-Lieb diagrammatical approach, first of all, we focus on the general construction of the representation of the tangle relations (4) of the BMW algebra.…”